[{"data":1,"prerenderedAt":20389},["ShallowReactive",2],{"site-info":3,"blog-list-zh-TW":99},{"id":4,"company":5,"extension":16,"meta":17,"services":18,"social":96,"stem":97,"__hash__":98},"site\u002Fsite.yml",{"name":6,"nameEn":7,"email":8,"address":9,"addressEn":10,"lineUrl":11,"lineId":12,"duns":13,"youtube":14,"youtubeId":15},"銡策數位有限公司","Jecision Digital Co., Ltd.","contact@jecision.com","231 新北市新店區北新路二段 262 號 3 樓之 3","3F.-3, No. 262, Sec. 2, Beixin Rd., Xindian Dist., New Taipei City 231, Taiwan","https:\u002F\u002Fline.me\u002FR\u002Fti\u002Fp\u002F@374qnumi","@374qnumi","963905705","https:\u002F\u002Fwww.youtube.com\u002Fshorts\u002F6UE0WTmyjsg","6UE0WTmyjsg","yml",{},[19,43,69],{"key":20,"zh":21,"en":32},"audit",{"title":22,"tagline":23,"summary":24,"points":25,"deliverable":29,"detail":30,"duration":31},"企業 AI 導入健檢","先看清楚，再花錢","60 分鐘線上訪談，盤點最耗時、最依賴人力的營運環節；我們會告訴你哪些工作現在就能交給 AI、哪些還不成熟。",[26,27,28],"現況盤點：梳理日常營運中最耗時、最依賴人力的環節","痛點診斷：判斷哪些工作現在就能交給 AI，哪些建議先保留人力","可行性評估：依你的業務特性與預算，篩選適合的模型與工具","《企業 AI 導入建議報告書》（PDF）","含 1–2 個「開發成本最低、成效最明顯」的優先項目、技術棧建議、預期節省的人力時間與下一步","訪談 60 分鐘，報告書 3 個工作天內交付",{"title":33,"tagline":34,"summary":35,"points":36,"deliverable":40,"detail":41,"duration":42},"AI Readiness Audit","Look before you spend","A 60-minute remote audit of the workflows that eat the most time and people-power — and an honest answer on what AI can take over today, and what it cannot.",[37,38,39],"Audit the daily workflows that consume the most time and headcount","Triage which tasks AI can own today, and which should stay human","Shortlist the models and tools that fit your budget and constraints","AI Adoption Recommendation Report (PDF)","One or two highest-ROI entry points, a recommended tech stack, expected hours saved, and the concrete next step","60-minute call, report delivered within 3 business days",{"key":44,"zh":45,"en":56},"prompt",{"title":46,"tagline":47,"summary":48,"points":49,"deliverable":53,"detail":54,"duration":55},"提示詞工程","讓 AI 穩定產出，而不是碰運氣","針對你的實際任務設計提示詞與多步驟工作流，並做極限測試，確保換了資料、換了人操作也不會走鐘。",[50,51,52],"客製提示詞：ChatGPT、Claude、Gemini、Midjourney 等主流工具的專屬模板","除錯與優化：修掉幻覺、格式跑掉、時好時壞的問題","工作流建置：把複雜任務拆成多步驟提示詞與可重複的 SOP","提示詞文件或設定檔","附使用說明與範例，並提供 1 對 1 教學，確保你與團隊能自己維護","依任務複雜度報價，通常 3–7 個工作天",{"title":57,"tagline":58,"summary":59,"points":60,"deliverable":66,"detail":67,"duration":68},"Prompt Engineering","Make AI produce reliably, not by luck","Prompt sets and multi-step workflows designed for your actual tasks, stress-tested so they keep working when the data or the operator changes.",[61,62,64],"Custom prompts for ChatGPT, Claude, Gemini, Midjourney and friends",{"Debugging and tuning":63},"kill hallucinations, broken formats and flaky output",{"Workflow design":65},"break complex work into repeatable multi-step SOPs","Prompt library or exportable config","Ships with usage notes and examples, plus a 1:1 walkthrough so your team can maintain it","Quoted by complexity, typically 3–7 business days",{"key":70,"zh":71,"en":82},"poc",{"title":72,"tagline":73,"summary":74,"points":75,"deliverable":79,"detail":80,"duration":81},"極速 PoC","五天，看到能跑的東西","不談架構藍圖，直接做一個能動的雛形。五個工作天內交付線上可操作的智能體，用你真實的資料驗證。",[76,77,78],"Day 1：15 分鐘對焦，鎖定單一痛點與資料盤點","Day 2–4：模型串接、邏輯設計與雛形開發","Day 5：Demo 展示與 AI 擴展建議書","可線上操作的智能體雛形","交付範圍為概念驗證與展示，不含後續維運或大規模伺服器部署；要進階成完整系統會另行報價","5 個工作天",{"title":83,"tagline":84,"summary":85,"points":86,"deliverable":93,"detail":94,"duration":95},"Rapid PoC","Five days to something that runs","No architecture decks — a working prototype built in five business days, validated against your real data.",[87,89,91],{"Day 1":88},"15-minute scoping call, one pain point, data inventory",{"Day 2–4":90},"model integration, logic design, prototype build",{"Day 5":92},"live demo plus an AI expansion recommendation","A runnable online agent prototype","Proof of concept and demo scope only; ongoing operations and large-scale deployment are quoted separately","5 business days",[],"site","XTwARfVAOcrXNiKehsM6V6cdkd2HXy4d5yZVrSWrI3Q",[100,339,5037,5428,5695,6404,10103,15745,16611,18212,18856,20137],{"id":101,"title":102,"author":103,"body":104,"cover":320,"coverAlt":321,"date":322,"description":323,"draft":324,"extension":325,"featured":326,"lang":327,"meta":328,"navigation":326,"path":329,"seo":330,"slug":331,"stem":332,"tags":333,"translationKey":331,"updated":337,"__hash__":338},"blog\u002Fblog\u002Fzh-TW\u002Fhello-jecision.md","搬家記：把部落格從一個沒人找得到的網域搬回來","Jecision",{"type":105,"value":106,"toc":310},"minimark",[107,111,115,134,141,144,147,154,169,172,175,196,214,220,223,235,245,248,262,268,271,298,301,307],[108,109,110],"h2",{"id":110},"起因是一行原始碼",[112,113,114],"p",{},"前陣子要把一篇舊文章轉給同事看，順手抓了頁面原始碼，然後愣住。",[112,116,117,121,122,125,126,129,130,133],{},[118,119,120],"code",{},"\u003Chtml>"," 沒有 ",[118,123,124],{},"lang","。沒有 ",[118,127,128],{},"\u003Ctitle>","。沒有 description，沒有 og，沒有 canonical，沒有一行 JSON-LD。公式的部分更妙——原始碼裡躺著的是 ",[118,131,132],{},"\\operatorname{Attention}"," 這串字，要等瀏覽器把 JavaScript 跑完，人才看得到排版好的數學。",[112,135,136,137],{},"也就是說：",[138,139,140],"strong",{},"對搜尋引擎來說，那篇文章從來沒有存在過。",[112,142,143],{},"文章寫得再好，放在一個沒有標題、沒有摘要、沒有結構化資料的頁面上，它就只是一段沒有人索引的文字。",[108,145,146],{"id":146},"當初為什麼會變這樣",[112,148,149,150,153],{},"不是不用心，是",[138,151,152],{},"工具用得比它設計的方式更用力","。",[112,155,156,157,160,161,164,165,168],{},"當時為了讓公式能顯示，我們在 Markdown 裡塞了 ",[118,158,159],{},"\u003Cdiv class=\"latex-block\">","，然後在 ",[118,162,163],{},"onMounted"," 裡用 KaTeX 把它的 ",[118,166,167],{},"textContent"," 換成公式。瀏覽器上看起來完全正常，所以也沒有人覺得有問題。",[112,170,171],{},"問題在於這個做法把兩件事綁在一起：內容要靠 JavaScript 才成立，而 SEO 需要的是「不跑 JavaScript 也成立」。",[108,173,174],{"id":174},"搬家的三個決定",[112,176,177,180,181,184,185,188,189,184,192,195],{},[138,178,179],{},"一、內容搬家，不是版面搬家。"," 12 篇文章全部重新轉換，公式改寫成 ",[118,182,183],{},"$...$"," 與 ",[118,186,187],{},"$$...$$","，交給 ",[118,190,191],{},"remark-math",[118,193,194],{},"rehype-katex"," 在 build 時處理。現在公式是 HTML 的一部分，不需要任何前端 JavaScript。",[112,197,198,201,202,205,206,209,210,213],{},[138,199,200],{},"二、網址由資料決定，不由檔名決定。"," 舊站的檔名是 ",[118,203,204],{},"2604.04921v1.md","，結果網址變成 ",[118,207,208],{},"\u002Fen\u002Fblog\u002F04921v1","——數字前綴被吃掉，檔名和網址永遠對不起來。新站改成 frontmatter 裡寫 ",[118,211,212],{},"slug","，檔名再亂都不影響網址。",[112,215,216,219],{},[138,217,218],{},"三、舊網址一條都不能斷。"," 已經被 Google 記住的東西，不該因為我們重寫而消失。28 條 301 對照全部列出來，包含尾斜線的版本。",[108,221,222],{"id":222},"三個差點讓上線失敗的意外",[112,224,225,228,229,231,232,234],{},[138,226,227],{},"第一個：slug 被吃掉。"," 這件事是在舊站發現的——",[118,230,204],{}," 變成 ",[118,233,208],{},"。如果沒有先抓一次線上頁面，我們會以為搬家只是複製檔案。",[112,236,237,240,241,244],{},[138,238,239],{},"第二個：hreflang 指到不存在的頁面。"," 新站預設會幫每個語系產生 alternate 連結，但只有中文版的文章，英文連結就是一個 404。跑 ",[118,242,243],{},"nuxt generate"," 的時候 prerender 直接掛掉，錯誤訊息長得像「Article not found，linked from \u002Fblog\u002Fxxx」。",[112,246,247],{},"解法不是關掉檢查，而是讓每一頁自己宣告「我實際有哪些語系版本」。只有真的存在的中英版本，才會出現在 hreflang 裡。",[112,249,250,257,258,261],{},[138,251,252,253,256],{},"第三個：",[118,254,255],{},"\u002Fsitemap.xml"," 是 HTML。"," 這件事最陰險。build 成功、檔案存在、打開也有內容——但它是一個 ",[118,259,260],{},"\u003Cmeta http-equiv=\"refresh\">"," 的轉址頁，不是 XML。Google 會直接判定 sitemap 無效。",[112,263,264,265,267],{},"後來在 build 後加了一步，把 sitemap index 的內容直接寫成 ",[118,266,255],{},"，才真的能提交。",[108,269,270],{"id":270},"現在的樣子",[272,273,274,278,285,292],"ul",{},[275,276,277],"li",{},"公式在伺服器端就排版完成，原始碼裡有 MathML",[275,279,280,281,284],{},"每一頁都有 title、description、canonical、hreflang、og，文章另有 ",[118,282,283],{},"BlogPosting"," 結構化資料",[275,286,287,288,291],{},"RSS、",[118,289,290],{},"llms.txt","、sitemap、301 對照都有",[275,293,294,295,297],{},"還寫了一支 ",[118,296,20],{}," 腳本，build 完直接檢查 25 個頁面的 SEO 與壞連結。目前是 0 錯誤 0 警告",[108,299,300],{"id":300},"一句話總結",[112,302,303,304,153],{},"搬家最花時間的不是搬，是",[138,305,306],{},"確認搬過去的東西真的還在",[112,308,309],{},"我們的做法是：把「應該要有」的東西寫成檢查，讓它每次 build 都自己驗一遍。人會忘記，腳本不會。",{"title":311,"searchDepth":312,"depth":312,"links":313},"",2,[314,315,316,317,318,319],{"id":110,"depth":312,"text":110},{"id":146,"depth":312,"text":146},{"id":174,"depth":312,"text":174},{"id":222,"depth":312,"text":222},{"id":270,"depth":312,"text":270},{"id":300,"depth":312,"text":300},"\u002Fimages\u002Fblog\u002Fhello-jecision.svg","從臨時網域搬回品牌網域的示意圖","2026-10-23","舊站的每一篇文章，原始碼裡都沒有 title、沒有 description、沒有 canonical，公式還是純文字。這篇記錄我們把部落格搬回 jecision.com 的過程，以及三個差點讓上線失敗的意外。",false,"md",true,"zh-TW",{},"\u002Fblog\u002Fzh-tw\u002Fhello-jecision",{"title":102,"description":323},"hello-jecision","blog\u002Fzh-TW\u002Fhello-jecision",[334,335,336],"nuxt","seo","重構",null,"eMzmxLsypk5GpqCf6q2_9_s4WWIcInwgrCKPRumQDZc",{"id":340,"title":341,"author":103,"body":342,"cover":5022,"coverAlt":5023,"date":5024,"description":5025,"draft":324,"extension":325,"featured":324,"lang":327,"meta":5026,"navigation":326,"path":5027,"seo":5028,"slug":5029,"stem":5030,"tags":5031,"translationKey":5029,"updated":5035,"__hash__":5036},"blog\u002Fblog\u002Fzh-TW\u002Fattention-as-kernel.md","Attention 其實是一個核平滑器",{"type":105,"value":343,"toc":5014},[344,347,350,353,356,752,1109,1143,2084,2303,2519,3063,3066,3069,3072,3078,3263,3903,4024,4403,4406,4409,4415,4841,4844,4847,4962,4994,4997,5000,5007,5010],[108,345,346],{"id":346},"一句話讓我想了很久",[112,348,349],{},"「Attention 就是一個核平滑器。」",[112,351,352],{},"第一次看到這句話的時候我以為是某種比喻，畢竟「核方法」聽起來像是 SVM 那個年代的事，跟 Transformer 沒什麼關係。後來自己把公式展開一次，才發現它不是在打比方。",[108,354,355],{"id":355},"把公式拆開",[112,357,358,359,498,499,599,600,751],{},"給定查詢矩陣 ",[360,361,364,409],"span",{"className":362},[363],"katex",[360,365,368],{"className":366},[367],"katex-mathml",[369,370,372],"math",{"xmlns":371},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[373,374,375,404],"semantics",{},[376,377,378,382,386],"mrow",{},[379,380,381],"mi",{},"Q",[383,384,385],"mo",{},"∈",[387,388,389,393],"msup",{},[379,390,392],{"mathvariant":391},"double-struck","R",[376,394,395,398,401],{},[379,396,397],{},"n",[383,399,400],{},"×",[379,402,403],{},"d",[405,406,408],"annotation",{"encoding":407},"application\u002Fx-tex","Q \\in \\mathbb{R}^{n \\times d}",[360,410,414,440],{"className":411,"ariaHidden":413},[412],"katex-html","true",[360,415,418,423,428,433,437],{"className":416},[417],"base",[360,419],{"className":420,"style":422},[421],"strut","height:0.8778em;vertical-align:-0.1944em;",[360,424,381],{"className":425},[426,427],"mord","mathnormal",[360,429],{"className":430,"style":432},[431],"mspace","margin-right:0.2778em;",[360,434,385],{"className":435},[436],"mrel",[360,438],{"className":439,"style":432},[431],[360,441,443,447],{"className":442},[417],[360,444],{"className":445,"style":446},[421],"height:0.8491em;",[360,448,450,454],{"className":449},[426],[360,451,392],{"className":452},[426,453],"mathbb",[360,455,458],{"className":456},[457],"msupsub",[360,459,462],{"className":460},[461],"vlist-t",[360,463,466],{"className":464},[465],"vlist-r",[360,467,470],{"className":468,"style":446},[469],"vlist",[360,471,473,478],{"style":472},"top:-3.063em;margin-right:0.05em;",[360,474],{"className":475,"style":477},[476],"pstrut","height:2.7em;",[360,479,485],{"className":480},[481,482,483,484],"sizing","reset-size6","size3","mtight",[360,486,488,491,495],{"className":487},[426,484],[360,489,397],{"className":490},[426,427,484],[360,492,400],{"className":493},[494,484],"mbin",[360,496,403],{"className":497},[426,427,484],"、鍵矩陣 ",[360,500,502,532],{"className":501},[363],[360,503,505],{"className":504},[367],[369,506,507],{"xmlns":371},[373,508,509,529],{},[376,510,511,514,516],{},[379,512,513],{},"K",[383,515,385],{},[387,517,518,520],{},[379,519,392],{"mathvariant":391},[376,521,522,525,527],{},[379,523,524],{},"m",[383,526,400],{},[379,528,403],{},[405,530,531],{"encoding":407},"K \\in \\mathbb{R}^{m \\times d}",[360,533,535,555],{"className":534,"ariaHidden":413},[412],[360,536,538,542,546,549,552],{"className":537},[417],[360,539],{"className":540,"style":541},[421],"height:0.7224em;vertical-align:-0.0391em;",[360,543,513],{"className":544,"style":545},[426,427],"margin-right:0.0715em;",[360,547],{"className":548,"style":432},[431],[360,550,385],{"className":551},[436],[360,553],{"className":554,"style":432},[431],[360,556,558,561],{"className":557},[417],[360,559],{"className":560,"style":446},[421],[360,562,564,567],{"className":563},[426],[360,565,392],{"className":566},[426,453],[360,568,570],{"className":569},[457],[360,571,573],{"className":572},[461],[360,574,576],{"className":575},[465],[360,577,579],{"className":578,"style":446},[469],[360,580,581,584],{"style":472},[360,582],{"className":583,"style":477},[476],[360,585,587],{"className":586},[481,482,483,484],[360,588,590,593,596],{"className":589},[426,484],[360,591,524],{"className":592},[426,427,484],[360,594,400],{"className":595},[494,484],[360,597,403],{"className":598},[426,427,484]," 與值矩陣 ",[360,601,603,638],{"className":602},[363],[360,604,606],{"className":605},[367],[369,607,608],{"xmlns":371},[373,609,610,635],{},[376,611,612,615,617],{},[379,613,614],{},"V",[383,616,385],{},[387,618,619,621],{},[379,620,392],{"mathvariant":391},[376,622,623,625,627],{},[379,624,524],{},[383,626,400],{},[628,629,630,632],"msub",{},[379,631,403],{},[379,633,634],{},"v",[405,636,637],{"encoding":407},"V \\in \\mathbb{R}^{m \\times 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= \\frac{\\sum_{j=1}^{m} \\exp\\!\\left(\\frac{q_i^{\\top} k_j}{\\sqrt{d}}\\right) v_j}{\\sum_{j=1}^{m} \\exp\\!\\left(\\frac{q_i^{\\top} k_j}{\\sqrt{d}}\\right)} 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426,484],[360,1582,1584,1587],{"className":1583},[426,484],[360,1585,1210],{"className":1586,"style":736},[426,427,484],[360,1588,1590],{"className":1589},[457],[360,1591,1593,1630],{"className":1592},[461,713],[360,1594,1596,1627],{"className":1595},[465],[360,1597,1600,1612],{"className":1598,"style":1599},[469],"height:0.8408em;",[360,1601,1603,1606],{"style":1602},"top:-2.1777em;margin-left:-0.0359em;margin-right:0.0714em;",[360,1604],{"className":1605,"style":727},[476],[360,1607,1609],{"className":1608},[481,731,732,484],[360,1610,1126],{"className":1611},[426,427,484],[360,1613,1615,1618],{"style":1614},"top:-2.8448em;margin-right:0.0714em;",[360,1616],{"className":1617,"style":727},[476],[360,1619,1621],{"className":1620},[481,731,732,484],[360,1622,1624],{"className":1623},[426,484],[360,1625,823],{"className":1626},[426,484],[360,1628,741],{"className":1629},[740],[360,1631,1633],{"className":1632},[465],[360,1634,1637],{"className":1635,"style":1636},[469],"height:0.3223em;",[360,1638],{},[360,1640,1642,1646],{"className":1641},[426,484],[360,1643,1219],{"className":1644,"style":1645},[426,427,484],"margin-right:0.0315em;",[360,1647,1649],{"className":1648},[457],[360,1650,1652,1674],{"className":1651},[461,713],[360,1653,1655,1671],{"className":1654},[465],[360,1656,1659],{"className":1657,"style":1658},[469],"height:0.3281em;",[360,1660,1662,1665],{"style":1661},"top:-2.357em;margin-left:-0.0315em;margin-right:0.0714em;",[360,1663],{"className":1664,"style":727},[476],[360,1666,1668],{"className":1667},[481,731,732,484],[360,1669,1181],{"className":1670,"style":1424},[426,427,484],[360,1672,741],{"className":1673},[740],[360,1675,1677],{"className":1676},[465],[360,1678,1681],{"className":1679,"style":1680},[469],"height:0.2819em;",[360,1682],{},[360,1684,741],{"className":1685},[740],[360,1687,1689],{"className":1688},[465],[360,1690,1693],{"className":1691,"style":1692},[469],"height:0.538em;",[360,1694],{},[360,1696],{"className":1697},[887,936],[360,1699,1701],{"className":1700,"style":925},[887,924],[360,1702,793],{"className":1703},[929,1479],[360,1705,1707,1710],{"style":1706},"top:-3.3891em;",[360,1708],{"className":1709,"style":1383},[476],[360,1711],{"className":1712,"style":1034},[1033],[360,1714,1716,1719],{"style":1715},"top:-4.1991em;",[360,1717],{"className":1718,"style":1383},[476],[360,1720,1722,1785,1788,1791,1794,1797,2021,2024],{"className":1721},[426],[360,1723,1725,1728],{"className":1724},[852],[360,1726,1176],{"className":1727,"style":1395},[852,1393,1394],[360,1729,1731],{"className":1730},[457],[360,1732,1734,1777],{"className":1733},[461,713],[360,1735,1737,1774],{"className":1736},[465],[360,1738,1740,1760],{"className":1739,"style":1408},[469],[360,1741,1742,1745],{"style":1411},[360,1743],{"className":1744,"style":477},[476],[360,1746,1748],{"className":1747},[481,482,483,484],[360,1749,1751,1754,1757],{"className":1750},[426,484],[360,1752,1181],{"className":1753,"style":1424},[426,427,484],[360,1755,796],{"className":1756},[436,484],[360,1758,1187],{"className":1759},[426,484],[360,1761,1762,1765],{"style":1433},[360,1763],{"className":1764,"style":477},[476],[360,1766,1768],{"className":1767},[481,482,483,484],[360,1769,1771],{"className":1770},[426,484],[360,1772,524],{"className":1773},[426,427,484],[360,1775,741],{"className":1776},[740],[360,1778,1780],{"className":1779},[465],[360,1781,1783],{"className":1782,"style":1455},[469],[360,1784],{},[360,1786],{"className":1787,"style":871},[431],[360,1789,1192],{"className":1790},[852],[360,1792],{"className":1793,"style":913},[431],[360,1795],{"className":1796,"style":871},[431],[360,1798,1800,1806,2015],{"className":1799},[920],[360,1801,1803],{"className":1802,"style":925},[860,924],[360,1804,779],{"className":1805},[929,1479],[360,1807,1809,1812,2012],{"className":1808},[426],[360,1810],{"className":1811},[860,936],[360,1813,1815],{"className":1814},[811],[360,1816,1818,2004],{"className":1817},[461,713],[360,1819,1821,2001],{"className":1820},[465],[360,1822,1825,1883,1891],{"className":1823,"style":1824},[469],"height:1.1591em;",[360,1826,1827,1830],{"style":1501},[360,1828],{"className":1829,"style":956},[476],[360,1831,1833],{"className":1832},[481,482,483,484],[360,1834,1836],{"className":1835},[426,484],[360,1837,1839],{"className":1838},[426,963,484],[360,1840,1842,1875],{"className":1841},[461,713],[360,1843,1845,1872],{"className":1844},[465],[360,1846,1848,1860],{"className":1847,"style":1523},[469],[360,1849,1851,1854],{"className":1850,"style":978},[977],[360,1852],{"className":1853,"style":956},[476],[360,1855,1857],{"className":1856,"style":985},[426,484],[360,1858,403],{"className":1859},[426,427,484],[360,1861,1862,1865],{"style":1538},[360,1863],{"className":1864,"style":956},[476],[360,1866,1868],{"className":1867,"style":999},[998,484],[1001,1869,1870],{"xmlns":1003,"width":1004,"height":1005,"viewBox":1006,"preserveAspectRatio":1007},[1009,1871],{"d":1011},[360,1873,741],{"className":1874},[740],[360,1876,1878],{"className":1877},[465],[360,1879,1881],{"className":1880,"style":1558},[469],[360,1882],{},[360,1884,1885,1888],{"style":1026},[360,1886],{"className":1887,"style":956},[476],[360,1889],{"className":1890,"style":1034},[1033],[360,1892,1894,1897],{"style":1893},"top:-3.5102em;",[360,1895],{"className":1896,"style":956},[476],[360,1898,1900],{"className":1899},[481,482,483,484],[360,1901,1903,1961],{"className":1902},[426,484],[360,1904,1906,1909],{"className":1905},[426,484],[360,1907,1210],{"className":1908,"style":736},[426,427,484],[360,1910,1912],{"className":1911},[457],[360,1913,1915,1952],{"className":1914},[461,713],[360,1916,1918,1949],{"className":1917},[465],[360,1919,1922,1934],{"className":1920,"style":1921},[469],"height:0.927em;",[360,1923,1925,1928],{"style":1924},"top:-2.214em;margin-left:-0.0359em;margin-right:0.0714em;",[360,1926],{"className":1927,"style":727},[476],[360,1929,1931],{"className":1930},[481,731,732,484],[360,1932,1126],{"className":1933},[426,427,484],[360,1935,1937,1940],{"style":1936},"top:-2.931em;margin-right:0.0714em;",[360,1938],{"className":1939,"style":727},[476],[360,1941,1943],{"className":1942},[481,731,732,484],[360,1944,1946],{"className":1945},[426,484],[360,1947,823],{"className":1948},[426,484],[360,1950,741],{"className":1951},[740],[360,1953,1955],{"className":1954},[465],[360,1956,1959],{"className":1957,"style":1958},[469],"height:0.286em;",[360,1960],{},[360,1962,1964,1967],{"className":1963},[426,484],[360,1965,1219],{"className":1966,"style":1645},[426,427,484],[360,1968,1970],{"className":1969},[457],[360,1971,1973,1993],{"className":1972},[461,713],[360,1974,1976,1990],{"className":1975},[465],[360,1977,1979],{"className":1978,"style":1658},[469],[360,1980,1981,1984],{"style":1661},[360,1982],{"className":1983,"style":727},[476],[360,1985,1987],{"className":1986},[481,731,732,484],[360,1988,1181],{"className":1989,"style":1424},[426,427,484],[360,1991,741],{"className":1992},[740],[360,1994,1996],{"className":1995},[465],[360,1997,1999],{"className":1998,"style":1680},[469],[360,2000],{},[360,2002,741],{"className":2003},[740],[360,2005,2007],{"className":2006},[465],[360,2008,2010],{"className":2009,"style":1692},[469],[360,2011],{},[360,2013],{"className":2014},[887,936],[360,2016,2018],{"className":2017,"style":925},[887,924],[360,2019,793],{"className":2020},[929,1479],[360,2022],{"className":2023,"style":871},[431],[360,2025,2027,2030],{"className":2026},[426],[360,2028,634],{"className":2029,"style":736},[426,427],[360,2031,2033],{"className":2032},[457],[360,2034,2036,2057],{"className":2035},[461,713],[360,2037,2039,2054],{"className":2038},[465],[360,2040,2042],{"className":2041,"style":1317},[469],[360,2043,2045,2048],{"style":2044},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[360,2046],{"className":2047,"style":477},[476],[360,2049,2051],{"className":2050},[481,482,483,484],[360,2052,1181],{"className":2053,"style":1424},[426,427,484],[360,2055,741],{"className":2056},[740],[360,2058,2060],{"className":2059},[465],[360,2061,2064],{"className":2062,"style":2063},[469],"height:0.2861em;",[360,2065],{},[360,2067,741],{"className":2068},[740],[360,2070,2072],{"className":2071},[465],[360,2073,2076],{"className":2074,"style":2075},[469],"height:1.69em;",[360,2077],{},[360,2079],{"className":2080},[887,936],[360,2082,835],{"className":2083},[426],[112,2085,2086,2087,2158,2159,184,2230,2302],{},"左邊的分子是一堆 ",[360,2088,2090,2108],{"className":2089},[363],[360,2091,2093],{"className":2092},[367],[369,2094,2095],{"xmlns":371},[373,2096,2097,2105],{},[376,2098,2099],{},[628,2100,2101,2103],{},[379,2102,634],{},[379,2104,1181],{},[405,2106,2107],{"encoding":407},"v_j",[360,2109,2111],{"className":2110,"ariaHidden":413},[412],[360,2112,2114,2118],{"className":2113},[417],[360,2115],{"className":2116,"style":2117},[421],"height:0.7167em;vertical-align:-0.2861em;",[360,2119,2121,2124],{"className":2120},[426],[360,2122,634],{"className":2123,"style":736},[426,427],[360,2125,2127],{"className":2126},[457],[360,2128,2130,2150],{"className":2129},[461,713],[360,2131,2133,2147],{"className":2132},[465],[360,2134,2136],{"className":2135,"style":1317},[469],[360,2137,2138,2141],{"style":2044},[360,2139],{"className":2140,"style":477},[476],[360,2142,2144],{"className":2143},[481,482,483,484],[360,2145,1181],{"className":2146,"style":1424},[426,427,484],[360,2148,741],{"className":2149},[740],[360,2151,2153],{"className":2152},[465],[360,2154,2156],{"className":2155,"style":2063},[469],[360,2157],{}," 的加權和，權重是 ",[360,2160,2162,2180],{"className":2161},[363],[360,2163,2165],{"className":2164},[367],[369,2166,2167],{"xmlns":371},[373,2168,2169,2177],{},[376,2170,2171],{},[628,2172,2173,2175],{},[379,2174,1210],{},[379,2176,1126],{},[405,2178,2179],{"encoding":407},"q_i",[360,2181,2183],{"className":2182,"ariaHidden":413},[412],[360,2184,2186,2190],{"className":2185},[417],[360,2187],{"className":2188,"style":2189},[421],"height:0.625em;vertical-align:-0.1944em;",[360,2191,2193,2196],{"className":2192},[426],[360,2194,1210],{"className":2195,"style":736},[426,427],[360,2197,2199],{"className":2198},[457],[360,2200,2202,2222],{"className":2201},[461,713],[360,2203,2205,2219],{"className":2204},[465],[360,2206,2208],{"className":2207,"style":1317},[469],[360,2209,2210,2213],{"style":2044},[360,2211],{"className":2212,"style":477},[476],[360,2214,2216],{"className":2215},[481,482,483,484],[360,2217,1126],{"className":2218},[426,427,484],[360,2220,741],{"className":2221},[740],[360,2223,2225],{"className":2224},[465],[360,2226,2228],{"className":2227,"style":1339},[469],[360,2229],{},[360,2231,2233,2251],{"className":2232},[363],[360,2234,2236],{"className":2235},[367],[369,2237,2238],{"xmlns":371},[373,2239,2240,2248],{},[376,2241,2242],{},[628,2243,2244,2246],{},[379,2245,1219],{},[379,2247,1181],{},[405,2249,2250],{"encoding":407},"k_j",[360,2252,2254],{"className":2253,"ariaHidden":413},[412],[360,2255,2257,2261],{"className":2256},[417],[360,2258],{"className":2259,"style":2260},[421],"height:0.9805em;vertical-align:-0.2861em;",[360,2262,2264,2267],{"className":2263},[426],[360,2265,1219],{"className":2266,"style":1645},[426,427],[360,2268,2270],{"className":2269},[457],[360,2271,2273,2294],{"className":2272},[461,713],[360,2274,2276,2291],{"className":2275},[465],[360,2277,2279],{"className":2278,"style":1317},[469],[360,2280,2282,2285],{"style":2281},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[360,2283],{"className":2284,"style":477},[476],[360,2286,2288],{"className":2287},[481,482,483,484],[360,2289,1181],{"className":2290,"style":1424},[426,427,484],[360,2292,741],{"className":2293},[740],[360,2295,2297],{"className":2296},[465],[360,2298,2300],{"className":2299,"style":2063},[469],[360,2301],{}," 的相似度；左邊的分母只是把權重歸一化，讓它們加起來等於 1。",[112,2304,2305,2306,2518],{},"如果把 ",[360,2307,2309,2363],{"className":2308},[363],[360,2310,2312],{"className":2311},[367],[369,2313,2314],{"xmlns":371},[373,2315,2316,2360],{},[376,2317,2318,2321,2323,2325,2327,2329,2331,2333,2335,2337,2339],{},[379,2319,2320],{},"κ",[383,2322,779],{"stretchy":778},[379,2324,1210],{},[383,2326,784],{"separator":413},[379,2328,1219],{},[383,2330,793],{"stretchy":778},[383,2332,796],{},[379,2334,1192],{},[383,2336,775],{},[803,2338,805],{},[376,2340,2341,2343,2349,2351,2354,2358],{},[383,2342,779],{"fence":413},[387,2344,2345,2347],{},[379,2346,1210],{},[379,2348,823],{"mathvariant":771},[379,2350,1219],{},[379,2352,2353],{"mathvariant":771},"\u002F",[825,2355,2356],{},[379,2357,403],{},[383,2359,793],{"fence":413},[405,2361,2362],{"encoding":407},"\\kappa(q, k) = \\exp\\!\\left(q^{\\top}k \u002F \\sqrt{d}\\right)",[360,2364,2366,2402],{"className":2365,"ariaHidden":413},[412],[360,2367,2369,2372,2375,2378,2381,2384,2387,2390,2393,2396,2399],{"className":2368},[417],[360,2370],{"className":2371,"style":848},[421],[360,2373,2320],{"className":2374},[426,427],[360,2376,779],{"className":2377},[860],[360,2379,1210],{"className":2380,"style":736},[426,427],[360,2382,784],{"className":2383},[867],[360,2385],{"className":2386,"style":871},[431],[360,2388,1219],{"className":2389,"style":1645},[426,427],[360,2391,793],{"className":2392},[887],[360,2394],{"className":2395,"style":432},[431],[360,2397,796],{"className":2398},[436],[360,2400],{"className":2401,"style":432},[431],[360,2403,2405,2409,2412,2415,2418],{"className":2404},[417],[360,2406],{"className":2407,"style":2408},[421],"height:1.8em;vertical-align:-0.65em;",[360,2410,1192],{"className":2411},[852],[360,2413],{"className":2414,"style":913},[431],[360,2416],{"className":2417,"style":871},[431],[360,2419,2421,2427,2459,2462,2465,2512],{"className":2420},[920],[360,2422,2424],{"className":2423,"style":925},[860,924],[360,2425,779],{"className":2426},[929,1479],[360,2428,2430,2433],{"className":2429},[426],[360,2431,1210],{"className":2432,"style":736},[426,427],[360,2434,2436],{"className":2435},[457],[360,2437,2439],{"className":2438},[461],[360,2440,2442],{"className":2441},[465],[360,2443,2445],{"className":2444,"style":446},[469],[360,2446,2447,2450],{"style":472},[360,2448],{"className":2449,"style":477},[476],[360,2451,2453],{"className":2452},[481,482,483,484],[360,2454,2456],{"className":2455},[426,484],[360,2457,823],{"className":2458},[426,484],[360,2460,1219],{"className":2461,"style":1645},[426,427],[360,2463,2353],{"className":2464},[426],[360,2466,2468],{"className":2467},[426,963],[360,2469,2471,2504],{"className":2470},[461,713],[360,2472,2474,2501],{"className":2473},[465],[360,2475,2477,2489],{"className":2476,"style":973},[469],[360,2478,2480,2483],{"className":2479,"style":978},[977],[360,2481],{"className":2482,"style":956},[476],[360,2484,2486],{"className":2485,"style":985},[426],[360,2487,403],{"className":2488},[426,427],[360,2490,2491,2494],{"style":991},[360,2492],{"className":2493,"style":956},[476],[360,2495,2497],{"className":2496,"style":999},[998],[1001,2498,2499],{"xmlns":1003,"width":1004,"height":1005,"viewBox":1006,"preserveAspectRatio":1007},[1009,2500],{"d":1011},[360,2502,741],{"className":2503},[740],[360,2505,2507],{"className":2506},[465],[360,2508,2510],{"className":2509,"style":1021},[469],[360,2511],{},[360,2513,2515],{"className":2514,"style":925},[887,924],[360,2516,793],{"className":2517},[929,1479]," 當成一個核函數，上面這一坨就是",[360,2520,2522],{"className":2521},[755],[360,2523,2525,2615],{"className":2524},[363],[360,2526,2528],{"className":2527},[367],[369,2529,2530],{"xmlns":371,"display":764},[373,2531,2532,2612],{},[376,2533,2534,2540,2542,2610],{},[628,2535,2536,2538],{},[379,2537,1162],{},[379,2539,1126],{},[383,2541,796],{},[811,2543,2544,2582],{},[376,2545,2546,2553,2555,2557,2563,2565,2571,2573,2576],{},[2547,2548,2549,2551],"munder",{},[383,2550,1176],{},[379,2552,1181],{},[379,2554,2320],{},[383,2556,779],{"stretchy":778},[628,2558,2559,2561],{},[379,2560,1210],{},[379,2562,1126],{},[383,2564,784],{"separator":413},[628,2566,2567,2569],{},[379,2568,1219],{},[379,2570,1181],{},[383,2572,793],{"stretchy":778},[803,2574,2575],{}," ",[628,2577,2578,2580],{},[379,2579,634],{},[379,2581,1181],{},[376,2583,2584,2590,2592,2594,2600,2602,2608],{},[2547,2585,2586,2588],{},[383,2587,1176],{},[379,2589,1181],{},[379,2591,2320],{},[383,2593,779],{"stretchy":778},[628,2595,2596,2598],{},[379,2597,1210],{},[379,2599,1126],{},[383,2601,784],{"separator":413},[628,2603,2604,2606],{},[379,2605,1219],{},[379,2607,1181],{},[383,2609,793],{"stretchy":778},[383,2611,784],{"separator":413},[405,2613,2614],{"encoding":407},"o_i = \\frac{\\sum_j \\kappa(q_i, k_j)\\, v_j}{\\sum_j \\kappa(q_i, k_j)} ,",[360,2616,2618,2673],{"className":2617,"ariaHidden":413},[412],[360,2619,2621,2624,2664,2667,2670],{"className":2620},[417],[360,2622],{"className":2623,"style":1298},[421],[360,2625,2627,2630],{"className":2626},[426],[360,2628,1162],{"className":2629},[426,427],[360,2631,2633],{"className":2632},[457],[360,2634,2636,2656],{"className":2635},[461,713],[360,2637,2639,2653],{"className":2638},[465],[360,2640,2642],{"className":2641,"style":1317},[469],[360,2643,2644,2647],{"style":1320},[360,2645],{"className":2646,"style":477},[476],[360,2648,2650],{"className":2649},[481,482,483,484],[360,2651,1126],{"className":2652},[426,427,484],[360,2654,741],{"className":2655},[740],[360,2657,2659],{"className":2658},[465],[360,2660,2662],{"className":2661,"style":1339},[469],[360,2663],{},[360,2665],{"className":2666,"style":432},[431],[360,2668,796],{"className":2669},[436],[360,2671],{"className":2672,"style":432},[431],[360,2674,2676,2680,3060],{"className":2675},[417],[360,2677],{"className":2678,"style":2679},[421],"height:2.6976em;vertical-align:-1.1218em;",[360,2681,2683,2686,3057],{"className":2682},[426],[360,2684],{"className":2685},[860,936],[360,2687,2689],{"className":2688},[811],[360,2690,2692,3048],{"className":2691},[461,713],[360,2693,2695,3045],{"className":2694},[465],[360,2696,2699,2847,2855],{"className":2697,"style":2698},[469],"height:1.5758em;",[360,2700,2702,2705],{"style":2701},"top:-2.314em;",[360,2703],{"className":2704,"style":956},[476],[360,2706,2708,2749,2752,2755,2758,2798,2801,2804,2844],{"className":2707},[426],[360,2709,2711,2714],{"className":2710},[852],[360,2712,1176],{"className":2713,"style":1395},[852,1393,1394],[360,2715,2717],{"className":2716},[457],[360,2718,2720,2741],{"className":2719},[461,713],[360,2721,2723,2738],{"className":2722},[465],[360,2724,2727],{"className":2725,"style":2726},[469],"height:0.162em;",[360,2728,2729,2732],{"style":1411},[360,2730],{"className":2731,"style":477},[476],[360,2733,2735],{"className":2734},[481,482,483,484],[360,2736,1181],{"className":2737,"style":1424},[426,427,484],[360,2739,741],{"className":2740},[740],[360,2742,2744],{"className":2743},[465],[360,2745,2747],{"className":2746,"style":1455},[469],[360,2748],{},[360,2750],{"className":2751,"style":871},[431],[360,2753,2320],{"className":2754},[426,427],[360,2756,779],{"className":2757},[860],[360,2759,2761,2764],{"className":2760},[426],[360,2762,1210],{"className":2763,"style":736},[426,427],[360,2765,2767],{"className":2766},[457],[360,2768,2770,2790],{"className":2769},[461,713],[360,2771,2773,2787],{"className":2772},[465],[360,2774,2776],{"className":2775,"style":1317},[469],[360,2777,2778,2781],{"style":2044},[360,2779],{"className":2780,"style":477},[476],[360,2782,2784],{"className":2783},[481,482,483,484],[360,2785,1126],{"className":2786},[426,427,484],[360,2788,741],{"className":2789},[740],[360,2791,2793],{"className":2792},[465],[360,2794,2796],{"className":2795,"style":1339},[469],[360,2797],{},[360,2799,784],{"className":2800},[867],[360,2802],{"className":2803,"style":871},[431],[360,2805,2807,2810],{"className":2806},[426],[360,2808,1219],{"className":2809,"style":1645},[426,427],[360,2811,2813],{"className":2812},[457],[360,2814,2816,2836],{"className":2815},[461,713],[360,2817,2819,2833],{"className":2818},[465],[360,2820,2822],{"className":2821,"style":1317},[469],[360,2823,2824,2827],{"style":2281},[360,2825],{"className":2826,"style":477},[476],[360,2828,2830],{"className":2829},[481,482,483,484],[360,2831,1181],{"className":2832,"style":1424},[426,427,484],[360,2834,741],{"className":2835},[740],[360,2837,2839],{"className":2838},[465],[360,2840,2842],{"className":2841,"style":2063},[469],[360,2843],{},[360,2845,793],{"className":2846},[887],[360,2848,2849,2852],{"style":1026},[360,2850],{"className":2851,"style":956},[476],[360,2853],{"className":2854,"style":1034},[1033],[360,2856,2858,2861],{"style":2857},"top:-3.8258em;",[360,2859],{"className":2860,"style":956},[476],[360,2862,2864,2904,2907,2910,2913,2953,2956,2959,2999,3002,3005],{"className":2863},[426],[360,2865,2867,2870],{"className":2866},[852],[360,2868,1176],{"className":2869,"style":1395},[852,1393,1394],[360,2871,2873],{"className":2872},[457],[360,2874,2876,2896],{"className":2875},[461,713],[360,2877,2879,2893],{"className":2878},[465],[360,2880,2882],{"className":2881,"style":2726},[469],[360,2883,2884,2887],{"style":1411},[360,2885],{"className":2886,"style":477},[476],[360,2888,2890],{"className":2889},[481,482,483,484],[360,2891,1181],{"className":2892,"style":1424},[426,427,484],[360,2894,741],{"className":2895},[740],[360,2897,2899],{"className":2898},[465],[360,2900,2902],{"className":2901,"style":1455},[469],[360,2903],{},[360,2905],{"className":2906,"style":871},[431],[360,2908,2320],{"className":2909},[426,427],[360,2911,779],{"className":2912},[860],[360,2914,2916,2919],{"className":2915},[426],[360,2917,1210],{"className":2918,"style":736},[426,427],[360,2920,2922],{"className":2921},[457],[360,2923,2925,2945],{"className":2924},[461,713],[360,2926,2928,2942],{"className":2927},[465],[360,2929,2931],{"className":2930,"style":1317},[469],[360,2932,2933,2936],{"style":2044},[360,2934],{"className":2935,"style":477},[476],[360,2937,2939],{"className":2938},[481,482,483,484],[360,2940,1126],{"className":2941},[426,427,484],[360,2943,741],{"className":2944},[740],[360,2946,2948],{"className":2947},[465],[360,2949,2951],{"className":2950,"style":1339},[469],[360,2952],{},[360,2954,784],{"className":2955},[867],[360,2957],{"className":2958,"style":871},[431],[360,2960,2962,2965],{"className":2961},[426],[360,2963,1219],{"className":2964,"style":1645},[426,427],[360,2966,2968],{"className":2967},[457],[360,2969,2971,2991],{"className":2970},[461,713],[360,2972,2974,2988],{"className":2973},[465],[360,2975,2977],{"className":2976,"style":1317},[469],[360,2978,2979,2982],{"style":2281},[360,2980],{"className":2981,"style":477},[476],[360,2983,2985],{"className":2984},[481,482,483,484],[360,2986,1181],{"className":2987,"style":1424},[426,427,484],[360,2989,741],{"className":2990},[740],[360,2992,2994],{"className":2993},[465],[360,2995,2997],{"className":2996,"style":2063},[469],[360,2998],{},[360,3000,793],{"className":3001},[887],[360,3003],{"className":3004,"style":871},[431],[360,3006,3008,3011],{"className":3007},[426],[360,3009,634],{"className":3010,"style":736},[426,427],[360,3012,3014],{"className":3013},[457],[360,3015,3017,3037],{"className":3016},[461,713],[360,3018,3020,3034],{"className":3019},[465],[360,3021,3023],{"className":3022,"style":1317},[469],[360,3024,3025,3028],{"style":2044},[360,3026],{"className":3027,"style":477},[476],[360,3029,3031],{"className":3030},[481,482,483,484],[360,3032,1181],{"className":3033,"style":1424},[426,427,484],[360,3035,741],{"className":3036},[740],[360,3038,3040],{"className":3039},[465],[360,3041,3043],{"className":3042,"style":2063},[469],[360,3044],{},[360,3046,741],{"className":3047},[740],[360,3049,3051],{"className":3050},[465],[360,3052,3055],{"className":3053,"style":3054},[469],"height:1.1218em;",[360,3056],{},[360,3058],{"className":3059},[887,936],[360,3061,784],{"className":3062},[867],[112,3064,3065],{},"——非參數回歸裡的 Nadaraya–Watson 估計。一模一樣。",[112,3067,3068],{},"差別只在於：這裡的「樣本」是模型自己學出來的，不是給定的資料集。",[108,3070,3071],{"id":3071},"知道這件事有什麼用",[112,3073,3074,3075,153],{},"不是為了數學上的優雅。是因為它會告訴你",[138,3076,3077],{},"哪裡可以砍",[112,3079,3080,3081,3111,3112,3262],{},"如果 ",[360,3082,3084,3098],{"className":3083},[363],[360,3085,3087],{"className":3086},[367],[369,3088,3089],{"xmlns":371},[373,3090,3091,3095],{},[376,3092,3093],{},[379,3094,2320],{},[405,3096,3097],{"encoding":407},"\\kappa",[360,3099,3101],{"className":3100,"ariaHidden":413},[412],[360,3102,3104,3108],{"className":3103},[417],[360,3105],{"className":3106,"style":3107},[421],"height:0.4306em;",[360,3109,2320],{"className":3110},[426,427]," 可以寫成有限維特徵映射的內積，也就是 ",[360,3113,3115,3163],{"className":3114},[363],[360,3116,3118],{"className":3117},[367],[369,3119,3120],{"xmlns":371},[373,3121,3122,3160],{},[376,3123,3124,3126,3128,3130,3132,3134,3136,3139,3142,3144,3146,3152,3154,3156,3158],{},[379,3125,2320],{},[383,3127,779],{"stretchy":778},[379,3129,1210],{},[383,3131,784],{"separator":413},[379,3133,1219],{},[383,3135,793],{"stretchy":778},[383,3137,3138],{},"≈",[379,3140,3141],{},"ϕ",[383,3143,779],{"stretchy":778},[379,3145,1210],{},[387,3147,3148,3150],{},[383,3149,793],{"stretchy":778},[379,3151,823],{"mathvariant":771},[379,3153,3141],{},[383,3155,779],{"stretchy":778},[379,3157,1219],{},[383,3159,793],{"stretchy":778},[405,3161,3162],{"encoding":407},"\\kappa(q,k) \\approx \\phi(q)^{\\top}\\phi(k)",[360,3164,3166,3202],{"className":3165,"ariaHidden":413},[412],[360,3167,3169,3172,3175,3178,3181,3184,3187,3190,3193,3196,3199],{"className":3168},[417],[360,3170],{"className":3171,"style":848},[421],[360,3173,2320],{"className":3174},[426,427],[360,3176,779],{"className":3177},[860],[360,3179,1210],{"className":3180,"style":736},[426,427],[360,3182,784],{"className":3183},[867],[360,3185],{"className":3186,"style":871},[431],[360,3188,1219],{"className":3189,"style":1645},[426,427],[360,3191,793],{"className":3192},[887],[360,3194],{"className":3195,"style":432},[431],[360,3197,3138],{"className":3198},[436],[360,3200],{"className":3201,"style":432},[431],[360,3203,3205,3209,3212,3215,3218,3250,3253,3256,3259],{"className":3204},[417],[360,3206],{"className":3207,"style":3208},[421],"height:1.0991em;vertical-align:-0.25em;",[360,3210,3141],{"className":3211},[426,427],[360,3213,779],{"className":3214},[860],[360,3216,1210],{"className":3217,"style":736},[426,427],[360,3219,3221,3224],{"className":3220},[887],[360,3222,793],{"className":3223},[887],[360,3225,3227],{"className":3226},[457],[360,3228,3230],{"className":3229},[461],[360,3231,3233],{"className":3232},[465],[360,3234,3236],{"className":3235,"style":446},[469],[360,3237,3238,3241],{"style":472},[360,3239],{"className":3240,"style":477},[476],[360,3242,3244],{"className":3243},[481,482,483,484],[360,3245,3247],{"className":3246},[426,484],[360,3248,823],{"className":3249},[426,484],[360,3251,3141],{"className":3252},[426,427],[360,3254,779],{"className":3255},[860],[360,3257,1219],{"className":3258,"style":1645},[426,427],[360,3260,793],{"className":3261},[887],"，那麼",[360,3264,3266],{"className":3265},[755],[360,3267,3269,3373],{"className":3268},[363],[360,3270,3272],{"className":3271},[367],[369,3273,3274],{"xmlns":371,"display":764},[373,3275,3276,3370],{},[376,3277,3278,3284,3286,3368],{},[628,3279,3280,3282],{},[379,3281,1162],{},[379,3283,1126],{},[383,3285,3138],{},[811,3287,3288,3332],{},[376,3289,3290,3292,3294,3300,3306,3312,3314,3316,3322,3324],{},[379,3291,3141],{},[383,3293,779],{"stretchy":778},[628,3295,3296,3298],{},[379,3297,1210],{},[379,3299,1126],{},[387,3301,3302,3304],{},[383,3303,793],{"stretchy":778},[379,3305,823],{"mathvariant":771},[2547,3307,3308,3310],{},[383,3309,1176],{},[379,3311,1181],{},[379,3313,3141],{},[383,3315,779],{"stretchy":778},[628,3317,3318,3320],{},[379,3319,1219],{},[379,3321,1181],{},[383,3323,793],{"stretchy":778},[1206,3325,3326,3328,3330],{},[379,3327,634],{},[379,3329,1181],{},[379,3331,823],{"mathvariant":771},[376,3333,3334,3336,3338,3344,3350,3356,3358,3360,3366],{},[379,3335,3141],{},[383,3337,779],{"stretchy":778},[628,3339,3340,3342],{},[379,3341,1210],{},[379,3343,1126],{},[387,3345,3346,3348],{},[383,3347,793],{"stretchy":778},[379,3349,823],{"mathvariant":771},[2547,3351,3352,3354],{},[383,3353,1176],{},[379,3355,1181],{},[379,3357,3141],{},[383,3359,779],{"stretchy":778},[628,3361,3362,3364],{},[379,3363,1219],{},[379,3365,1181],{},[383,3367,793],{"stretchy":778},[379,3369,835],{"mathvariant":771},[405,3371,3372],{"encoding":407},"o_i \\approx \\frac{\\phi(q_i)^{\\top} \\sum_j \\phi(k_j) v_j^{\\top}}{\\phi(q_i)^{\\top} \\sum_j \\phi(k_j)} .",[360,3374,3376,3432],{"className":3375,"ariaHidden":413},[412],[360,3377,3379,3383,3423,3426,3429],{"className":3378},[417],[360,3380],{"className":3381,"style":3382},[421],"height:0.6331em;vertical-align:-0.15em;",[360,3384,3386,3389],{"className":3385},[426],[360,3387,1162],{"className":3388},[426,427],[360,3390,3392],{"className":3391},[457],[360,3393,3395,3415],{"className":3394},[461,713],[360,3396,3398,3412],{"className":3397},[465],[360,3399,3401],{"className":3400,"style":1317},[469],[360,3402,3403,3406],{"style":1320},[360,3404],{"className":3405,"style":477},[476],[360,3407,3409],{"className":3408},[481,482,483,484],[360,3410,1126],{"className":3411},[426,427,484],[360,3413,741],{"className":3414},[740],[360,3416,3418],{"className":3417},[465],[360,3419,3421],{"className":3420,"style":1339},[469],[360,3422],{},[360,3424],{"className":3425,"style":432},[431],[360,3427,3138],{"className":3428},[436],[360,3430],{"className":3431,"style":432},[431],[360,3433,3435,3439,3900],{"className":3434},[417],[360,3436],{"className":3437,"style":3438},[421],"height:2.7967em;vertical-align:-1.1218em;",[360,3440,3442,3445,3897],{"className":3441},[426],[360,3443],{"className":3444},[860,936],[360,3446,3448],{"className":3447},[811],[360,3449,3451,3889],{"className":3450},[461,713],[360,3452,3454,3886],{"className":3453},[465],[360,3455,3458,3641,3649],{"className":3456,"style":3457},[469],"height:1.6749em;",[360,3459,3460,3463],{"style":2701},[360,3461],{"className":3462,"style":956},[476],[360,3464,3466,3469,3472,3512,3546,3549,3589,3592,3595,3598,3638],{"className":3465},[426],[360,3467,3141],{"className":3468},[426,427],[360,3470,779],{"className":3471},[860],[360,3473,3475,3478],{"className":3474},[426],[360,3476,1210],{"className":3477,"style":736},[426,427],[360,3479,3481],{"className":3480},[457],[360,3482,3484,3504],{"className":3483},[461,713],[360,3485,3487,3501],{"className":3486},[465],[360,3488,3490],{"className":3489,"style":1317},[469],[360,3491,3492,3495],{"style":2044},[360,3493],{"className":3494,"style":477},[476],[360,3496,3498],{"className":3497},[481,482,483,484],[360,3499,1126],{"className":3500},[426,427,484],[360,3502,741],{"className":3503},[740],[360,3505,3507],{"className":3506},[465],[360,3508,3510],{"className":3509,"style":1339},[469],[360,3511],{},[360,3513,3515,3518],{"className":3514},[887],[360,3516,793],{"className":3517},[887],[360,3519,3521],{"className":3520},[457],[360,3522,3524],{"className":3523},[461],[360,3525,3527],{"className":3526},[465],[360,3528,3531],{"className":3529,"style":3530},[469],"height:0.7751em;",[360,3532,3534,3537],{"style":3533},"top:-2.989em;margin-right:0.05em;",[360,3535],{"className":3536,"style":477},[476],[360,3538,3540],{"className":3539},[481,482,483,484],[360,3541,3543],{"className":3542},[426,484],[360,3544,823],{"className":3545},[426,484],[360,3547],{"className":3548,"style":871},[431],[360,3550,3552,3555],{"className":3551},[852],[360,3553,1176],{"className":3554,"style":1395},[852,1393,1394],[360,3556,3558],{"className":3557},[457],[360,3559,3561,3581],{"className":3560},[461,713],[360,3562,3564,3578],{"className":3563},[465],[360,3565,3567],{"className":3566,"style":2726},[469],[360,3568,3569,3572],{"style":1411},[360,3570],{"className":3571,"style":477},[476],[360,3573,3575],{"className":3574},[481,482,483,484],[360,3576,1181],{"className":3577,"style":1424},[426,427,484],[360,3579,741],{"className":3580},[740],[360,3582,3584],{"className":3583},[465],[360,3585,3587],{"className":3586,"style":1455},[469],[360,3588],{},[360,3590],{"className":3591,"style":871},[431],[360,3593,3141],{"className":3594},[426,427],[360,3596,779],{"className":3597},[860],[360,3599,3601,3604],{"className":3600},[426],[360,3602,1219],{"className":3603,"style":1645},[426,427],[360,3605,3607],{"className":3606},[457],[360,3608,3610,3630],{"className":3609},[461,713],[360,3611,3613,3627],{"className":3612},[465],[360,3614,3616],{"className":3615,"style":1317},[469],[360,3617,3618,3621],{"style":2281},[360,3619],{"className":3620,"style":477},[476],[360,3622,3624],{"className":3623},[481,482,483,484],[360,3625,1181],{"className":3626,"style":1424},[426,427,484],[360,3628,741],{"className":3629},[740],[360,3631,3633],{"className":3632},[465],[360,3634,3636],{"className":3635,"style":2063},[469],[360,3637],{},[360,3639,793],{"className":3640},[887],[360,3642,3643,3646],{"style":1026},[360,3644],{"className":3645,"style":956},[476],[360,3647],{"className":3648,"style":1034},[1033],[360,3650,3651,3654],{"style":2857},[360,3652],{"className":3653,"style":956},[476],[360,3655,3657,3660,3663,3703,3735,3738,3778,3781,3784,3787,3827,3830],{"className":3656},[426],[360,3658,3141],{"className":3659},[426,427],[360,3661,779],{"className":3662},[860],[360,3664,3666,3669],{"className":3665},[426],[360,3667,1210],{"className":3668,"style":736},[426,427],[360,3670,3672],{"className":3671},[457],[360,3673,3675,3695],{"className":3674},[461,713],[360,3676,3678,3692],{"className":3677},[465],[360,3679,3681],{"className":3680,"style":1317},[469],[360,3682,3683,3686],{"style":2044},[360,3684],{"className":3685,"style":477},[476],[360,3687,3689],{"className":3688},[481,482,483,484],[360,3690,1126],{"className":3691},[426,427,484],[360,3693,741],{"className":3694},[740],[360,3696,3698],{"className":3697},[465],[360,3699,3701],{"className":3700,"style":1339},[469],[360,3702],{},[360,3704,3706,3709],{"className":3705},[887],[360,3707,793],{"className":3708},[887],[360,3710,3712],{"className":3711},[457],[360,3713,3715],{"className":3714},[461],[360,3716,3718],{"className":3717},[465],[360,3719,3721],{"className":3720,"style":446},[469],[360,3722,3723,3726],{"style":472},[360,3724],{"className":3725,"style":477},[476],[360,3727,3729],{"className":3728},[481,482,483,484],[360,3730,3732],{"className":3731},[426,484],[360,3733,823],{"className":3734},[426,484],[360,3736],{"className":3737,"style":871},[431],[360,3739,3741,3744],{"className":3740},[852],[360,3742,1176],{"className":3743,"style":1395},[852,1393,1394],[360,3745,3747],{"className":3746},[457],[360,3748,3750,3770],{"className":3749},[461,713],[360,3751,3753,3767],{"className":3752},[465],[360,3754,3756],{"className":3755,"style":2726},[469],[360,3757,3758,3761],{"style":1411},[360,3759],{"className":3760,"style":477},[476],[360,3762,3764],{"className":3763},[481,482,483,484],[360,3765,1181],{"className":3766,"style":1424},[426,427,484],[360,3768,741],{"className":3769},[740],[360,3771,3773],{"className":3772},[465],[360,3774,3776],{"className":3775,"style":1455},[469],[360,3777],{},[360,3779],{"className":3780,"style":871},[431],[360,3782,3141],{"className":3783},[426,427],[360,3785,779],{"className":3786},[860],[360,3788,3790,3793],{"className":3789},[426],[360,3791,1219],{"className":3792,"style":1645},[426,427],[360,3794,3796],{"className":3795},[457],[360,3797,3799,3819],{"className":3798},[461,713],[360,3800,3802,3816],{"className":3801},[465],[360,3803,3805],{"className":3804,"style":1317},[469],[360,3806,3807,3810],{"style":2281},[360,3808],{"className":3809,"style":477},[476],[360,3811,3813],{"className":3812},[481,482,483,484],[360,3814,1181],{"className":3815,"style":1424},[426,427,484],[360,3817,741],{"className":3818},[740],[360,3820,3822],{"className":3821},[465],[360,3823,3825],{"className":3824,"style":2063},[469],[360,3826],{},[360,3828,793],{"className":3829},[887],[360,3831,3833,3836],{"className":3832},[426],[360,3834,634],{"className":3835,"style":736},[426,427],[360,3837,3839],{"className":3838},[457],[360,3840,3842,3877],{"className":3841},[461,713],[360,3843,3845,3874],{"className":3844},[465],[360,3846,3848,3860],{"className":3847,"style":446},[469],[360,3849,3851,3854],{"style":3850},"top:-2.4413em;margin-left:-0.0359em;margin-right:0.05em;",[360,3852],{"className":3853,"style":477},[476],[360,3855,3857],{"className":3856},[481,482,483,484],[360,3858,1181],{"className":3859,"style":1424},[426,427,484],[360,3861,3862,3865],{"style":472},[360,3863],{"className":3864,"style":477},[476],[360,3866,3868],{"className":3867},[481,482,483,484],[360,3869,3871],{"className":3870},[426,484],[360,3872,823],{"className":3873},[426,484],[360,3875,741],{"className":3876},[740],[360,3878,3880],{"className":3879},[465],[360,3881,3884],{"className":3882,"style":3883},[469],"height:0.3948em;",[360,3885],{},[360,3887,741],{"className":3888},[740],[360,3890,3892],{"className":3891},[465],[360,3893,3895],{"className":3894,"style":3054},[469],[360,3896],{},[360,3898],{"className":3899},[887,936],[360,3901,835],{"className":3902},[426],[112,3904,3905,3906,3955,3956,4023],{},"右邊的兩個累加量可以事先算好，每個查詢只要查表。複雜度從 ",[360,3907,3909,3932],{"className":3908},[363],[360,3910,3912],{"className":3911},[367],[369,3913,3914],{"xmlns":371},[373,3915,3916,3929],{},[376,3917,3918,3921,3923,3925,3927],{},[379,3919,3920],{},"O",[383,3922,779],{"stretchy":778},[379,3924,397],{},[379,3926,524],{},[383,3928,793],{"stretchy":778},[405,3930,3931],{"encoding":407},"O(nm)",[360,3933,3935],{"className":3934,"ariaHidden":413},[412],[360,3936,3938,3941,3945,3948,3952],{"className":3937},[417],[360,3939],{"className":3940,"style":848},[421],[360,3942,3920],{"className":3943,"style":3944},[426,427],"margin-right:0.0278em;",[360,3946,779],{"className":3947},[860],[360,3949,3951],{"className":3950},[426,427],"nm",[360,3953,793],{"className":3954},[887]," 掉到 ",[360,3957,3959,3984],{"className":3958},[363],[360,3960,3962],{"className":3961},[367],[369,3963,3964],{"xmlns":371},[373,3965,3966,3981],{},[376,3967,3968,3970,3972,3974,3977,3979],{},[379,3969,3920],{},[383,3971,779],{"stretchy":778},[379,3973,397],{},[383,3975,3976],{},"+",[379,3978,524],{},[383,3980,793],{"stretchy":778},[405,3982,3983],{"encoding":407},"O(n + m)",[360,3985,3987,4011],{"className":3986,"ariaHidden":413},[412],[360,3988,3990,3993,3996,3999,4002,4005,4008],{"className":3989},[417],[360,3991],{"className":3992,"style":848},[421],[360,3994,3920],{"className":3995,"style":3944},[426,427],[360,3997,779],{"className":3998},[860],[360,4000,397],{"className":4001},[426,427],[360,4003],{"className":4004,"style":650},[431],[360,4006,3976],{"className":4007},[494],[360,4009],{"className":4010,"style":650},[431],[360,4012,4014,4017,4020],{"className":4013},[417],[360,4015],{"className":4016,"style":848},[421],[360,4018,524],{"className":4019},[426,427],[360,4021,793],{"className":4022},[887],"：",[360,4025,4027],{"className":4026},[755],[360,4028,4030,4135],{"className":4029},[363],[360,4031,4033],{"className":4032},[367],[369,4034,4035],{"xmlns":371,"display":764},[373,4036,4037,4132],{},[4038,4039,4043,4082],"mtable",{"rowspacing":4040,"columnalign":4041,"columnspacing":4042},"0.25em","right left","0em",[4044,4045,4046,4056],"mtr",{},[4047,4048,4049],"mtd",{},[4050,4051,4053],"mstyle",{"scriptlevel":4052,"displaystyle":413},"0",[803,4054,4055],{},"標準 attention",[4047,4057,4058],{},[4050,4059,4060],{"scriptlevel":4052,"displaystyle":413},[376,4061,4062,4064,4067,4070,4072,4074,4076,4078,4080],{},[376,4063],{},[383,4065,4066],{},":",[431,4068],{"width":4069},"1em",[379,4071,3920],{},[383,4073,779],{"stretchy":778},[379,4075,397],{},[379,4077,524],{},[379,4079,403],{},[383,4081,793],{"stretchy":778},[4044,4083,4084,4091],{},[4047,4085,4086],{},[4050,4087,4088],{"scriptlevel":4052,"displaystyle":413},[803,4089,4090],{},"線性 attention",[4047,4092,4093],{},[4050,4094,4095],{"scriptlevel":4052,"displaystyle":413},[376,4096,4097,4099,4101,4103,4105,4107],{},[376,4098],{},[383,4100,4066],{},[431,4102],{"width":4069},[379,4104,3920],{},[803,4106,805],{},[376,4108,4109,4111,4113,4120,4122,4124,4130],{},[383,4110,779],{"fence":413},[379,4112,397],{},[387,4114,4115,4117],{},[379,4116,403],{},[1185,4118,4119],{},"2",[383,4121,3976],{},[379,4123,524],{},[387,4125,4126,4128],{},[379,4127,403],{},[1185,4129,4119],{},[383,4131,793],{"fence":413},[405,4133,4134],{"encoding":407},"\\begin{aligned}\n\\text{標準 attention} &: \\quad O(n m d) \\\\\n\\text{線性 attention} &: \\quad O\\!\\left(n d^2 + m d^2\\right)\n\\end{aligned}",[360,4136,4138],{"className":4137,"ariaHidden":413},[412],[360,4139,4141,4145],{"className":4140},[417],[360,4142],{"className":4143,"style":4144},[421],"height:2.7241em;vertical-align:-1.1121em;",[360,4146,4148],{"className":4147},[426],[360,4149,4151,4218],{"className":4150},[4038],[360,4152,4155],{"className":4153},[4154],"col-align-r",[360,4156,4158,4209],{"className":4157},[461,713],[360,4159,4161,4206],{"className":4160},[465],[360,4162,4165,4187],{"className":4163,"style":4164},[469],"height:1.6121em;",[360,4166,4168,4171],{"style":4167},"top:-3.7721em;",[360,4169],{"className":4170,"style":956},[476],[360,4172,4174],{"className":4173},[426],[360,4175,4178,4183],{"className":4176},[426,4177],"text",[360,4179,4182],{"className":4180},[426,4181],"cjk_fallback","標準",[360,4184,4186],{"className":4185},[426]," attention",[360,4188,4190,4193],{"style":4189},"top:-2.2479em;",[360,4191],{"className":4192,"style":956},[476],[360,4194,4196],{"className":4195},[426],[360,4197,4199,4203],{"className":4198},[426,4177],[360,4200,4202],{"className":4201},[426,4181],"線性",[360,4204,4186],{"className":4205},[426],[360,4207,741],{"className":4208},[740],[360,4210,4212],{"className":4211},[465],[360,4213,4216],{"className":4214,"style":4215},[469],"height:1.1121em;",[360,4217],{},[360,4219,4222],{"className":4220},[4221],"col-align-l",[360,4223,4225,4395],{"className":4224},[461,713],[360,4226,4228,4392],{"className":4227},[465],[360,4229,4231,4270],{"className":4230,"style":4164},[469],[360,4232,4233,4236],{"style":4167},[360,4234],{"className":4235,"style":956},[476],[360,4237,4239,4242,4245,4248,4252,4255,4258,4261,4264,4267],{"className":4238},[426],[360,4240],{"className":4241},[426],[360,4243],{"className":4244,"style":432},[431],[360,4246,4066],{"className":4247},[436],[360,4249],{"className":4250,"style":4251},[431],"margin-right:1em;",[360,4253],{"className":4254,"style":432},[431],[360,4256,3920],{"className":4257,"style":3944},[426,427],[360,4259,779],{"className":4260},[860],[360,4262,3951],{"className":4263},[426,427],[360,4265,403],{"className":4266},[426,427],[360,4268,793],{"className":4269},[887],[360,4271,4272,4275],{"style":4189},[360,4273],{"className":4274,"style":956},[476],[360,4276,4278,4281,4284,4287,4290,4293,4296,4299,4302],{"className":4277},[426],[360,4279],{"className":4280},[426],[360,4282],{"className":4283,"style":432},[431],[360,4285,4066],{"className":4286},[436],[360,4288],{"className":4289,"style":4251},[431],[360,4291],{"className":4292,"style":432},[431],[360,4294,3920],{"className":4295,"style":3944},[426,427],[360,4297],{"className":4298,"style":913},[431],[360,4300],{"className":4301,"style":871},[431],[360,4303,4305,4311,4314,4345,4348,4351,4354,4357,4386],{"className":4304},[920],[360,4306,4308],{"className":4307,"style":925},[860,924],[360,4309,779],{"className":4310},[929,732],[360,4312,397],{"className":4313},[426,427],[360,4315,4317,4320],{"className":4316},[426],[360,4318,403],{"className":4319},[426,427],[360,4321,4323],{"className":4322},[457],[360,4324,4326],{"className":4325},[461],[360,4327,4329],{"className":4328},[465],[360,4330,4333],{"className":4331,"style":4332},[469],"height:0.8641em;",[360,4334,4336,4339],{"style":4335},"top:-3.113em;margin-right:0.05em;",[360,4337],{"className":4338,"style":477},[476],[360,4340,4342],{"className":4341},[481,482,483,484],[360,4343,4119],{"className":4344},[426,484],[360,4346],{"className":4347,"style":650},[431],[360,4349,3976],{"className":4350},[494],[360,4352],{"className":4353,"style":650},[431],[360,4355,524],{"className":4356},[426,427],[360,4358,4360,4363],{"className":4359},[426],[360,4361,403],{"className":4362},[426,427],[360,4364,4366],{"className":4365},[457],[360,4367,4369],{"className":4368},[461],[360,4370,4372],{"className":4371},[465],[360,4373,4375],{"className":4374,"style":4332},[469],[360,4376,4377,4380],{"style":4335},[360,4378],{"className":4379,"style":477},[476],[360,4381,4383],{"className":4382},[481,482,483,484],[360,4384,4119],{"className":4385},[426,484],[360,4387,4389],{"className":4388,"style":925},[887,924],[360,4390,793],{"className":4391},[929,732],[360,4393,741],{"className":4394},[740],[360,4396,4398],{"className":4397},[465],[360,4399,4401],{"className":4400,"style":4215},[469],[360,4402],{},[112,4404,4405],{},"這就是 linear attention 那一整家族在做的事。",[108,4407,4408],{"id":4408},"代價在分母",[112,4410,4411,4412],{},"這裡有個很容易被忽略的地方：",[138,4413,4414],{},"誤差最大的地方不是分子，是分母。",[112,4416,4417,4418,4650,4651,4840],{},"分子是「加權和」，近似得差一點，輸出只是稍微偏掉；分母是「歸一化常數」，它一錯，整個輸出會跟著縮放。序列越長，",[360,4419,4421,4467],{"className":4420},[363],[360,4422,4424],{"className":4423},[367],[369,4425,4426],{"xmlns":371},[373,4427,4428,4464],{},[376,4429,4430,4432,4434,4440,4446,4452,4454,4456,4462],{},[379,4431,3141],{},[383,4433,779],{"stretchy":778},[628,4435,4436,4438],{},[379,4437,1210],{},[379,4439,1126],{},[387,4441,4442,4444],{},[383,4443,793],{"stretchy":778},[379,4445,823],{"mathvariant":771},[628,4447,4448,4450],{},[383,4449,1176],{},[379,4451,1181],{},[379,4453,3141],{},[383,4455,779],{"stretchy":778},[628,4457,4458,4460],{},[379,4459,1219],{},[379,4461,1181],{},[383,4463,793],{"stretchy":778},[405,4465,4466],{"encoding":407},"\\phi(q_i)^{\\top}\\sum_j \\phi(k_j)",[360,4468,4470],{"className":4469,"ariaHidden":413},[412],[360,4471,4473,4477,4480,4483,4523,4555,4558,4598,4601,4604,4607,4647],{"className":4472},[417],[360,4474],{"className":4475,"style":4476},[421],"height:1.2849em;vertical-align:-0.4358em;",[360,4478,3141],{"className":4479},[426,427],[360,4481,779],{"className":4482},[860],[360,4484,4486,4489],{"className":4485},[426],[360,4487,1210],{"className":4488,"style":736},[426,427],[360,4490,4492],{"className":4491},[457],[360,4493,4495,4515],{"className":4494},[461,713],[360,4496,4498,4512],{"className":4497},[465],[360,4499,4501],{"className":4500,"style":1317},[469],[360,4502,4503,4506],{"style":2044},[360,4504],{"className":4505,"style":477},[476],[360,4507,4509],{"className":4508},[481,482,483,484],[360,4510,1126],{"className":4511},[426,427,484],[360,4513,741],{"className":4514},[740],[360,4516,4518],{"className":4517},[465],[360,4519,4521],{"className":4520,"style":1339},[469],[360,4522],{},[360,4524,4526,4529],{"className":4525},[887],[360,4527,793],{"className":4528},[887],[360,4530,4532],{"className":4531},[457],[360,4533,4535],{"className":4534},[461],[360,4536,4538],{"className":4537},[465],[360,4539,4541],{"className":4540,"style":446},[469],[360,4542,4543,4546],{"style":472},[360,4544],{"className":4545,"style":477},[476],[360,4547,4549],{"className":4548},[481,482,483,484],[360,4550,4552],{"className":4551},[426,484],[360,4553,823],{"className":4554},[426,484],[360,4556],{"className":4557,"style":871},[431],[360,4559,4561,4564],{"className":4560},[852],[360,4562,1176],{"className":4563,"style":1395},[852,1393,1394],[360,4565,4567],{"className":4566},[457],[360,4568,4570,4590],{"className":4569},[461,713],[360,4571,4573,4587],{"className":4572},[465],[360,4574,4576],{"className":4575,"style":2726},[469],[360,4577,4578,4581],{"style":1411},[360,4579],{"className":4580,"style":477},[476],[360,4582,4584],{"className":4583},[481,482,483,484],[360,4585,1181],{"className":4586,"style":1424},[426,427,484],[360,4588,741],{"className":4589},[740],[360,4591,4593],{"className":4592},[465],[360,4594,4596],{"className":4595,"style":1455},[469],[360,4597],{},[360,4599],{"className":4600,"style":871},[431],[360,4602,3141],{"className":4603},[426,427],[360,4605,779],{"className":4606},[860],[360,4608,4610,4613],{"className":4609},[426],[360,4611,1219],{"className":4612,"style":1645},[426,427],[360,4614,4616],{"className":4615},[457],[360,4617,4619,4639],{"className":4618},[461,713],[360,4620,4622,4636],{"className":4621},[465],[360,4623,4625],{"className":4624,"style":1317},[469],[360,4626,4627,4630],{"style":2281},[360,4628],{"className":4629,"style":477},[476],[360,4631,4633],{"className":4632},[481,482,483,484],[360,4634,1181],{"className":4635,"style":1424},[426,427,484],[360,4637,741],{"className":4638},[740],[360,4640,4642],{"className":4641},[465],[360,4643,4645],{"className":4644,"style":2063},[469],[360,4646],{},[360,4648,793],{"className":4649},[887]," 和真正的 ",[360,4652,4654,4692],{"className":4653},[363],[360,4655,4657],{"className":4656},[367],[369,4658,4659],{"xmlns":371},[373,4660,4661,4689],{},[376,4662,4663,4669,4671,4673,4679,4681,4687],{},[628,4664,4665,4667],{},[383,4666,1176],{},[379,4668,1181],{},[379,4670,2320],{},[383,4672,779],{"stretchy":778},[628,4674,4675,4677],{},[379,4676,1210],{},[379,4678,1126],{},[383,4680,784],{"separator":413},[628,4682,4683,4685],{},[379,4684,1219],{},[379,4686,1181],{},[383,4688,793],{"stretchy":778},[405,4690,4691],{"encoding":407},"\\sum_j \\kappa(q_i,k_j)",[360,4693,4695],{"className":4694,"ariaHidden":413},[412],[360,4696,4698,4702,4742,4745,4748,4751,4791,4794,4797,4837],{"className":4697},[417],[360,4699],{"className":4700,"style":4701},[421],"height:1.1858em;vertical-align:-0.4358em;",[360,4703,4705,4708],{"className":4704},[852],[360,4706,1176],{"className":4707,"style":1395},[852,1393,1394],[360,4709,4711],{"className":4710},[457],[360,4712,4714,4734],{"className":4713},[461,713],[360,4715,4717,4731],{"className":4716},[465],[360,4718,4720],{"className":4719,"style":2726},[469],[360,4721,4722,4725],{"style":1411},[360,4723],{"className":4724,"style":477},[476],[360,4726,4728],{"className":4727},[481,482,483,484],[360,4729,1181],{"className":4730,"style":1424},[426,427,484],[360,4732,741],{"className":4733},[740],[360,4735,4737],{"className":4736},[465],[360,4738,4740],{"className":4739,"style":1455},[469],[360,4741],{},[360,4743],{"className":4744,"style":871},[431],[360,4746,2320],{"className":4747},[426,427],[360,4749,779],{"className":4750},[860],[360,4752,4754,4757],{"className":4753},[426],[360,4755,1210],{"className":4756,"style":736},[426,427],[360,4758,4760],{"className":4759},[457],[360,4761,4763,4783],{"className":4762},[461,713],[360,4764,4766,4780],{"className":4765},[465],[360,4767,4769],{"className":4768,"style":1317},[469],[360,4770,4771,4774],{"style":2044},[360,4772],{"className":4773,"style":477},[476],[360,4775,4777],{"className":4776},[481,482,483,484],[360,4778,1126],{"className":4779},[426,427,484],[360,4781,741],{"className":4782},[740],[360,4784,4786],{"className":4785},[465],[360,4787,4789],{"className":4788,"style":1339},[469],[360,4790],{},[360,4792,784],{"className":4793},[867],[360,4795],{"className":4796,"style":871},[431],[360,4798,4800,4803],{"className":4799},[426],[360,4801,1219],{"className":4802,"style":1645},[426,427],[360,4804,4806],{"className":4805},[457],[360,4807,4809,4829],{"className":4808},[461,713],[360,4810,4812,4826],{"className":4811},[465],[360,4813,4815],{"className":4814,"style":1317},[469],[360,4816,4817,4820],{"style":2281},[360,4818],{"className":4819,"style":477},[476],[360,4821,4823],{"className":4822},[481,482,483,484],[360,4824,1181],{"className":4825,"style":1424},[426,427,484],[360,4827,741],{"className":4828},[740],[360,4830,4832],{"className":4831},[465],[360,4833,4835],{"className":4834,"style":2063},[469],[360,4836],{},[360,4838,793],{"className":4839},[887]," 差得越多，輸出就越不可信。",[112,4842,4843],{},"所以我們在評估這類方法時，看的第一個數字不是整體 perplexity，而是分母的相對誤差。",[108,4845,4846],{"id":4846},"實務上的三個檢查點",[4848,4849,4850,4863],"table",{},[4851,4852,4853],"thead",{},[4854,4855,4856,4860],"tr",{},[4857,4858,4859],"th",{},"檢查",[4857,4861,4862],{},"為什麼",[4864,4865,4866,4875,4954],"tbody",{},[4854,4867,4868,4872],{},[4869,4870,4871],"td",{},"分母的相對誤差",[4869,4873,4874],{},"比分子更致命，因為它會整體縮放輸出",[4854,4876,4877,4951],{},[4869,4878,4879,4880],{},"特徵維度 ",[360,4881,4883,4901],{"className":4882},[363],[360,4884,4886],{"className":4885},[367],[369,4887,4888],{"xmlns":371},[373,4889,4890,4898],{},[376,4891,4892],{},[628,4893,4894,4896],{},[379,4895,403],{},[379,4897,3141],{},[405,4899,4900],{"encoding":407},"d_\\phi",[360,4902,4904],{"className":4903,"ariaHidden":413},[412],[360,4905,4907,4910],{"className":4906},[417],[360,4908],{"className":4909,"style":2260},[421],[360,4911,4913,4916],{"className":4912},[426],[360,4914,403],{"className":4915},[426,427],[360,4917,4919],{"className":4918},[457],[360,4920,4922,4943],{"className":4921},[461,713],[360,4923,4925,4940],{"className":4924},[465],[360,4926,4929],{"className":4927,"style":4928},[469],"height:0.3361em;",[360,4930,4931,4934],{"style":1320},[360,4932],{"className":4933,"style":477},[476],[360,4935,4937],{"className":4936},[481,482,483,484],[360,4938,3141],{"className":4939},[426,427,484],[360,4941,741],{"className":4942},[740],[360,4944,4946],{"className":4945},[465],[360,4947,4949],{"className":4948,"style":2063},[469],[360,4950],{},[4869,4952,4953],{},"太少則近似粗糙，太多則省不下計算",[4854,4955,4956,4959],{},[4869,4957,4958],{},"序列長度分佈",[4869,4960,4961],{},"短序列幾乎沒有收益，長序列才值得換",[4963,4964,4968],"pre",{"className":4965,"code":4966,"language":4967,"meta":311,"style":311},"language-python shiki shiki-themes material-theme-lighter material-theme material-theme-palenight","# 概念示意：先把 K、V 的累加量算好\nkv = einsum(\"m d, m e -> d e\", phi(K), V)   # O(m d^2)\nz  = phi(K).sum(dim=0)                       # O(m d)\nout = einsum(\"n d, d e -> n e\", phi(Q), kv) \u002F einsum(\"n d, d -> n\", phi(Q), z)\n","python",[118,4969,4970,4977,4982,4988],{"__ignoreMap":311},[360,4971,4974],{"class":4972,"line":4973},"line",1,[360,4975,4976],{},"# 概念示意：先把 K、V 的累加量算好\n",[360,4978,4979],{"class":4972,"line":312},[360,4980,4981],{},"kv = einsum(\"m d, m e -> d e\", phi(K), V)   # O(m d^2)\n",[360,4983,4985],{"class":4972,"line":4984},3,[360,4986,4987],{},"z  = phi(K).sum(dim=0)                       # O(m d)\n",[360,4989,4991],{"class":4972,"line":4990},4,[360,4992,4993],{},"out = einsum(\"n d, d e -> n e\", phi(Q), kv) \u002F einsum(\"n d, d -> n\", phi(Q), z)\n",[108,4995,4996],{"id":4996},"給下一個要調這個模型的人",[112,4998,4999],{},"如果你正在為長序列煩惱，我會建議先把 attention 當成核平滑器看一遍，再決定要不要換模型。",[112,5001,5002,5003,5006],{},"因為換成 linear attention 不是「更快的 attention」，而是",[138,5004,5005],{},"用特徵映射換掉精確的歸一化","——你付出的是準確度，不是免費的午餐。",[112,5008,5009],{},"知道自己在付什麼，比知道自己在用什麼重要。",[5011,5012,5013],"style",{},"html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}html.dark .shiki span {color: var(--shiki-dark);background: var(--shiki-dark-bg);font-style: var(--shiki-dark-font-style);font-weight: var(--shiki-dark-font-weight);text-decoration: var(--shiki-dark-text-decoration);}",{"title":311,"searchDepth":312,"depth":312,"links":5015},[5016,5017,5018,5019,5020,5021],{"id":346,"depth":312,"text":346},{"id":355,"depth":312,"text":355},{"id":3071,"depth":312,"text":3071},{"id":4408,"depth":312,"text":4408},{"id":4846,"depth":312,"text":4846},{"id":4996,"depth":312,"text":4996},"\u002Fimages\u002Fblog\u002Fattention-kernel.svg","Attention 與核函數對應關係的示意圖","2026-10-20","第一次看到「attention 是核方法」這個說法時，我以為那只是類比。自己推過一次才發現不是——而這件事直接決定了我們該在哪裡砍計算量。",{},"\u002Fblog\u002Fzh-tw\u002Fattention-as-kernel",{"title":341,"description":5025},"attention-as-kernel","blog\u002Fzh-TW\u002Fattention-as-kernel",[5032,5033,5034],"transformer","數學","推論","2026-10-22","ipCLaHTj0DVQiZoXlJ6DPzX7hl68udhNuMH4wa4gke8",{"id":5038,"title":5039,"author":103,"body":5040,"cover":5415,"coverAlt":5039,"date":5416,"description":5417,"draft":324,"extension":325,"featured":324,"lang":327,"meta":5418,"navigation":326,"path":5419,"seo":5420,"slug":5421,"stem":5422,"tags":5423,"translationKey":5421,"updated":337,"__hash__":5427},"blog\u002Fblog\u002Fzh-TW\u002Farxiv-2510-27688v1.md","連續自迴歸語言模型 CALM 技術解析",{"type":105,"value":5041,"toc":5405},[5042,5052,5060,5066,5072,5076,5079,5082,5096,5102,5106,5112,5118,5123,5179,5188,5192,5245,5265,5272,5286,5290,5293,5325,5329,5355,5359,5382,5397],[112,5043,5044,5047,5048,5051],{},[138,5045,5046],{},"研究單位","：騰訊微信 AI × 清華大學\n",[138,5049,5050],{},"核心命題","：傳統大模型逐字（Token-by-token）生成的效率已觸及物理天花板。CALM 將離散 Token 轉換為連續向量（Continuous Vectors），將生成步數縮減至原來的 1\u002FK 為大語言模型的擴充開闢了全新的效率路徑。",[112,5053,5054],{},[5055,5056,5057],"a",{"href":5057,"rel":5058},"https:\u002F\u002Farxiv.org\u002Fhtml\u002F2510.27688v1",[5059],"nofollow",[112,5061,5062],{},[5055,5063,5064],{"href":5064,"rel":5065},"https:\u002F\u002Fshaochenze.github.io\u002Fblog\u002F2025\u002FCALM\u002F",[5059],[112,5067,5068],{},[5055,5069,5070],{"href":5070,"rel":5071},"https:\u002F\u002Fgithub.com\u002Fmahmoodlab\u002Fclam",[5059],[108,5073,5075],{"id":5074},"一背景與動機離散生成的效率瓶頸","一、背景與動機：離散生成的效率瓶頸",[112,5077,5078],{},"現有的大型語言模型（LLMs）都依賴「逐字預測」的舊範式。每個 Token 都需要進行一次完整的前向計算，當序列變長時，計算成本呈線性甚至平方級增長。",[112,5080,5081],{},"儘管文本表徵已經從字符集進化到子詞集（如 BPE），但這種「離散表示」已觸及物理極限：",[272,5083,5084,5090],{},[275,5085,5086,5089],{},[138,5087,5088],{},"詞表膨脹的詛咒","：為了涵蓋更多語義，詞表規模從 32,000 膨脹至 256,000，導致最後一層的計算與記憶體成本飆升。",[275,5091,5092,5095],{},[138,5093,5094],{},"資訊密度極低","：每個離散詞元僅攜帶約 15–18 bit 的資訊，潛力已然耗盡。模型擁有龐大的算力，卻被困在一次只能預測一個低資訊量 Token 的勞力密集工作中。",[112,5097,5098,5101],{},[138,5099,5100],{},"結論","：必須轉向高維連續路線，從根本上改變生成機制。",[108,5103,5105],{"id":5104},"二calm-核心思路與原理","二、CALM 核心思路與原理",[112,5107,5108,5111],{},[138,5109,5110],{},"基本機制","：利用自編碼器(Autoencoder)將 K 個離散 Token 壓縮為「一個」高維連續向量。這使得模型的序列長度從 T 降至 T\u002FK，生成步數直接縮減為原來的 K 分之一。",[112,5113,5114,5117],{},[138,5115,5116],{},"本質","：以更高的「語義頻寬」(Semantic Bandwidth)換取更少的生成步驟，實現算力與速度的跨越式突破。",[5119,5120,5122],"h3",{"id":5121},"傳統範式-vs-calm-連續模型","傳統範式 vs. CALM 連續模型",[4848,5124,5125,5138],{},[4851,5126,5127],{},[4854,5128,5129,5132,5135],{},[4857,5130,5131],{},"維度",[4857,5133,5134],{},"傳統離散模型 (Discrete)",[4857,5136,5137],{},"CALM 連續模型 (Continuous)",[4864,5139,5140,5153,5166],{},[4854,5141,5142,5147,5150],{},[4869,5143,5144],{},[138,5145,5146],{},"擴容方式",[4869,5148,5149],{},"詞表指數級膨脹，算力爆炸",[4869,5151,5152],{},"增加向量維度，算力平滑增長",[4854,5154,5155,5160,5163],{},[4869,5156,5157],{},[138,5158,5159],{},"計算目標",[4869,5161,5162],{},"從數十萬詞中選一個機率",[4869,5164,5165],{},"在連續空間中直接預測精確座標",[4854,5167,5168,5173,5176],{},[4869,5169,5170],{},[138,5171,5172],{},"序列長度",[4869,5174,5175],{},"極長序列 T",[4869,5177,5178],{},"短序列 T\u002FK，步數銳減",[5180,5181,5182],"blockquote",{},[112,5183,5184,5187],{},[138,5185,5186],{},"通俗比喻","：傳統生成就像在「擁有數十萬個抽屜的倉庫中盲目尋寶」，而 CALM 則是「直接繪製高維座標」，徹底擺脫了詞表膨脹的物理限制。",[108,5189,5191],{"id":5190},"三calm-的四大技術支柱","三、CALM 的四大技術支柱",[5193,5194,5195,5231,5240],"ol",{},[275,5196,5197,5200,5201,5230],{},[138,5198,5199],{},"高保真自編碼器 (Autoencoder)","\n將 ",[360,5202,5204,5217],{"className":5203},[363],[360,5205,5207],{"className":5206},[367],[369,5208,5209],{"xmlns":371},[373,5210,5211,5215],{},[376,5212,5213],{},[379,5214,513],{},[405,5216,513],{"encoding":407},[360,5218,5220],{"className":5219,"ariaHidden":413},[412],[360,5221,5223,5227],{"className":5222},[417],[360,5224],{"className":5225,"style":5226},[421],"height:0.6833em;",[360,5228,513],{"className":5229,"style":545},[426,427]," 個離散詞元壓縮為 128 或更高維度的連續向量。在約 7500 萬參數的輕量級開銷下，重構準確率高達 > 99.9%。它是實現離散與連續間無損轉換的基石。",[275,5232,5233,5236,5237,153],{},[138,5234,5235],{},"能量生成頭與無似然框架","\n摒棄了傳統的 Softmax 機率計算，在連續空間中直接透過迴歸（Regression）預測下一個目標向量，無需像擴散模型（Diffusion）那樣進行數十上百次的去噪迭代，實現",[138,5238,5239],{},"單步生成",[275,5241,5242],{},[138,5243,5244],{},"連續空間魯棒性三大魔法",[272,5246,5247,5253,5259],{},[275,5248,5249,5252],{},[138,5250,5251],{},"變分正則化 (Variational Regularization)","：熨平潛空間（Latent space）中的褶皺與斷層。",[275,5254,5255,5258],{},[138,5256,5257],{},"KL 裁減 (KL Clipping)","：防止潛空間維度嚴重塌陷。",[275,5260,5261,5264],{},[138,5262,5263],{},"雙重 Dropout","：提供額外的抗噪能力。\n這三者共同構建了平滑且富有彈性的高維地形，確保微小的生成偏差不會導致亂碼。",[5193,5266,5267],{"start":4990},[275,5268,5269],{},[138,5270,5271],{},"無機率評估體系 BrierLM & 溫度控制",[272,5273,5274,5280],{},[275,5275,5276,5279],{},[138,5277,5278],{},"Brier 評估 (Brier Score)","：傳統的困惑度 (Perplexity) 在連續模型中失效。CALM 引入了無似然的 Brier 分數（準確率 − 碰撞率）作為專屬標尺，與交叉熵損失呈極強負相關。",[275,5281,5282,5285],{},[138,5283,5284],{},"黑盒溫度採樣","：因為沒有傳統機率分母可調，CALM 以「批量大小 N」(Batch Size) 作為新型的溫度旋鈕。N 小則低溫高確定性，N 大則高溫高多樣性。搭配拒絕採樣機制，效果與傳統溫度調節高度一致。",[108,5287,5289],{"id":5288},"四流程總覽","四、流程總覽",[112,5291,5292],{},"整個運作閉環如下，速度遠超傳統的逐字生成：",[5193,5294,5295,5301,5307,5313,5319],{},[275,5296,5297,5300],{},[138,5298,5299],{},"輸入","：離散 Token 序列。",[275,5302,5303,5306],{},[138,5304,5305],{},"壓縮","：編碼器將每 K 個 Token 壓縮為 1 個連續向量。",[275,5308,5309,5312],{},[138,5310,5311],{},"處理","：Transformer 骨幹網絡在極短的連續向量序列上進行深層注意力計算。",[275,5314,5315,5318],{},[138,5316,5317],{},"預測","：生成頭直接單步預測出下一個目標連續向量。",[275,5320,5321,5324],{},[138,5322,5323],{},"還原","：解碼器將預測出的連續向量無損還原為 K 個離散 Token。",[108,5326,5328],{"id":5327},"五實驗結果與性能優勢","五、實驗結果與性能優勢",[272,5330,5331,5337,5343,5349],{},[275,5332,5333,5336],{},[138,5334,5335],{},"極致的算力節省","：在 K=4 的設置下，訓練算力降低 44%，推理算力降低 34%，生成步數從 T 驟降至 T\u002F4。",[275,5338,5339,5342],{},[138,5340,5341],{},"打破「更大必定更耗算力」的認知","：CALM 參數達 3.71 億（略大於對照組），卻能在打平甚至超越基線性能的情況下，消耗更少的整體算力。",[275,5344,5345,5348],{},[138,5346,5347],{},"全新縮放維度「資訊密度」","：引入了全新的 Scaling 維度。當語義頻寬提高時，CALM 能以更低的算力達到更高的 Brier 分數，擊穿了原有的算力-性能邊界。",[275,5350,5351,5354],{},[138,5352,5353],{},"「後積薄發」的訓練動態","：呈現特殊的訓練曲線。模型在 20 萬步後性能會迎來急劇上升並超越傳統模型，顯示掌握高維連續規律需要前期積累，但後續爆發力極強。",[108,5356,5358],{"id":5357},"六總結與展望","六、總結與展望",[112,5360,5361,5362,5365,5366,5369,5370,5373,5374,5377,5378,5381],{},"CALM 成功構建了從離散到連續的完整基礎設施，其組件協同形成了一個強大的生態閉環：\n",[138,5363,5364],{},"魯棒自編碼器"," (高保真壓縮) → ",[138,5367,5368],{},"頻寬倍增"," (步數縮短至 1\u002FK) → ",[138,5371,5372],{},"單步生成引擎"," (極速推理) → ",[138,5375,5376],{},"拒絕採樣"," (多樣性可調) → ",[138,5379,5380],{},"BrierLM"," (公正尺標)。",[112,5383,5384,5385,5388,5389,5392,5393,5396],{},"這項研究證明了語言模型的未來可能不在離散的像素與詞表中，而是在連續的波瀾裡。這為未來的",[138,5386,5387],{},"多模態連續生成","、",[138,5390,5391],{},"強化學習整合","以及",[138,5394,5395],{},"終端設備部署","開啟了全新的紀元。",[5180,5398,5399],{},[112,5400,5401,5404],{},[138,5402,5403],{},"開放資源","：相關論文與程式碼已在 GitHub 開源，標誌著大語言模型從離散向連續遷徙的開端。",{"title":311,"searchDepth":312,"depth":312,"links":5406},[5407,5408,5411,5412,5413,5414],{"id":5074,"depth":312,"text":5075},{"id":5104,"depth":312,"text":5105,"children":5409},[5410],{"id":5121,"depth":4984,"text":5122},{"id":5190,"depth":312,"text":5191},{"id":5288,"depth":312,"text":5289},{"id":5327,"depth":312,"text":5328},{"id":5357,"depth":312,"text":5358},"\u002Fimages\u002Fblog\u002F2510.27688v1.webp","2026-08-06","傳統大模型逐字（Token-by-token）生成的效率已觸及物理天花板。CALM 將離散 Token 轉換為連續向量（Continuous Vectors），將生成步數縮減至原來的 1\u002FK，為大語言模型的擴充開闢了全新的效率路徑。",{},"\u002Fblog\u002Fzh-tw\u002Farxiv-2510-27688v1",{"title":5039,"description":5417},"arxiv-2510-27688v1","blog\u002Fzh-TW\u002Farxiv-2510-27688v1",[5424,5425,5426],"paper","llm","efficiency","YXovptigvOJLPbjjtVgdCae3S6_ZcSZxJw3mH-BUBnc",{"id":5429,"title":5430,"author":103,"body":5431,"cover":5683,"coverAlt":5430,"date":5684,"description":5685,"draft":324,"extension":325,"featured":324,"lang":327,"meta":5686,"navigation":326,"path":5687,"seo":5688,"slug":5689,"stem":5690,"tags":5691,"translationKey":5689,"updated":337,"__hash__":5694},"blog\u002Fblog\u002Fzh-TW\u002Fheadroom-context-compression.md","大語言模型上下文壓縮技術：Headroom 底層邏輯與架構原理分析",{"type":105,"value":5432,"toc":5671},[5433,5437,5440,5445,5453,5462,5465,5472,5478,5481,5484,5488,5491,5526,5530,5533,5605,5609,5612,5630,5634,5652,5656],[5119,5434,5436],{"id":5435},"headroom-context-optimization-for-llm-agents","Headroom — Context Optimization for LLM Agents",[108,5438,5439],{"id":5439},"開發團隊",[112,5441,5442],{},[138,5443,5444],{},"Headroom Labs 與開源社群",[5180,5446,5447],{},[112,5448,5449],{},[5450,5451,5452],"em",{},"專案授權: 遵循 Apache-2.0 開源授權",[5454,5455,5457,5458],"h4",{"id":5456},"專案位置httpsgithubcomheadroomlabs-aiheadroom","專案位置：",[5055,5459,5460],{"href":5460,"rel":5461},"https:\u002F\u002Fgithub.com\u002Fheadroomlabs-ai\u002Fheadroom",[5059],[108,5463,5464],{"id":5464},"摘要",[112,5466,5467,5468,5471],{},"在人工智慧代理（AI Agents）執行檢索增強生成（RAG）、資料庫查詢或讀取日誌時，其獲取的工具輸出往往包含高達 70% 至 95% 的冗餘雜訊與樣板文件。這不僅導致推論成本高昂，更會引發模型「迷失在中間（Lost in the Middle）」效應，降低推理準確度。本文解析了開源上下文最佳化與壓縮層 ",[138,5469,5470],{},"Headroom"," 的底層邏輯。此框架引入了由 Rust 與 Python 混合編寫的三階段管線架構，並結合最具突破性的「可逆壓縮機制（CCR）」，能在不改變模型最終回答準確度的前提下，將傳輸至大語言模型（LLM）的 Token 數量縮減 60% 至 95%。",[112,5473,5474,5477],{},[138,5475,5476],{},"關鍵字","：大語言模型 (LLM)、上下文壓縮、AI 代理 (AI Agents)、可逆壓縮 (CCR)、抽象語法樹 (AST)",[5479,5480],"hr",{},[108,5482,5483],{"id":5483},"核心貢獻與重點",[5119,5485,5487],{"id":5486},"_1-三階段自動化壓縮管線","1. 三階段自動化壓縮管線",[112,5489,5490],{},"Headroom 建構了一條全自動化、具備內容感知能力的資料處理管線，無需開發者手動配置。",[272,5492,5493,5498,5501,5504,5509,5512,5515,5520,5523],{},[275,5494,5495],{},[138,5496,5497],{},"階段一：CacheAligner (快取對齊)",[275,5499,5500],{},"將系統提示詞中的動態變數（如時間戳記、Session ID）抽離至訊息尾端。",[275,5502,5503],{},"確保前綴在位元組層級（Byte-identical）保持絕對一致，以最大化命中模型提供商（如 Anthropic、OpenAI）的提示詞快取。",[275,5505,5506],{},[138,5507,5508],{},"階段二：ContentRouter (內容路由)",[275,5510,5511],{},"整合機器學習檢測器（如 Magika），自動判別傳入的異質資料結構。",[275,5513,5514],{},"將資料精準路由至最佳的專屬壓縮引擎。",[275,5516,5517],{},[138,5518,5519],{},"階段三：IntelligentContext (智慧預算管理)",[275,5521,5522],{},"透過六個核心維度（包含時間近期性、語意相似度等）計算歷史訊息的保留價值。",[275,5524,5525],{},"避免傳統滑動窗口直接丟棄記憶的缺點，將低分訊息轉存本地快取並替換為微型檢索標記。",[5119,5527,5529],{"id":5528},"_2-專屬內容壓縮引擎","2. 專屬內容壓縮引擎",[112,5531,5532],{},"針對不同資料結構，Headroom 提供深度定製的壓縮演算法，兼顧資訊保留率與效能：",[4848,5534,5535,5551],{},[4851,5536,5537],{},[4854,5538,5539,5542,5545,5548],{},[4857,5540,5541],{},"引擎名稱",[4857,5543,5544],{},"處理內容",[4857,5546,5547],{},"底層技術與策略",[4857,5549,5550],{},"Token 節省率",[4864,5552,5553,5569,5589],{},[4854,5554,5555,5560,5563,5566],{},[4869,5556,5557],{},[138,5558,5559],{},"SmartCrusher",[4869,5561,5562],{},"JSON 陣列",[4869,5564,5565],{},"基於雙連詞覆蓋率計算與 Kneedle 演算法尋找最佳截斷點，並強制保留錯誤狀態與統計異常值。",[4869,5567,5568],{},"70% – 90%",[4854,5570,5571,5576,5579,5586],{},[4869,5572,5573],{},[138,5574,5575],{},"CodeAwareCompressor",[4869,5577,5578],{},"原始碼",[4869,5580,5581,5582,5585],{},"透過 ",[118,5583,5584],{},"tree-sitter"," 解析抽象語法樹 (AST)，強制保留函式簽章與模組匯入，僅壓縮內部實作細節。",[4869,5587,5588],{},"40% – 70%",[4854,5590,5591,5596,5599,5602],{},[4869,5592,5593],{},[138,5594,5595],{},"LogCompressor",[4869,5597,5598],{},"系統日誌",[4869,5600,5601],{},"保留錯誤堆疊與警告，透過模式匹配消除大量重複的成功輸出。",[4869,5603,5604],{},"85% – 95%",[5119,5606,5608],{"id":5607},"_3-核心創新可逆壓縮架構-ccr","3. 核心創新：可逆壓縮架構 (CCR)",[112,5610,5611],{},"打破傳統降維帶來的資料遺失風險，實現 100% 資訊無損性（Lossless）。",[272,5613,5614,5620],{},[275,5615,5616,5619],{},[138,5617,5618],{},"本地儲存與標記："," 原始未壓縮資料會存入本地 LRU 快取，並生成雜湊鍵嵌入給 LLM 的壓縮文本中。",[275,5621,5622,5625,5626,5629],{},[138,5623,5624],{},"工具注入與攔截："," 系統動態向 LLM 注入 ",[118,5627,5628],{},"headroom_retrieve"," 虛擬工具。當 LLM 需要細節時呼叫此工具，代理伺服器會在 1 毫秒內於本地攔截並直接回傳原始資料。",[5119,5631,5633],{"id":5632},"_4-輸出端縮減與學習機制","4. 輸出端縮減與學習機制",[272,5635,5636,5642],{},[275,5637,5638,5641],{},[138,5639,5640],{},"事前干預 (Output Shaper)："," 在提示詞尾端動態附加精簡指令（冗長度引導），並根據任務難度動態調整模型的推論力度（Effort Routing），消除無意義的重複輸出。",[275,5643,5644,5651],{},[138,5645,5646,5647,5650],{},"離線失敗學習 (",[118,5648,5649],{},"headroom learn",")："," 自動探勘歷史紀錄，識別代理的無效迴圈，並生成防範規則寫入本地設定檔，避免未來重蹈覆轍。",[5119,5653,5655],{"id":5654},"_5-系統效能與應用邊界","5. 系統效能與應用邊界",[112,5657,5658,5659,5662,5663,5666,5667,5670],{},"✅ ",[138,5660,5661],{},"極低延遲","：底層 Rust 核心處理極快，整體管線中位數開銷僅約 52 ms。\n✅ ",[138,5664,5665],{},"高度彈性","：提供 Library、HTTP Proxy、Agent Wrap 及 MCP 伺服器等多種無縫部署方式。\n⚠️ ",[138,5668,5669],{},"適用場景限制","：最適合處理龐大日誌、大型原始碼庫的 AI 代理長期工作階段；對於單一輪次短對話或純粹的程式碼生成任務，壓縮效益較低。",{"title":311,"searchDepth":312,"depth":312,"links":5672},[5673,5674,5675,5676],{"id":5435,"depth":4984,"text":5436},{"id":5439,"depth":312,"text":5439},{"id":5464,"depth":312,"text":5464},{"id":5483,"depth":312,"text":5483,"children":5677},[5678,5679,5680,5681,5682],{"id":5486,"depth":4984,"text":5487},{"id":5528,"depth":4984,"text":5529},{"id":5607,"depth":4984,"text":5608},{"id":5632,"depth":4984,"text":5633},{"id":5654,"depth":4984,"text":5655},"\u002Fimages\u002Fblog\u002Fheadroom-architecture.webp","2026-07-14","專為 LLM 與 AI 代理設計的開源上下文壓縮層，透過三階段管線與可逆壓縮機制（CCR），在維持推論準確度的前提下縮減 60%-95% 的 Token 消耗",{},"\u002Fblog\u002Fzh-tw\u002Fheadroom-context-compression",{"title":5430,"description":5685},"headroom-context-compression","blog\u002Fzh-TW\u002Fheadroom-context-compression",[5425,5692,5693],"context","架構","d0vI9hYJDRBRd5wzvqIKa77efCPwJEXn5Tp7y3D73KM",{"id":5696,"title":5697,"author":103,"body":5698,"cover":6392,"coverAlt":5697,"date":6393,"description":6394,"draft":324,"extension":325,"featured":324,"lang":327,"meta":6395,"navigation":326,"path":6396,"seo":6397,"slug":6398,"stem":6399,"tags":6400,"translationKey":6398,"updated":337,"__hash__":6403},"blog\u002Fblog\u002Fzh-TW\u002Farxiv-2604-04921v1.md","TriAttention：程式碼與推理智能體之三角函數自適應上下文修剪機制與理論深度分析報告",{"type":105,"value":5699,"toc":6368},[5700,5705,5711,5716,5719,5722,5725,5730,5733,5739,5742,5748,5751,5754,5757,5763,5766,5772,5775,5778,5783,5786,5797,5814,5819,5827,5837,5859,5862,5873,5885,5904,5909,5912,5918,5921,6047,6050,6053,6073,6079,6082,6085,6099,6105,6108,6122,6127,6130,6136,6139,6142,6145,6151,6154,6157,6171,6174,6179,6182,6189,6192,6197,6200,6203,6206,6212],[108,5701,5703],{"id":5702},"參考文獻與專案資源",[138,5704,5702],{},[112,5706,5707],{},[5055,5708,5709],{"href":5709,"rel":5710},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2604.04921",[5059],[108,5712,5714],{"id":5713},"執行摘要與前瞻視野",[138,5715,5713],{},[112,5717,5718],{},"隨著人工智慧在軟體工程與複雜邏輯運算領域的快速演進，大語言模型（Large Language Models, LLMs）的應用範疇已經從早期的單純文本補全（Code Completion），跨越至具備自主決策與長鏈條推理能力的代理系統（Autonomous Agents）。在此發展軌跡中，無論是處理數學競賽等級的複雜推導，還是執行大規模程式碼庫的代理操作，系統皆無可避免地撞上了一道物理與認知的邊界——「上下文記憶體牆」（Context Memory Wall）1。這道牆的本質在於，當模型進行長序列生成時，必須在圖形處理器（GPU）的視訊記憶體（VRAM）中持續維護龐大的鍵值快取（Key-Value Cache, KV Cache），導致記憶體消耗隨著生成長度的增加而呈現線性甚至超線性增長，最終引發系統記憶體耗盡（Out-of-Memory, OOM）的災難性中斷 2。",[112,5720,5721],{},"為了解決這項阻礙長篇推理落地的系統性瓶頸，由麻省理工學院（MIT）、NVIDIA 與浙江大學（Zhejiang University）所組成的聯合研究團隊，提出了一項名為 TriAttention 的革命性 KV 快取壓縮框架 4。不同於過去依賴啟發式規則或線上注意力分數觀察的傳統壓縮技術，TriAttention 首度將視角深入至 Transformer 注意力機制的底層幾何空間，揭露了旋轉位置編碼（Rotary Position Embedding, RoPE）應用前的「Pre-RoPE 空間」中，查詢（Query, Q）與鍵（Key, K）向量存在高度集中的分佈特性 2。研究團隊發現，這種物理上的集中現象，允許我們透過純粹的數學演繹——具體而言是三角函數級數（Trigonometric Series）——來預測上下文標記（Tokens）在未來生成過程中的相對重要性，從而達成極致精準的自適應上下文修剪 5。",[112,5723,5724],{},"實證數據顯示，TriAttention 架構在保留關鍵推理狀態的能力上達到了前所未有的高度。在極具挑戰性的數學推理基準（如 AIME25）中，該技術不僅能在維持與全注意力機制（Full Attention）完全一致的推理準確率下運作，更實現了高達 10.7 倍的 KV 記憶體縮減與 2.5 倍的整體吞吐量（Throughput）提升 2。更具深遠影響的是其對邊緣計算（Edge Computing）與消費級硬體生態的重塑。藉由與 OpenClaw 網關等框架的深度整合，TriAttention 成功打破了高階運算硬體的物理藩籬，使得擁有高達 32B 參數級別的複雜推理模型，得以流暢且無損地部署於僅具備 24GB 顯示記憶體的單張消費級 NVIDIA RTX 4090 顯示卡上 2。結合 Apple Silicon（MLX 架構）與 llama.cpp 等開源生態的廣泛支援與效能疊加，本報告將深入剖析 TriAttention 的理論基礎、數學機制與實證表現，並勾勒其在未來程式碼與推理智能體架構中的關鍵基礎設施地位。",[108,5726,5728],{"id":5727},"理論背景與上下文危機的實證分析",[138,5729,5727],{},[112,5731,5732],{},"要深刻理解 TriAttention 所帶來的技術突破，我們必須先解構當前大語言模型在長上下文推理市場所面臨的系統性缺陷，以及為何現有主流的壓縮演算法會在深度的邏輯推演與遞迴任務中頻繁面臨結構性的崩潰。",[5119,5734,5736],{"id":5735},"認知負載與-kv-快取的成本分析",[138,5737,5738],{},"認知負載與 KV 快取的成本分析",[112,5740,5741],{},"當代大語言模型的核心運算機制基於 Transformer 架構中的自注意力機制（Self-Attention Mechanism）。在生成式推理（Autoregressive Generation）的過程中，模型必須計算當下生成的查詢向量與過去所有歷史標記的鍵向量之間的內積，以衡量上下文的語意關聯性 6。為了避免針對歷史標記進行重複的矩陣乘法運算，系統架構會將歷史的鍵向量與值向量（Value Vectors）暫存於高速的圖形記憶體中。這種以空間換取時間的策略雖然有效降低了計算延遲，卻也將系統推向了記憶體容量的極限邊緣。在數學推理、程式碼重構或代理化環境探索等任務中，智能體往往需要讀取並生成長達數萬個標記的脈絡（例如 32K Tokens 的生成長度），這使得 KV 快取的實體記憶體佔用量急遽膨脹，甚至反客為主，超越了模型權重本身的儲存需求 3。這種高昂的記憶體成本不僅大幅限制了批次處理（Batch Processing）的規模，更使得長序列推理無法在邊緣設備上普及。",[5119,5743,5745],{"id":5744},"傳統-post-rope-壓縮機制的結構性失效",[138,5746,5747],{},"傳統 Post-RoPE 壓縮機制的結構性失效",[112,5749,5750],{},"為應對上述的上下文危機，學術界與產業界近年來提出了諸多 KV 快取壓縮方法。然而，過往的主流技術路徑幾乎全數落入「Post-RoPE 壓縮方法」（Post-RoPE Compression Methods）的範疇。這些方法的共同核心邏輯，是透過觀察模型近期生成的查詢向量所產生的注意力分數（Attention Scores），來推斷哪些歷史標記具有較高的重要性，並將低分標記從快取中剔除。",[112,5752,5753],{},"具體而言，諸如 SnapKV 演算法透過在一個局部的觀察視窗內計算注意力分佈，並聚合這些分數以預測未來生成時可能需要的「重度命中者」（Heavy Hitters）2；Scissorhands 則基於「重要性持久」（Persistence of Importance）的假說，利用歷史累積的注意力分佈來引導快取的剔除決策 2；而近期針對推理模型優化的 R-KV 演算法，亦是結合最近期查詢的注意力分數與冗餘狀態檢測機制來為快取標記進行評分 2；此外，LazyEviction 透過追蹤觀察視窗內標記重要性的重現頻率，採取延遲剔除的保守策略 2。另外如 Ada-KV 等方法，雖引入了自適應預算分配的概念，允許不同注意力頭擁有不同的快取大小，但其底層判斷基準依然未脫離線上注意力分數的框架 13。",[112,5755,5756],{},"儘管這些方法在短文本或一般摘要任務中展現了成效，但在面對長鏈條推理時卻暴露出致命的缺陷。其根本原因在於旋轉位置編碼（Rotary Position Embedding, RoPE）對向量空間所造成的幾何擾動 2。RoPE 的數學本質是在複數平面上，依據標記的絕對位置對向量進行頻率遞增的旋轉。這意味著，當一個查詢向量隨著生成過程在序列中不斷向後移動時，其在高維空間中的指向會持續發生劇烈的旋轉變化 2。因此，在套用 RoPE 之後的空間（即 Post-RoPE 空間）中，近期的查詢向量完全無法穩定地代表未來的查詢向量。當系統依賴這些極度不穩定的近期查詢來評估全域重要性時，會面臨「具代表性的查詢數量極少」的統計困境，導致演算法頻繁選錯 Top-K 關鍵標記 2。在嚴密的邏輯推理中，提前剔除任何一個關鍵的「中間推理狀態」，都會切斷邏輯鏈條，進而在模型的後端輸出引發災難性的邏輯崩潰與幻覺。",[108,5758,5760],{"id":5759},"triattention-系統架構與數學機制深度剖析",[138,5761,5762],{},"TriAttention 系統架構與數學機制深度剖析",[112,5764,5765],{},"面對 Post-RoPE 方法在長篇推理上的系統性極限，TriAttention 團隊選擇了一條截然不同的理論路徑。該架構並非在現有的注意力評分機制上進行邊緣微調，而是回歸矩陣運算與幾何分佈的本質，進行了徹底的數學重構。",[5119,5767,5769],{"id":5768},"pre-rope-空間中的-qk-集中現象",[138,5770,5771],{},"Pre-RoPE 空間中的 Q\u002FK 集中現象",[112,5773,5774],{},"TriAttention 的核心突破源自於一個基礎但過去未被充分重視的觀察：研究團隊將檢視的視角從不斷動態旋轉的 Post-RoPE 空間，回溯至尚未套用位置編碼的初始狀態，亦即「Pre-RoPE 空間」2。在這個未經幾何扭曲的原始張量空間中，研究人員揭露了一個令人驚訝的模型內部屬性：在絕大多數的注意力頭（Attention Heads）之中，查詢向量（Q）與鍵向量（K）並非呈現隨機或均勻分佈，而是高度且緊密地集中在某些固定的非零中心點（Fixed Non-zero Centers）周圍 2。",[112,5776,5777],{},"這種物理上的向量集中現象具有極高的全域穩定性。它完全不受標記在序列中的絕對位置影響，也不因輸入上下文的具體語意內容或提示詞結構的改變而發生位移 9。換言之，這些中心點是模型權重在海量數據預訓練過程中，所內化形成的一種內在結構特徵（Intrinsic Property）。為精確量化這種集中程度的穩定性，研究引入了方向統計學（Directional Statistics）中的「平均合成向量長度」（Mean Resultant Length，通常記為 R）作為指標 5。數值 R 的範圍介於 0 與 1 之間，當 R 趨近於 1 時，代表該空間內的向量群聚程度極高、變異數極小。實證測量資料明確指出，在諸如 Qwen3-8B 等當代大型語言模型中，高達 90% 的注意力頭其 R 值皆超過了 0.95，這強而有力地證實了 Q\u002FK 集中現象的普遍存在與高度可靠性 5。",[5119,5779,5781],{"id":5780},"三角函數級數轉換與位置偏好預測",[138,5782,5780],{},[112,5784,5785],{},"當我們確認了 Q 與 K 向量在 Pre-RoPE 空間中高度集中於穩定中心點後，原本複雜且高度動態的注意力對數幾率（Attention Logit，即送入 Softmax 函數之前的原始關聯分數）計算，便迎來了大幅度化簡的數學契機 5。",[112,5787,5788,5789,5796],{},"由於 Pre-RoPE 的中心點是固定不變的，當我們在其上套用旋轉位置編碼（RoPE）時，其數學等價於對一個固定的基底向量進行已知頻率與角度的旋轉變換。這項特質使得未來的注意力模式不再需要依賴消耗算力的即時（Online）矩陣內積來計算，而是可以直接透過解析解方程式進行嚴謹的數學演繹預測。具體而言，該研究證明注意力 Logit 可以被精確地近似展開為一個僅取決於查詢與鍵之間相對位置距離（Positional Distance，記為 ",[5790,5791],"img",{"className":5792,"src":5794,"alt":5795},[5793],"inline-math","\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage1.png","公式","）的三角函數級數（Trigonometric Series）2。",[112,5798,5799,5800,5804,5805,184,5809,5813],{},"在上述公式中，頻率 ",[5790,5801],{"className":5802,"src":5803,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage2.png"," 繼承自 RoPE 的設計，而展開式中的係數 ",[5790,5806],{"className":5807,"src":5808,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage3.png",[5790,5810],{"className":5811,"src":5812,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage4.png"," 則完全由 Pre-RoPE 空間中固定的 Q\u002FK 中心點座標所決定。由於這些中心點在模型部署前即可視為靜態常數，研究團隊得以透過離線的校準數據（Calibration Data）預先計算並提取這些統計特徵 5。這種將線上運算負擔轉移至離線統計的策略，徹底讓系統擺脫了對近期即時查詢分數的依賴，從根本上解決了位置旋轉帶來的評估失準問題。",[5119,5815,5817],{"id":5816},"雙軌評分機制與自適應融合演算法",[138,5818,5816],{},[112,5820,5821,5822,5826],{},"在實際執行 KV 快取剔除操作時，TriAttention 運用了上述的理論基礎，構建了一套無需觀察即時查詢的預測型評分演算法。當模型的生成長度觸及預先設定的 KV 快取預算上限時，系統會為每一個快取中的鍵向量 ",[5790,5823],{"className":5824,"src":5825,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage5.png"," 賦予一個綜合的重要性分數。為確保在不同特性的注意力頭中皆能保持高精度，該機制包含了兩大核心運算模組：",[5454,5828,5830],{"id":5829},"_1-三角函數級數分數-trigonometric-series-score",[138,5831,5832,5833,793],{},"1. 三角函數級數分數 (Trigonometric Series Score, ",[5790,5834],{"className":5835,"src":5836,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage6.png",[112,5838,5839,5840,5843,5844,5848,5849,5853,5854,5858],{},"針對那些集中度極高的注意力頭，",[5790,5841],{"className":5842,"src":5836,"alt":5795},[5793]," 模組扮演了主要的篩選角色。它利用離線預先計算好的 Q 向量中心（",[5790,5845],{"className":5846,"src":5847,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage7.png","）以及實際寫入快取的鍵向量特徵（",[5790,5850],{"className":5851,"src":5852,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage8.png","），來估計該鍵向量在未來遭遇位於不同距離 ",[5790,5855],{"className":5856,"src":5857,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage9.png"," 的查詢時，將會獲得多少注意力關注 5。由於一個被保留的鍵向量在未來可能會被位於各種不同偏移位置的查詢所關注，為了確保評估的全域穩健性，TriAttention 會針對一組透過幾何間距（Geometric Spacing）分佈的未來偏移量集合（Future Offsets）進行加權平均計算 5。其評估函數可數學表示為：",[112,5860,5861],{},"此計算式深刻地利用了特定頻段（Frequency Bands）與相對距離之間的餘弦映射關係，賦予模型強大且可預測的位置偏好（Distance Preferences）分析能力，例如模型如何優先關注鄰近的標記或特定週期的結構化標記 2。",[5454,5863,5865],{"id":5864},"_2-基於範數的調節分數-norm-based-score-與自適應權重",[138,5866,5867,5868,5872],{},"2. 基於範數的調節分數 (Norm-Based Score, ",[5790,5869],{"className":5870,"src":5871,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage10.png",") 與自適應權重",[112,5874,5875,5876,5880,5881,5884],{},"儘管絕大多數的注意力頭表現出高度的集中，但仍有少數注意力頭的 ",[5790,5877],{"className":5878,"src":5879,"alt":5795},[5793],"\u002Fimages\u002Fblog\u002Farxiv-2604-04921v1\u002Fimage11.png"," 值較低，呈現出較為發散的分佈。在這些發散的空間中，單純依賴幾何中心點進行三角函數預測會引入顯著的估計誤差。為了彌補這項短板，TriAttention 引入了 ",[5790,5882],{"className":5883,"src":5871,"alt":5795},[5793]," 模組。該模組不依賴嚴格的方向性中心，而是藉由預期的查詢範數（Expected Query Norm）大小，針對不同的頻率帶進行加權，以此作為評估鍵向量重要性的額外輔助訊號 3。",[112,5886,5887,5888,5891,5892,5895,5896,5899,5900,5903],{},"在整合階段，系統巧妙地利用了先前計算的平均合成向量長度 ",[5790,5889],{"className":5890,"src":5879,"alt":5795},[5793]," 作為自適應權重（Adaptive Weight）。對於 ",[5790,5893],{"className":5894,"src":5879,"alt":5795},[5793]," 接近 1 的注意力頭，系統會賦予 ",[5790,5897],{"className":5898,"src":5836,"alt":5795},[5793]," 極高的權重；反之，對於較為發散的注意力頭，系統則自動增加 ",[5790,5901],{"className":5902,"src":5871,"alt":5795},[5793]," 的影響力。這種動態融合機制確保了模型在所有的網路深度與不同的特徵提取通道中，皆能維持最優的快取壓縮精度 3。",[108,5905,5907],{"id":5906},"實證分析與推理基準性能表現",[138,5908,5906],{},[112,5910,5911],{},"理論架構的優越性最終必須透過嚴苛的經驗數據來檢驗。TriAttention 在多項涵蓋複雜數學推導、深度遞迴模擬以及長上下文泛用檢索的基準測試中，皆展現了壓倒性的效能優勢，不僅大幅超越了現有的 Post-RoPE 壓縮技術，更在極端壓縮比之下逼近了無損（Lossless）的全注意力機制表現。",[5119,5913,5915],{"id":5914},"數學推理與吞吐量極限測試-aime24-aime25-math-500",[138,5916,5917],{},"數學推理與吞吐量極限測試 (AIME24, AIME25, MATH-500)",[112,5919,5920],{},"數學推理任務被學界視為檢驗 KV 快取壓縮演算法保留關鍵資訊能力的「終極試金石」。這是因為數學解題過程中的邏輯狀態轉移是高度嚴密的，任何一個前置條件或中間計算結果的錯誤剔除，都會導致後續推導的全面潰敗。實驗數據顯示，TriAttention 成功在極端的記憶體壓縮率與推論準確率之間取得了前所未見的完美平衡。",[4848,5922,5923,5952],{},[4851,5924,5925],{},[4854,5926,5927,5931,5934,5937,5940,5943,5946,5949],{},[4857,5928,5930],{"align":5929},"left","基準測試項目 (Benchmark)",[4857,5932,5933],{"align":5929},"總上下文限制",[4857,5935,5936],{"align":5929},"TriAttention KV 預算",[4857,5938,5939],{"align":5929},"全注意力準確率 (Full Acc)",[4857,5941,5942],{"align":5929},"TriAttention 準確率",[4857,5944,5945],{"align":5929},"全注意力吞吐量",[4857,5947,5948],{"align":5929},"TriAttention 吞吐量",[4857,5950,5951],{"align":5929},"速度提升比",[4864,5953,5954,5986,6017],{},[4854,5955,5956,5961,5964,5967,5970,5975,5978,5981],{},[4869,5957,5958],{"align":5929},[138,5959,5960],{},"MATH-500",[4869,5962,5963],{"align":5929},"32,768 Tokens",[4869,5965,5966],{"align":5929},"1,024 Tokens",[4869,5968,5969],{"align":5929},"69.6%",[4869,5971,5972],{"align":5929},[138,5973,5974],{},"68.4%",[4869,5976,5977],{"align":5929},"222.8 tok\u002Fs",[4869,5979,5980],{"align":5929},"1,405.2 tok\u002Fs",[4869,5982,5983],{"align":5929},[138,5984,5985],{},"6.3x",[4854,5987,5988,5993,5996,5999,6002,6007,6009,6012],{},[4869,5989,5990],{"align":5929},[138,5991,5992],{},"AIME24",[4869,5994,5995],{"align":5929},"-",[4869,5997,5998],{"align":5929},"4,096 Tokens",[4869,6000,6001],{"align":5929},"57.1%",[4869,6003,6004],{"align":5929},[138,6005,6006],{},"54.6%",[4869,6008,5977],{"align":5929},[4869,6010,6011],{"align":5929},"413.9 tok\u002Fs",[4869,6013,6014],{"align":5929},[138,6015,6016],{},"1.9x",[4854,6018,6019,6024,6027,6030,6033,6037,6039,6042],{},[4869,6020,6021],{"align":5929},[138,6022,6023],{},"AIME25",[4869,6025,6026],{"align":5929},"32,000+ Tokens",[4869,6028,6029],{"align":5929},"3,072 Tokens",[4869,6031,6032],{"align":5929},"40.8%",[4869,6034,6035],{"align":5929},[138,6036,6032],{},[4869,6038,5977],{"align":5929},[4869,6040,6041],{"align":5929},"563.5 tok\u002Fs",[4869,6043,6044],{"align":5929},[138,6045,6046],{},"2.5x",[112,6048,6049],{},"數據來源彙整自 TriAttention 官方基準測試報告與開源社群分析 5",[112,6051,6052],{},"深入分析上述效能矩陣，我們可以提煉出幾項關鍵的洞察：",[5193,6054,6055,6061,6067],{},[275,6056,6057,6060],{},[138,6058,6059],{},"極端預算環境下的高認知韌性："," 在 MATH-500 測試中，系統僅被賦予了極度苛刻的 1,024 個 Tokens 的快取空間（佔最大可用上下文空間 32,768 Tokens 的約 3.1%），但 TriAttention 卻能維持高達 68.4% 的推理準確率，與未經任何壓縮的全注意力機制（69.6%）僅有微乎其微的差距，並同時創造了 6.3 倍的驚人吞吐量加速表現 5。",[275,6062,6063,6066],{},[138,6064,6065],{},"長鏈條無損壓縮的實踐："," 在難度極高且生成長度可達 32K 的 AIME25 測試環境中，TriAttention 憑藉僅 3,072 Tokens 的快取預算，完美追平了全注意力機制的 40.8% 準確率。這意味著系統達成了高達 10.7 倍的 KV 記憶體實質縮減，並實現了 2.5 倍的速度躍升 2。相較之下，在同等的壓縮效率要求下，現行領先的其他基準壓縮方法其準確率通常僅剩下一半 2。",[275,6068,6069,6072],{},[138,6070,6071],{},"對比同儕演算法的壓倒性勝出："," 在 AIME24 的比較基準中，當使用 Qwen3-8B 模型並設定 2,048 Tokens 預算限制時，TriAttention 取得了 42.1% 的準確率。而同為針對推理優化的競爭演算法 R-KV 僅能勉強達到 25.4%，兩者之間存在高達 16.7 個百分點的巨大效能鴻溝 5。",[5119,6074,6076],{"id":6075},"結構性記憶保留測試recursive-state-query-benchmark",[138,6077,6078],{},"結構性記憶保留測試：Recursive State Query Benchmark",[112,6080,6081],{},"為了更精確地診斷並量化各式壓縮演算法在處理長鏈條邏輯運算時的「狀態遺忘」問題，研究團隊創新地設計並引入了「遞迴狀態查詢」（Recursive State Query）基準測試 5。這項測試的設計邏輯建構於深度優先搜尋（Depth-First Search, DFS）的遞迴探索模擬之上，強制要求大語言模型在生成冗長序列的同時，必須精確維護並在後續步驟中回溯至極早期的中間分岔節點狀態 2。",[112,6083,6084],{},"在 DFS 遞迴的脈絡中，任何一個父節點狀態的錯誤剔除，都會使得模型在需要回溯（Backtracking）時失去路徑導航，進而導致後續搜索空間的整體崩潰。這項嚴苛的測試徹底揭示了傳統 Post-RoPE 方法的致命盲點：",[272,6086,6087,6093],{},[275,6088,6089,6092],{},[138,6090,6091],{},"R-KV 等方法的災難性衰退："," 在遞迴深度（Depth）較淺的情況下，R-KV 尚能應付；當深度達到 14 時，其仍能維持約 61% 的準確率。然而，當遞迴深度進一步被推升至 16，跨越了其依賴近期查詢的保留閾值時，準確率隨即呈現懸崖式的暴跌，僅存 31% 5。這清楚且無情地證明了，基於近期注意力分數的機制，會因為在生成深層子節點的過程中較少關注到父節點，而無意間將這些至關重要的歷史遞迴狀態標記為「低重要性」並予以捨棄。",[275,6094,6095,6098],{},[138,6096,6097],{},"TriAttention 的拓撲結構完整性："," 即使在施加中度至重度的記憶體限制壓力下，TriAttention 於深度達 16 的複雜遞迴環境中，依然展現出與全注意力機制高度匹敵的穩定記憶表現 5。這強烈印證了其基於三角函數的先驗預測模型，能夠在不依賴線上查詢的情況下，有效保護並鎖定跨越漫長時序的關鍵拓撲與邏輯結構 15。",[5119,6100,6102],{"id":6101},"泛用語言理解與高維度檢索任務-longbench-ruler",[138,6103,6104],{},"泛用語言理解與高維度檢索任務 (LongBench & RULER)",[112,6106,6107],{},"為了證明 TriAttention 底層的 Q\u002FK 集中現象不僅是針對數學解題的特化現象，研究團隊亦將其應用於通用自然語言處理與海量資訊檢索任務中。",[272,6109,6110,6116],{},[275,6111,6112,6115],{},[138,6113,6114],{},"LongBench 綜合測試："," 在涵蓋問答系統、文本摘要、少樣本分類、檢索增強與程式碼分析等 16 個子任務的 LongBench 綜合評測中，於 Qwen3-8B 模型施加 50% KV 預算限制的條件下，TriAttention 取得了全場最高的 48.1 平均分。其在 16 個子任務中強勢贏得 11 項第一，並領先由 Ada-KV 與 SnapKV 組合的最佳複合基線達 2.5 分 5。",[275,6117,6118,6121],{},[138,6119,6120],{},"RULER 檢索基準："," 在專注於長篇幅大海撈針（Needle-in-a-Haystack）的 RULER 檢索測試中，於 4K 上下文長度下，TriAttention 斬獲了 66.1 分的高分，與主要競爭對手 SnapKV 之間拉開了 10.5 分的驚人差距 5。這些數據無可辯駁地證實了，無需依賴即時查詢的頻域壓縮策略，在處理高雜訊環境下的資訊檢索時具有無可比擬的優勢 5。",[108,6123,6125],{"id":6124},"邊緣硬體部署與開源生態系擴張情報",[138,6126,6124],{},[112,6128,6129],{},"TriAttention 的理論突破並非僅停留在學術殿堂的數據報表上，其核心價值更在於迅速且猛烈地衝擊了開源社群，並徹底重塑了消費級硬體與邊緣運算設備部署大型模型的實體邊界。以下為本報告深入彙整當前開發者社群在各式硬體平台上的最新實踐情報。",[5119,6131,6133],{"id":6132},"openclaw-網關整合與單卡-32b-模型的消費級革命",[138,6134,6135],{},"OpenClaw 網關整合與單卡 32B 模型的消費級革命",[112,6137,6138],{},"回顧過往的系統架構實踐，當模型的生成上下文長度逐漸逼近 GPU 的物理記憶體極限時，推理引擎必須被迫啟動卸載（Offloading）機制，將部分 KV 狀態移轉至系統主記憶體（System RAM）中。然而，由於 PCIe 匯流排的頻寬與延遲限制，即便系統配備了最頂級的 DDR5 記憶體，這種頻繁跨越硬體邊界的資料搬運依舊會引發極度嚴重的延遲突波，這被開發者社群戲稱為「卸載地獄」（Offload Hell）11。",[112,6140,6141],{},"透過與專注於本地安全運行時的 OpenClaw 網關框架進行深度且原生的整合 2，TriAttention 徹底斬斷了對 PCIe 頻寬的依賴。高達 10.7 倍的壓縮率使得所有的 KV 狀態得以被牢牢鎖定在 GPU 的超高速 VRAM 內部 11。來自社群的極限基準測試報告指出，藉由這項技術，包含 Gemma 4 31B（或同等級 32B 參數規模）在內的龐大推理模型，如今已能毫無卡頓地順暢運行於僅擁有 24GB 顯示記憶體的消費級 NVIDIA RTX 4090 顯示卡上 5。",[112,6143,6144],{},"在社群所執行的複雜 SQL 生成測試場景中（涵蓋針對混亂資料庫結構所進行的嵌套 JOIN 語法模擬），搭配輕量級 E2B 草稿模型進行投機解碼（Speculative Decoding）時，包含主模型權重、草稿模型以及壓縮後的 KV 快取，整體的 VRAM 佔用量被精準控制在 21.5GB 的安全範圍內 11。在如此緊湊的記憶體預算下，SQL 查詢語句的生成呈現近乎瞬時的超低延遲響應，展現出極具侵略性與精準度的指令遵循能力，且在長達 15K Tokens 的提示詞輸入下，亦未曾發生結構性的幻覺崩潰 11。這項成果標誌著 24GB 顯存的硬體「物理天花板」，已被純粹軟體演算法層面的顛覆性優化所大幅抬升 11。",[5119,6146,6148],{"id":6147},"apple-silicon-mlx-統一記憶體生態系的擴展",[138,6149,6150],{},"Apple Silicon (MLX) 統一記憶體生態系的擴展",[112,6152,6153],{},"為了解決在 MacBook 等行動設備上部署強大 AI 助理的迫切需求，開源社群的活躍開發者（@DeadByDawn101）以極快的速度為 TriAttention 貢獻了對 Apple Silicon 生態系的 MLX 框架支援（涵蓋 M1 至最新 M5 世代晶片）7。MLX 是專為 Apple 獨特的統一記憶體架構（Unified Memory）與神經加速器（Neural Accelerators）所量身打造的高效張量運算庫 19。",[112,6155,6156],{},"在配備 MLX 的最新 Mac 設備上執行的基準測試，進一步驗證了 TriAttention 在非 CUDA 邊緣運算環境下的卓越適應性：",[272,6158,6159,6165],{},[275,6160,6161,6164],{},[138,6162,6163],{},"高階專業級配置："," 搭載 M4 Max 晶片（具備 128GB 統一記憶體）的工作站，在執行 Gemma 4 E4B 模型並設定 4096 的寬裕 KV 預算下，可達到約高達 35 Tokens\u002Fs 的極速吞吐量 18。",[275,6166,6167,6170],{},[138,6168,6169],{},"中階消費級配置："," 搭載 M2 Pro 晶片（具備 32GB 統一記憶體）的筆記型電腦，在執行 Gemma 4 E2B 模型並將 KV 預算壓縮至 2048 Tokens 時，依然能維持約 28 Tokens\u002Fs 的流暢生成體驗 18。",[112,6172,6173],{},"這項跨平台的底層架構整合，不僅成功打破了長久以來由 CUDA 生態系所主導的壟斷局面，更為「本地優先」（Local-first）的邊緣 AI 代理應用，提供了堅實且具備高度隱私保護能力的基礎設施。",[5119,6175,6177],{"id":6176},"多元推理後端與正交量化技術的幾何級數疊加效應",[138,6178,6176],{},[112,6180,6181],{},"社群層面的另一項重大工程里程碑，是由開發者 @domvox 所主導完成的 C\u002Fggml 架構移植專案。該專案將 TriAttention 的底層邏輯成功導入了廣受歡迎的 llama.cpp 框架，這不僅解鎖了對 AMD GPU（基於 HIP\u002FROCm 運算架構）的全面支援 7，更促成了一項驚人的技術化學反應。",[112,6183,6184,6185,6188],{},"具體的情報指出，TriAttention 具備與低位元權重量化技術（例如 TurboQuant）進行",[138,6186,6187],{},"正交疊加","的獨特屬性。TurboQuant 致力於壓縮模型權重與快取元素的精度（例如將儲存格式降至 3-bit 或 4-bit），而 TriAttention 則專注於壓縮序列的物理長度。當這兩項技術在 llama.cpp 框架中被結合運用時，架構層面的自適應剔除加上資料型態層面的位元壓縮，實現了高達 30 倍以上的整體 KV 快取記憶體幾何級數縮減（例如 75% 序列修剪乘以 1.33x 至 5.12x 不等的精度提升）1。這意味著開發團隊現在能夠在極度貧乏的記憶體環境中，維持超大型模型深邃的脈絡感知與上下文連結能力。",[112,6190,6191],{},"此外，TriAttention 官方與社群聯合推動的開發藍圖（Roadmap）也正以極快的節奏持續推進。目前專案已確認整合進 SGLang 等高效能分散式推理後端 7，並有社群成員（@dscain）正著手進行針對 NVIDIA DGX Spark (GB10) 企業級伺服器架構的深度適配 7。在多模態應用方面，TriAttention 更成功跨足至擴增實境（AR）與長篇影片生成領域（透過與 LongLive 影片生成專案的結合），進一步驗證了該三角函數壓縮演算法在處理高維度、高密度時序視覺資料上的強大泛用能力與未來發展潛力 7。",[108,6193,6195],{"id":6194},"結論與基礎設施的演進趨勢",[138,6196,6194],{},[112,6198,6199],{},"總結上述之深度分析，TriAttention 的問世不僅僅是演算法效能榜單上的一次常規刷新，更深刻揭示了大語言模型底層架構研究思維的一次關鍵性轉折。傳統的壓縮思維往往陷入系統層面「硬體資源調度」的泥沼，或過度依賴統計層面高度不穩定的「後驗概率」（如計算生成過程中的即時注意力分數），而 TriAttention 則展現了極致的數學優雅，將解決問題的視角拉高並回歸至張量空間的「先驗幾何結構」——即 Pre-RoPE 空間中的 Q\u002FK 集中現象。",[112,6201,6202],{},"透過將難以捉摸、不斷旋轉的動態查詢，轉譯為具備高度規律性與預測性的嚴謹三角函數級數，該架構成功且徹底地將 KV 快取的生命週期管理，從耗費龐大算力的「線上觀測」模式，昇華為精準的「離線數學預測」。這種運算範式的轉移帶來了無可比擬的邊際效益：在諸如 AIME25 這種無情考驗機器認知極限的深層推理任務中，能夠達成 10.7 倍的實質記憶體縮減與 2.5 倍的吞吐量躍升，並且完全不伴隨任何準確率的折損，這無疑是對現有所有 Post-RoPE 演算法效能的一次降維打擊。",[112,6204,6205],{},"從工程實踐與產業落地的宏觀視角來評估，TriAttention 正在強勢重新定義本地端推理硬體的價值衡量基準。當一張要價相對低廉的 24GB RTX 4090 顯示卡，或是具備 32GB 統一記憶體的消費級 Apple 設備，都能夠在免除 PCIe 頻寬瓶頸且毫無延遲的狀態下，順暢乘載 32B 參數級別的複雜智能體代理任務時，意味著長上下文運算的特權將不再被超大規模的雲端 GPU 叢集所壟斷。隨著 MLX、llama.cpp、vLLM、SGLang 以及 OpenClaw 等活躍開源生態後端的全面無縫整合，以三角函數為核心數學引擎的預測型快取壓縮架構，預期將在短時間內成為推動下一代個人化邊緣 AI、自主化軟體工程智能體，以及分散式推理基礎設施的標準核心配備。",[5454,6207,6209],{"id":6208},"works-cited",[138,6210,6211],{},"Works cited",[5193,6213,6214,6221,6228,6234,6241,6248,6255,6262,6269,6276,6283,6290,6297,6305,6312,6319,6326,6333,6340,6347,6354,6361],{},[275,6215,6216,6217],{},"TriAttention: Efficient Long Reasoning with Trigonometric KV Compression - Hugging Face, accessed May 11, 2026, ",[5055,6218,6219],{"href":6219,"rel":6220},"https:\u002F\u002Fhuggingface.co\u002Fpapers\u002F2604.04921",[5059],[275,6222,6223,6224],{},"TriAttention: Efficient Long Reasoning with Trigonometric KV Compression - arXiv, accessed May 11, 2026, ",[5055,6225,6226],{"href":6226,"rel":6227},"https:\u002F\u002Farxiv.org\u002Fhtml\u002F2604.04921v1",[5059],[275,6229,6230,6231],{},"[2604.04921] TriAttention: Efficient Long Reasoning with Trigonometric KV Compression, accessed May 11, 2026, ",[5055,6232,5709],{"href":5709,"rel":6233},[5059],[275,6235,6236,6237],{},"[R] TriAttention: Efficient KV Cache Compression for Long-Context ..., accessed May 11, 2026, ",[5055,6238,6239],{"href":6239,"rel":6240},"https:\u002F\u002Fwww.reddit.com\u002Fr\u002FMachineLearning\u002Fcomments\u002F1serby2\u002Fr_triattention_efficient_kv_cache_compression_for\u002F",[5059],[275,6242,6243,6244],{},"Researchers from MIT, NVIDIA, and Zhejiang University Propose TriAttention: A KV Cache Compression Method That Matches Full Attention at 2.5× Higher Throughput - MarkTechPost, accessed May 11, 2026, ",[5055,6245,6246],{"href":6246,"rel":6247},"https:\u002F\u002Fwww.marktechpost.com\u002F2026\u002F04\u002F11\u002Fresearchers-from-mit-nvidia-and-zhejiang-university-propose-triattention-a-kv-cache-compression-method-that-matches-full-attention-at-2-5x-higher-throughput\u002F",[5059],[275,6249,6250,6251],{},"TriAttention: Efficient Long Reasoning with Trigonometric KV Compression | alphaXiv, accessed May 11, 2026, ",[5055,6252,6253],{"href":6253,"rel":6254},"https:\u002F\u002Fwww.alphaxiv.org\u002Foverview\u002F2604.04921",[5059],[275,6256,6257,6258],{},"TriAttention — Efficient long reasoning with trigonometric KV cache compression. Enables OpenClaw local deployment on memory-constrained GPUs. - GitHub, accessed May 11, 2026, ",[5055,6259,6260],{"href":6260,"rel":6261},"https:\u002F\u002Fgithub.com\u002FWeianMao\u002Ftriattention",[5059],[275,6263,6264,6265],{},"TriAttention: Efficient Long Reasoning with Trigonometric KV Compression - YouTube, accessed May 11, 2026, ",[5055,6266,6267],{"href":6267,"rel":6268},"https:\u002F\u002Fwww.youtube.com\u002Fwatch?v=BOog_w2N2as",[5059],[275,6270,6271,6272],{},"Researchers from MIT, NVIDIA, and Zhejiang University Propose TriAttention: A KV Cache Compression Method That Matches Full Attention at 2.5× Higher Throughput - Reddit, accessed May 11, 2026, ",[5055,6273,6274],{"href":6274,"rel":6275},"https:\u002F\u002Fwww.reddit.com\u002Fr\u002Fmachinelearningnews\u002Fcomments\u002F1sittj6\u002Fresearchers_from_mit_nvidia_and_zhejiang\u002F",[5059],[275,6277,6278,6279],{},"TriAttention | Efficient KV Cache Compression for Long-Context Reasoning - Weian Mao, accessed May 11, 2026, ",[5055,6280,6281],{"href":6281,"rel":6282},"https:\u002F\u002Fweianmao.github.io\u002Ftri-attention-project-page\u002F",[5059],[275,6284,6285,6286],{},"A 32B model on a single RTX 4090? I benchmarked inference latency after the TriAttention drop. : r\u002FAskClaw - Reddit, accessed May 11, 2026, ",[5055,6287,6288],{"href":6288,"rel":6289},"https:\u002F\u002Fwww.reddit.com\u002Fr\u002FAskClaw\u002Fcomments\u002F1sjxxpr\u002Fa_32b_model_on_a_single_rtx_4090_i_benchmarked\u002F",[5059],[275,6291,6292,6293],{},"MoE-nD: Per-Layer Mixture-of-Experts Routing for Multi-Axis KV Cache Compression - arXiv, accessed May 11, 2026, ",[5055,6294,6295],{"href":6295,"rel":6296},"https:\u002F\u002Farxiv.org\u002Fhtml\u002F2604.17695v1",[5059],[275,6298,6299,6300],{},"Daily Papers - Hugging Face, accessed May 11, 2026, ",[5055,6301,6304],{"href":6302,"rel":6303},"https:\u002F\u002Fhuggingface.co\u002Fpapers?q=KV+cache+capacity",[5059],"https:\u002F\u002Fhuggingface.co\u002Fpapers?q=KV%20cache%20capacity",[275,6306,6299,6307],{},[5055,6308,6311],{"href":6309,"rel":6310},"https:\u002F\u002Fhuggingface.co\u002Fpapers?q=KV+cache+compression",[5059],"https:\u002F\u002Fhuggingface.co\u002Fpapers?q=KV%20cache%20compression",[275,6313,6314,6315],{},"triattention\u002Fdocs\u002Fresults.md at main · WeianMao\u002Ftriattention · GitHub, accessed May 11, 2026, ",[5055,6316,6317],{"href":6317,"rel":6318},"https:\u002F\u002Fgithub.com\u002FWeianMao\u002Ftriattention\u002Fblob\u002Fmain\u002Fdocs\u002Fresults.md",[5059],[275,6320,6321,6322],{},"TriAttention lets a single RTX 4090 run 32B. But what does full local inference latency actually look like? - Reddit, accessed May 11, 2026, ",[5055,6323,6324],{"href":6324,"rel":6325},"https:\u002F\u002Fwww.reddit.com\u002Fr\u002FAskClaw\u002Fcomments\u002F1skv9k1\u002Ftriattention_lets_a_single_rtx_4090_run_32b_but\u002F",[5059],[275,6327,6328,6329],{},"Yukang Chen - CatalyzeX, accessed May 11, 2026, ",[5055,6330,6331],{"href":6331,"rel":6332},"https:\u002F\u002Fwww.catalyzex.com\u002Fauthor\u002FYukang%20Chen",[5059],[275,6334,6335,6336],{},"triattention\u002Fdocs\u002Fmlx.md at main - GitHub, accessed May 11, 2026, ",[5055,6337,6338],{"href":6338,"rel":6339},"https:\u002F\u002Fgithub.com\u002FWeianMao\u002Ftriattention\u002Fblob\u002Fmain\u002Fdocs\u002Fmlx.md",[5059],[275,6341,6342,6343],{},"Exploring LLMs with MLX and the Neural Accelerators in the M5 GPU, accessed May 11, 2026, ",[5055,6344,6345],{"href":6345,"rel":6346},"https:\u002F\u002Fmachinelearning.apple.com\u002Fresearch\u002Fexploring-llms-mlx-m5",[5059],[275,6348,6349,6350],{},"[2510.18921] Benchmarking On-Device Machine Learning on Apple Silicon with MLX - arXiv, accessed May 11, 2026, ",[5055,6351,6352],{"href":6352,"rel":6353},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2510.18921",[5059],[275,6355,6356,6357],{},"How Fast Is MLX? A Comprehensive Benchmark on 8 Apple Silicon Chips and 4 CUDA GPUs | Towards Data Science, accessed May 11, 2026, ",[5055,6358,6359],{"href":6359,"rel":6360},"https:\u002F\u002Ftowardsdatascience.com\u002Fhow-fast-is-mlx-a-comprehensive-benchmark-on-8-apple-silicon-chips-and-4-cuda-gpus-378a0ae356a0\u002F",[5059],[275,6362,6363,6364],{},"The Memory Wall Is Coming Down — What It Means for Coding, accessed May 11, 2026, ",[5055,6365,6366],{"href":6366,"rel":6367},"https:\u002F\u002Falexlavaee.me\u002Fblog\u002Fattention-memory-coding-agents\u002F",[5059],{"title":311,"searchDepth":312,"depth":312,"links":6369},[6370,6371,6372,6376,6381,6386,6391],{"id":5702,"depth":312,"text":5702},{"id":5713,"depth":312,"text":5713},{"id":5727,"depth":312,"text":5727,"children":6373},[6374,6375],{"id":5735,"depth":4984,"text":5738},{"id":5744,"depth":4984,"text":5747},{"id":5759,"depth":312,"text":5762,"children":6377},[6378,6379,6380],{"id":5768,"depth":4984,"text":5771},{"id":5780,"depth":4984,"text":5780},{"id":5816,"depth":4984,"text":5816},{"id":5906,"depth":312,"text":5906,"children":6382},[6383,6384,6385],{"id":5914,"depth":4984,"text":5917},{"id":6075,"depth":4984,"text":6078},{"id":6101,"depth":4984,"text":6104},{"id":6124,"depth":312,"text":6124,"children":6387},[6388,6389,6390],{"id":6132,"depth":4984,"text":6135},{"id":6147,"depth":4984,"text":6150},{"id":6176,"depth":4984,"text":6176},{"id":6194,"depth":312,"text":6194},"\u002Fimages\u002Fblog\u002F2604.04921.webp","2026-05-11","隨著人工智慧在軟體工程與複雜邏輯運算領域的快速演進，大語言模型（Large Language Models, LLMs）的應用範疇已經從早期的單純文本補全（Code Completion），跨越至具備自主決策與長鏈條推理能力的代理系統（Autonomous Agents）",{},"\u002Fblog\u002Fzh-tw\u002Farxiv-2604-04921v1",{"title":5697,"description":6394},"arxiv-2604-04921v1","blog\u002Fzh-TW\u002Farxiv-2604-04921v1",[5424,6401,6402],"agent","attention","rSnvxJU_oS2Zq43Rx-tp4xFtuCXoltI5PUo00hyQ9m0",{"id":6405,"title":6406,"author":103,"body":6407,"cover":6413,"coverAlt":6406,"date":10094,"description":10095,"draft":324,"extension":325,"featured":324,"lang":327,"meta":10096,"navigation":326,"path":10097,"seo":10098,"slug":10099,"stem":10100,"tags":10101,"translationKey":10099,"updated":337,"__hash__":10102},"blog\u002Fblog\u002Fzh-TW\u002Farxiv-2601-16746v2.md","SWE-Pruner：程式碼智能體之自適應上下文修剪機制與理論深度分析報告",{"type":105,"value":6408,"toc":10078},[6409,6414,6417,6424,6426,6429,6432,6435,6438,6440,6443,6457,6460,6463,6468,6471,6476,6479,6482,6485,6488,6491,6494,6497,6501,6504,6510,6513,6516,6528,6531,6534,6537,6541,6544,6747,6750,7187,7190,7194,7197,7200,8093,8104,8431,8434,9031,9034,9394,9397,9400,9403,9406,9409,9412,9415,9419,9422,9425,9428,9431,9434,9438,9441,9444,9543,9546,9549,9552,9661,9664,9667,9670,9673,9676,9821,9824,9827,9830,9834,9837,9931,9934,9938,9944,9947,9957,9961,9964,9968,9979,9989,10003,10007,10014,10021,10028,10031,10034,10048,10051,10054,10057,10060,10063,10066,10069,10072,10075],[112,6410,6411],{},[5790,6412],{"alt":4052,"src":6413},"\u002Fimages\u002Fblog\u002F2601.16746v2.webp",[108,6415,6406],{"id":6416},"swe-pruner程式碼智能體之自適應上下文修剪機制與理論深度分析報告",[5119,6418,6420],{"id":6419},"httpsarxivorgabs260116746",[5055,6421,6422],{"href":6422,"rel":6423},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2601.16746",[5059],[5119,6425,5713],{"id":5713},[112,6427,6428],{},"在當代軟體工程的演進歷程中，大型語言模型（Large Language Models, LLMs）的應用已經跨越了單純的程式碼片段補全與靜態語義分析，正式邁入能夠自主導航龐大程式碼庫、執行複雜測試指令並端到端提交修補程式的互動式智能體（Coding Agents）時代 。諸如 Claude Code 與 Gemini CLI 等先進工具鏈的問世，賦予了模型操作終端機與檔案系統的權限，使其具備了執行多步邏輯推理的潛力 。然而，隨著智能體處理真實世界軟體專案複雜度的指數級提升，其效能無可避免地撞上了一堵被稱為「上下文之牆」（Context Wall）的物理與認知邊界 。",[112,6430,6431],{},"在長程、多輪的軟體除錯與開發互動中，不斷累積的歷史上下文不僅帶來了高昂的應用程式介面（API）呼叫成本與極高的運算延遲，更引入了海量的語義雜訊 。這些雜訊導致模型產生嚴重的注意力稀釋（Attention Dilution）與幻覺現象（Hallucinations），使得智能體在關鍵決策節點迷失方向 。傳統的上下文壓縮技術多數針對自然語言的機率分佈設計，依賴困惑度（Perplexity）等靜態且任務不可知的指標進行詞塊（Token）級別的無差別修剪 。將這些方法強行應用於具備嚴格抽象語法樹（AST）的程式碼時，往往會引發災難性的後果，不僅破壞了程式語言嚴謹的句法結構，更經常丟失除錯過程中至關重要的字元級實作細節 。",[112,6433,6434],{},"為徹底突破此一理論與實務瓶頸，本報告深入解析一種專為程式碼智能體量身打造的自適應上下文修剪框架——SWE-Pruner 。該框架的設計哲學深刻汲取了人類資深工程師在開發與除錯時的「選擇性略讀」（Selective Skimming）認知行為模式 。SWE-Pruner 允許智能體在每輪環境互動中，基於當下的推理狀態動態生成明確的「目標提示」（Goal Hint，例如指示系統「專注於 MRO 解析邏輯」），並透過一個僅有 0.6B 參數的輕量級神經網絡（Neural Skimmer），在嚴格保持程式碼行級（Line-level）結構完整性的前提下，精準篩選出與目標高度相關的上下文片段 。",[112,6436,6437],{},"詳盡的實證數據表明，SWE-Pruner 在嚴苛的多輪互動基準測試（如 SWE-Bench Verified）中，能為 Claude Sonnet 4.5 及 GLM 4.6 等業界頂尖模型減少 23% 至 54% 的絕對詞塊消耗，同時逆勢提升高達 1.4 個百分點的任務成功率，並減少最多 26% 的無效互動輪數 。在 LongCodeQA 等單輪長上下文問答任務中，該框架更實現了高達 14.84 倍的驚人壓縮率，且對核心效能的影響微乎其微 。此一突破性進展，不僅大幅降低了 AI 輔助軟體工程的營運成本，更為構建具備長效記憶與高認知專注力的自主智能系統奠定了關鍵的架構基礎。",[5119,6439,5727],{"id":5727},[5454,6441,6442],{"id":6442},"程式碼智能體的認知負荷與成本結構解構",[112,6444,6445,6446,6449,6450,6449,6453,6456],{},"在真實世界的企業級軟體開發場景中，程式碼智能體通常被投放於一個完全陌生且體積龐大的程式碼庫中。為了建立對系統全域架構的理解並精確定位潛在的軟體缺陷，智能體必須採用粗粒度的檔案操作工具（如 ",[118,6447,6448],{},"grep",", ",[118,6451,6452],{},"cat",[118,6454,6455],{},"head"," 等）進行廣泛且反覆的探索 。這種探索策略雖然在邏輯上對於建立全域認知至關重要，但其物理代價是會將成千上萬行的原始檔案或日誌串流，毫無保留地載入語言模型的上下文視窗中 。",[112,6458,6459],{},"為了量化這種行為模式帶來的資源消耗，研究團隊對 Mini-SWE-Agent 代理框架在 SWE-Bench Verified 基準測試上的互動軌跡進行了深度的實證分析 。該分析以 Claude Sonnet 4.5 為驅動核心，將智能體的所有工具呼叫行為嚴格分類為三大類型：讀取操作（涵蓋檔案與目錄的檢索檢視）、執行操作（涵蓋編譯與測試腳本的運行）以及編輯操作（涵蓋程式碼的就地修改） 。",[112,6461,6462],{},"分析結果揭示了一個極端不平衡且令人震驚的詞塊預算分配結構。數據表明，讀取類型的操作佔據了絕對的統治地位，消耗了高達 76.1% 的總詞塊（絕對數量約為 4.38M tokens） 。相較之下，真正推進任務進度的執行操作僅佔 12.1%（約 0.69M tokens），而直接修復問題的編輯操作佔比更低至 11.8%（約 0.68M tokens） 。",[112,6464,6465],{},[5790,6466],{"alt":1187,"src":6467},"\u002Fimages\u002Fblog\u002F2601.16746v2_01.webp",[112,6469,6470],{},"為驗證此一現象是否受到特定模型架構的偏差影響，研究團隊將相同的軌跡追蹤方法應用於架構與訓練範式截然不同的開源模型 GLM-4.6 上 。實驗數據呈現出高度一致的物理規律：GLM-4.6 的讀取操作同樣主導了詞塊消耗，佔比高達 67.5%（約 2.89M tokens），而編輯與執行操作則分別佔據 18.5% 與 14.0% 。",[112,6472,6473],{},[5790,6474],{"alt":4119,"src":6475},"\u002Fimages\u002Fblog\u002F2601.16746v2_02.webp",[112,6477,6478],{},"這種跨越模型架構的驚人一致性，深刻揭示了程式碼智能體運作機制中的一個根本性缺陷：智能體絕大部分的算力、金錢成本與上下文空間，都不可避免地浪費在處理探索階段所引入的冗餘環境觀察中 。隨著多輪互動的持續推進，早期檢索到但後續不再需要的非關鍵程式碼，會在其歷史紀錄中不斷堆疊。這種「上下文累積效應」不僅拖垮了系統的反應速度，更讓模型在龐雜的程式碼迷宮中逐漸喪失對核心問題的推理聚焦能力 。",[5454,6480,6481],{"id":6481},"傳統壓縮策略在程式碼領域的結構性失效",[112,6483,6484],{},"面對日益嚴峻的上下文過載危機，學界與業界曾提出多種試圖緩解該問題的壓縮方案。然而，當這些源自自然語言處理（NLP）領域的技術被強行移植到具有高度形式化特徵的軟體工程領域時，皆顯露出致命的理論與應用缺陷 。",[112,6486,6487],{},"第一類主流方法為詞塊級的靜態修剪技術（Token-level Pruning），其代表性框架包括 Selective-Context 與 LLMLingua2 。這類方法的核心運作邏輯是基於資訊理論中的自信息量（Self-information）或語言模型計算出的困惑度，將機率較高、看似「可預測」的詞塊視為低資訊量冗餘並予以刪除 。然而，程式碼與自然語言有著本質的差異，程式碼依賴嚴格的抽象語法樹（AST）來維持邏輯運算。以詞塊為單位的隨機刪除，會輕易且不可逆地破壞變數宣告的完整性、控制流的縮排範圍以及函式區塊的括號匹配 。實證數據無情地指出，經過 LLMLingua2 壓縮後的程式碼，其 AST 正確性僅存 0.29%，這意味著壓縮後的文本在編譯器或語言模型眼中，已退化為毫無語法意義的字元碎片 。",[112,6489,6490],{},"第二類方法為粗粒度檢索（Coarse-grained Retrieval）與生成式摘要（Generative Summarization） 。檢索增強生成（RAG）透過將程式碼切分為函式區塊並計算嵌入向量相似度來提取內容。儘管這保留了局部的語法完整性，但在複雜的除錯任務中，一個導致系統崩潰的關鍵異常處理邏輯可能僅隱藏在某一兩行程式碼中，粗粒度的向量比對極易將這些微小但致命的細節判定為低相關性而予以遺漏 。另一方面，使用大型語言模型（如 LLM Summarize）對程式碼檔案生成抽象摘要，雖然能有效提取模組的大意，卻會徹底抹除字元級別的實作資訊，剝奪了智能體進行精確代碼編輯的基礎，同時摘要生成的過程本身也引入了令人難以接受的高昂延遲 。",[112,6492,6493],{},"第三類方法則聚焦於智能體歷史軌跡的管理（Agent History Compression），如 SUPO、COMPASS 或 Agent Fold 等框架 。這些技術透過強化學習或主動折疊策略來壓縮智能體過去的對話與反思紀錄。儘管這些方法有效控制了對話輪次的增長，但它們完全迴避了「環境觀察（即檔案內容）本身過於龐大」的根本問題，因此無法真正解決因讀取大型檔案所造成的即時上下文溢位 。",[112,6495,6496],{},"綜上所述，一個真正適用於程式碼智能體的理想壓縮機制，必須同時滿足三個嚴苛的條件：其一，必須是動態且具備任務感知能力（Task-aware），能夠隨著智能體不斷變化的推理目標調整過濾標準；其二，必須是結構保留的（Structure-preserving），絕對不能破壞程式碼的語法有效性；其三，必須具備極致的輕量化特徵（Lightweight），以確保壓縮過程不會成為系統新的效能瓶頸 。這三大理論訴求，正是孕育 SWE-Pruner 架構設計的核心基石。",[5119,6498,6500],{"id":6499},"swe-pruner-系統架構與數學機制深度剖析","SWE-Pruner 系統架構與數學機制深度剖析",[112,6502,6503],{},"為了解決上述理論衝突與技術瓶頸，研究團隊提出了 SWE-Pruner，這是一個運作於程式碼智能體與其執行環境之間的中介軟體（Middleware）框架 。當智能體透過終端機發出讀取檔案的系統指令時，SWE-Pruner 會主動攔截可能包含數千行原始碼的龐大上下文 。系統不會直接將這些充滿雜訊的資料灌入智能體的記憶體，而是要求智能體提供一個描述當下意圖的「目標提示」，接著透過輕量級神經網絡進行精準過濾，最終僅放行高度相關且語法完整的程式碼行 。",[112,6505,6506],{},[5790,6507],{"alt":6508,"src":6509},"3","\u002Fimages\u002Fblog\u002F2601.16746v2_03.webp",[5454,6511,6512],{"id":6512},"意圖驅動的目標提示生成機制",[112,6514,6515],{},"SWE-Pruner 運作的第一步，在於精確捕捉智能體當下的認知需求。與傳統依賴死板關鍵字比對的過濾系統不同，SWE-Pruner 透過精密的提示詞工程（Prompt Engineering），引導目標智能體將其內部推理狀態轉化為結構化的自然語言問句 。",[112,6517,6518,6519,6449,6521,6523,6524,6527],{},"在具體實作上，系統對標準的 Unix 檔案操作工具（如 ",[118,6520,6452],{},[118,6522,6448],{},"）進行了無侵入式的封裝，為這些工具新增了一個名為 ",[118,6525,6526],{},"context_focus_question"," 的可選參數 。智能體被指示，當預期指令輸出可能過大時，應提供一個完整且自包含的問題（例如：「系統如何確保並行寫入時的資料一致性？」），以此作為引導過濾器的語義信號 。當此參數被賦值時，擷取到的原始輸出與該問題將一同被送入神經略讀器進行裁切；若智能體判定需要檢視完整結構而省略此參數，系統則會優雅地降級為原始行為，直接回傳未經修改的完整內容 。這種極具彈性的輕量級包裝設計，確保了 SWE-Pruner 能夠在不破壞現有智能體工作流程的前提下，實現平滑的向下相容與系統整合 。",[5454,6529,6530],{"id":6530},"輕量級神經略讀器之雙頭架構設計",[112,6532,6533],{},"系統的心臟是一個參數規模僅 0.6B 的神經略讀器（Neural Skimmer），其底層採用 Qwen3-Reranker-0.6B 作為特徵提取骨幹，並進行了深度架構改造，擴充為包含「修剪頭」（Pruning Head）與「重排頭」（Reranking Head）的雙任務學習網絡 。",[112,6535,6536],{},"為了讓如此小巧的模型能夠理解深層的程式碼語義，系統實作了多層特徵融合（Multi-Layer Feature Fusion）機制 。模型不僅依賴最終層的輸出，更主動提取並串聯了骨幹網絡中第 7、14 與 28 層的隱藏狀態。這些跨越不同語義深度的特徵融合後，會依序通過一個自注意力區塊與一個具有 8 個注意力頭、隱藏層維度為 256 的多頭注意力層進行精煉，從而賦予模型極其敏銳的上下文洞察力 。",[6538,6539,6540],"h5",{"id":6540},"詞塊到行級的聚合演算法",[112,6542,6543],{},"在評估階段，模型將上下文修剪定義為一個條件排序問題。給定智能體接收到的原始上下文序列與查詢目標，模型首先透過神經網絡參數計算每個單一詞塊的局部相關性分數 。",[360,6545,6547],{"className":6546},[755],[360,6548,6550,6602],{"className":6549},[363],[360,6551,6553],{"className":6552},[367],[369,6554,6555],{"xmlns":371,"display":764},[373,6556,6557,6599],{},[376,6558,6559,6566,6568,6572,6574,6576,6578,6585,6588,6591,6594,6597],{},[628,6560,6561,6564],{},[379,6562,6563],{},"s",[379,6565,1126],{},[383,6567,796],{},[379,6569,6571],{"mathvariant":6570},"script","F",[383,6573,779],{"stretchy":778},[379,6575,1210],{},[383,6577,784],{"separator":413},[628,6579,6580,6583],{},[379,6581,6582],{},"x",[379,6584,1126],{},[379,6586,6587],{"mathvariant":771},"∣",[379,6589,6590],{},"C",[383,6592,6593],{"separator":413},";",[379,6595,6596],{},"θ",[383,6598,793],{"stretchy":778},[405,6600,6601],{"encoding":407},"s_{i}=\\mathcal{F}(q,x_{i}|C;\\theta)",[360,6603,6605,6663],{"className":6604,"ariaHidden":413},[412],[360,6606,6608,6611,6654,6657,6660],{"className":6607},[417],[360,6609],{"className":6610,"style":1298},[421],[360,6612,6614,6617],{"className":6613},[426],[360,6615,6563],{"className":6616},[426,427],[360,6618,6620],{"className":6619},[457],[360,6621,6623,6646],{"className":6622},[461,713],[360,6624,6626,6643],{"className":6625},[465],[360,6627,6629],{"className":6628,"style":1317},[469],[360,6630,6631,6634],{"style":1320},[360,6632],{"className":6633,"style":477},[476],[360,6635,6637],{"className":6636},[481,482,483,484],[360,6638,6640],{"className":6639},[426,484],[360,6641,1126],{"className":6642},[426,427,484],[360,6644,741],{"className":6645},[740],[360,6647,6649],{"className":6648},[465],[360,6650,6652],{"className":6651,"style":1339},[469],[360,6653],{},[360,6655],{"className":6656,"style":432},[431],[360,6658,796],{"className":6659},[436],[360,6661],{"className":6662,"style":432},[431],[360,6664,6666,6669,6674,6677,6680,6683,6686,6729,6732,6735,6738,6741,6744],{"className":6665},[417],[360,6667],{"className":6668,"style":848},[421],[360,6670,6571],{"className":6671,"style":6673},[426,6672],"mathcal","margin-right:0.0993em;",[360,6675,779],{"className":6676},[860],[360,6678,1210],{"className":6679,"style":736},[426,427],[360,6681,784],{"className":6682},[867],[360,6684],{"className":6685,"style":871},[431],[360,6687,6689,6692],{"className":6688},[426],[360,6690,6582],{"className":6691},[426,427],[360,6693,6695],{"className":6694},[457],[360,6696,6698,6721],{"className":6697},[461,713],[360,6699,6701,6718],{"className":6700},[465],[360,6702,6704],{"className":6703,"style":1317},[469],[360,6705,6706,6709],{"style":1320},[360,6707],{"className":6708,"style":477},[476],[360,6710,6712],{"className":6711},[481,482,483,484],[360,6713,6715],{"className":6714},[426,484],[360,6716,1126],{"className":6717},[426,427,484],[360,6719,741],{"className":6720},[740],[360,6722,6724],{"className":6723},[465],[360,6725,6727],{"className":6726,"style":1339},[469],[360,6728],{},[360,6730,6587],{"className":6731},[426],[360,6733,6590],{"className":6734,"style":545},[426,427],[360,6736,6593],{"className":6737},[867],[360,6739],{"className":6740,"style":871},[431],[360,6742,6596],{"className":6743,"style":3944},[426,427],[360,6745,793],{"className":6746},[887],[112,6748,6749],{},"公式中，分數計算不僅考量了查詢條件與當前詞塊，更將整個上下文的宏觀狀態納入條件機率的評估中 。然而，直接依賴詞塊分數進行修剪正是導致 LLMLingua 等前人工作失敗的主因。為跨越這道鴻溝，SWE-Pruner 導入了行級聚合（Line-level Aggregation）邏輯。系統將連續的上下文拆解為以「行」為單位的集合，並計算每一行所包含之所有詞塊分數的算術平均值，以此作為該行的最終保留指標 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。",[360,9035,9037],{"className":9036},[755],[360,9038,9040,9131],{"className":9039},[363],[360,9041,9043],{"className":9042},[367],[369,9044,9045],{"xmlns":371,"display":764},[373,9046,9047,9128],{},[376,9048,9049,9066,9068,9070,9072,9074,9077,9079,9082,9104,9106,9108,9110],{},[628,9050,9051,9053],{},[379,9052,8453],{"mathvariant":6570},[376,9054,9055,9057,9059,9061,9063],{},[379,9056,6804],{},[379,9058,1162],{},[379,9060,6804],{},[379,9062,5055],{},[379,9064,9065],{},"l",[383,9067,796],{},[383,9069,779],{"stretchy":778},[1185,9071,1187],{},[383,9073,7367],{},[379,9075,9076],{},"λ",[383,9078,793],{"stretchy":778},[383,9080,9081],{},"⋅",[628,9083,9084,9086],{},[379,9085,8453],{"mathvariant":6570},[376,9087,9088,9090,9092,9094,9096,9098,9100,9102],{},[379,9089,7219],{},[379,9091,1162],{},[379,9093,524],{},[379,9095,112],{},[379,9097,7224],{},[379,9099,7227],{},[379,9101,6563],{},[379,9103,6563],{},[383,9105,3976],{},[379,9107,9076],{},[383,9109,9081],{},[628,9111,9112,9114],{},[379,9113,8453],{"mathvariant":6570},[376,9115,9116,9118,9120,9122,9124,9126],{},[379,9117,7224],{},[379,9119,7227],{},[379,9121,7224],{},[379,9123,5055],{},[379,9125,397],{},[379,9127,1219],{},[405,9129,9130],{"encoding":407},"\\mathcal{L}_{total}=(1-\\lambda)\\cdot\\mathcal{L}_{compress}+\\lambda\\cdot\\mathcal{L}_{rerank}",[360,9132,9134,9206,9227,9248,9318,9337],{"className":9133,"ariaHidden":413},[412],[360,9135,9137,9141,9197,9200,9203],{"className":9136},[417],[360,9138],{"className":9139,"style":9140},[421],"height:0.8333em;vertical-align:-0.15em;",[360,9142,9144,9147],{"className":9143},[426],[360,9145,8453],{"className":9146},[426,6672],[360,9148,9150],{"className":9149},[457],[360,9151,9153,9189],{"className":9152},[461,713],[360,9154,9156,9186],{"className":9155},[465],[360,9157,9159],{"className":9158,"style":4928},[469],[360,9160,9161,9164],{"style":1320},[360,9162],{"className":9163,"style":477},[476],[360,9165,9167],{"className":9166},[481,482,483,484],[360,9168,9170,9173,9176,9179,9182],{"className":9169},[426,484],[360,9171,6804],{"className":9172},[426,427,484],[360,9174,1162],{"className":9175},[426,427,484],[360,9177,6804],{"className":9178},[426,427,484],[360,9180,5055],{"className":9181},[426,427,484],[360,9183,9065],{"className":9184,"style":9185},[426,427,484],"margin-right:0.0197em;",[360,9187,741],{"className":9188},[740],[360,9190,9192],{"className":9191},[465],[360,9193,9195],{"className":9194,"style":1339},[469],[360,9196],{},[360,9198],{"className":9199,"style":432},[431],[360,9201,796],{"className":9202},[436],[360,9204],{"className":9205,"style":432},[431],[360,9207,9209,9212,9215,9218,9221,9224],{"className":9208},[417],[360,9210],{"className":9211,"style":848},[421],[360,9213,779],{"className":9214},[860],[360,9216,1187],{"className":9217},[426],[360,9219],{"className":9220,"style":650},[431],[360,9222,7367],{"className":9223},[494],[360,9225],{"className":9226,"style":650},[431],[360,9228,9230,9233,9236,9239,9242,9245],{"className":9229},[417],[360,9231],{"className":9232,"style":848},[421],[360,9234,9076],{"className":9235},[426,427],[360,9237,793],{"className":9238},[887],[360,9240],{"className":9241,"style":650},[431],[360,9243,9081],{"className":9244},[494],[360,9246],{"className":9247,"style":650},[431],[360,9249,9251,9254,9309,9312,9315],{"className":9250},[417],[360,9252],{"className":9253,"style":8556},[421],[360,9255,9257,9260],{"className":9256},[426],[360,9258,8453],{"className":9259},[426,6672],[360,9261,9263],{"className":9262},[457],[360,9264,9266,9301],{"className":9265},[461,713],[360,9267,9269,9298],{"className":9268},[465],[360,9270,9272],{"className":9271,"style":7469},[469],[360,9273,9274,9277],{"style":1320},[360,9275],{"className":9276,"style":477},[476],[360,9278,9280],{"className":9279},[481,482,483,484],[360,9281,9283,9286,9289,9292,9295],{"className":9282},[426,484],[360,9284,8589],{"className":9285},[426,427,484],[360,9287,524],{"className":9288},[426,427,484],[360,9290,112],{"className":9291},[426,427,484],[360,9293,7224],{"className":9294,"style":3944},[426,427,484],[360,9296,8602],{"className":9297},[426,427,484],[360,9299,741],{"className":9300},[740],[360,9302,9304],{"className":9303},[465],[360,9305,9307],{"className":9306,"style":2063},[469],[360,9308],{},[360,9310],{"className":9311,"style":650},[431],[360,9313,3976],{"className":9314},[494],[360,9316],{"className":9317,"style":650},[431],[360,9319,9321,9325,9328,9331,9334],{"className":9320},[417],[360,9322],{"className":9323,"style":9324},[421],"height:0.6944em;",[360,9326,9076],{"className":9327},[426,427],[360,9329],{"className":9330,"style":650},[431],[360,9332,9081],{"className":9333},[494],[360,9335],{"className":9336,"style":650},[431],[360,9338,9340,9343],{"className":9339},[417],[360,9341],{"className":9342,"style":9140},[421],[360,9344,9346,9349],{"className":9345},[426],[360,9347,8453],{"className":9348},[426,6672],[360,9350,9352],{"className":9351},[457],[360,9353,9355,9386],{"className":9354},[461,713],[360,9356,9358,9383],{"className":9357},[465],[360,9359,9361],{"className":9360,"style":4928},[469],[360,9362,9363,9366],{"style":1320},[360,9364],{"className":9365,"style":477},[476],[360,9367,9369],{"className":9368},[481,482,483,484],[360,9370,9372,9375,9379],{"className":9371},[426,484],[360,9373,7224],{"className":9374,"style":3944},[426,427,484],[360,9376,9378],{"className":9377,"style":3944},[426,427,484],"er",[360,9380,9382],{"className":9381,"style":1645},[426,427,484],"ank",[360,9384,741],{"className":9385},[740],[360,9387,9389],{"className":9388},[465],[360,9390,9392],{"className":9391,"style":1339},[469],[360,9393],{},[5119,9395,9396],{"id":9396},"多維度訓練範式與高品質多語種資料集構建",[112,9398,9399],{},"要訓練出具備強大結構感知與任務導向能力的 CRF 神經略讀器，高品質且包含行級標註的監督資料是不可或缺的。然而，縱觀開源社群與學術界，皆不存在滿足此一嚴苛要求的現成資料集 。為此，研究團隊設計並實作了一套基於「教師-學生範式」（Teacher-Student Paradigm）的合成資料庫構建管線 。",[5454,9401,9402],{"id":9402},"程式碼源的甄選與正規化",[112,9404,9405],{},"資料構建的基石來自於 GitHub Code 2025 數據集，這是一個包含超過 150 萬個優質儲存庫的龐大語料庫 。研究團隊從中精心抽樣了 5,945 個活躍專案中的 195,370 個源代碼檔案 。為確保資料的純淨度，所有被選中的檔案都經歷了極其嚴苛的預處理與正規化流程，徹底剔除了二進位構建產物、自動生成的配置雜訊以及經過混淆壓縮的微型代碼（Minified Code） 。這一過程確保了留存下來的資料，皆是具有高度語義價值且跨越多種程式語言範式（Polyglot）的原始結構，為後續訓練提供了卓越的泛化基礎 。",[5454,9407,9408],{"id":9408},"涵蓋完整生命週期的智能體任務分類學",[112,9410,9411],{},"為了讓神經略讀器能夠從容應對真實世界中千變萬化的智能體開發情境，研究團隊建構了一套包含九大維度的「智能體任務分類學」（Agentic Task Taxonomy），以此引導教師模型生成高度擬真的自然語言查詢 。這九大任務精確映射了軟體工程師在日常開發中的認知軌跡：",[112,9413,9414],{},"這九個維度分別為：程式碼摘要（模擬系統整合前的高階模組審查）、重構建議（聚焦於提升演算法的空間複雜度或架構模組化程度）、相關片段定位（要求模型在茫茫代碼海中精確指出特定業務邏輯的實作點）、程式碼優化（尋求效能瓶頸的突破）、程式碼定位（針對錯誤日誌或崩潰報告尋找發源地）、原理解釋（探討特定設計模式或數學公式的運用動機）、程式碼除錯（尋求針對邊緣案例異常的具體修復方針）、特徵添加（在不破壞現有狀態機的前提下無縫擴充新功能），以及特殊的程式碼補全任務（以代碼片段本身作為查詢，模擬即時開發環境中的上下文感知補全） 。透過強制在這些多元任務上進行訓練，模型學會了根據不同意圖動態調整修剪的寬容度與聚焦深度。",[5454,9416,9418],{"id":9417},"大模型作為裁判llm-as-a-judge之極致品質過濾","大模型作為裁判（LLM-as-a-Judge）之極致品質過濾",[112,9420,9421],{},"在生成階段，團隊動用了具有 300 億參數的 Qwen3-Coder-30B-A3B-Instruct 作為教師模型，負責基於原始碼與任務分類生成對應的查詢及行級二元遮罩（Binary Line-level Mask） 。然而，機器生成的行級標註不可避免地會出現過度修剪或邏輯斷層的瑕疵。",[112,9423,9424],{},"為確保黃金標準的資料品質，管線引入了具有 800 億參數且具備深度推理能力的 Qwen3-Next-80B-A3B-Thinking 作為終極裁判（Judge） 。該裁判模型被賦予了嚴格的評估準則，從三個正交維度對每一筆合成資料進行嚴格審查：第一是「查詢品質」，拒絕任何語意模糊或缺乏具體開發意圖的問題；第二是「刪除關聯性」，檢驗被標記為刪除的程式碼是否確實為無關雜訊；第三，也是最核心的「語義留存度」，嚴厲把關剩餘的代碼區塊是否依然維持句法正確與邏輯可讀 。",[112,9426,9427],{},"經過這套猶如煉油廠般嚴密的過濾機制，僅有約六分之一的候選樣本通過考驗。團隊最終提煉出 61,184 筆具有精實驗證行級標註的高品質訓練四元組 。這些數據的平均查詢長度達到 39.98 個詞彙，完美捕捉了真實開發者在複雜除錯場景中細膩且具體的資訊訴求，為神經略讀器的卓越表現注入了靈魂 。",[5119,9429,9430],{"id":9430},"多輪智能體任務的效能評估與實證突破",[112,9432,9433],{},"SWE-Pruner 的真正價值，必須在最貼近真實軟體開發流程的動態環境中接受檢驗。為此，研究團隊在涵蓋多輪互動場景的旗艦級基準測試上展開了詳盡的評估。",[5454,9435,9437],{"id":9436},"swe-bench-verified效率與成功率的雙重躍升","SWE-Bench Verified：效率與成功率的雙重躍升",[112,9439,9440],{},"在評估多輪除錯能力的 SWE-Bench Verified 測試中（包含 500 個真實世界 GitHub Issue 的端到端修復），系統被無縫整合至 Mini SWE Agent 框架中，並分別以 Claude Sonnet 4.5 及 GLM-4.6 作為心智引擎進行雙盲測試 。",[112,9442,9443],{},"實驗結果呈現了令人振奮的雙軌突破。對於 Claude Sonnet 4.5 而言，引入 SWE-Pruner 後，平均每個任務實例的詞塊消耗從 0.911M 銳減至 0.701M（降低 23.1%），更令人驚豔的是，其任務解決率不僅沒有因為資訊減少而衰退，反而逆勢從 70.6% 提升至 72.0% 。而在 GLM-4.6 模型的表現上，這種「少即是多」的效應更加猛烈：詞塊消耗狂降 38.3%（從 0.791M 降至 0.488M），任務成功率從 55.4% 攀升至 56.6%，同時智能體的平均互動輪數大幅縮減了 25.7%（從 49.3 輪下降至 36.6 輪） 。",[4848,9445,9446,9465],{},[4851,9447,9448],{},[4854,9449,9450,9453,9456,9459,9462],{},[4857,9451,9452],{},"智能體與模型配置",[4857,9454,9455],{},"互動輪數",[4857,9457,9458],{},"成功率 (%)",[4857,9460,9461],{},"總詞塊消耗 (M)",[4857,9463,9464],{},"API 預估成本 ($)",[4864,9466,9467,9484,9505,9522],{},[4854,9468,9469,9472,9475,9478,9481],{},[4869,9470,9471],{},"Mini SWE Agent (Claude 4.5)",[4869,9473,9474],{},"51.0",[4869,9476,9477],{},"70.6",[4869,9479,9480],{},"0.911",[4869,9482,9483],{},"0.504",[4854,9485,9486,9491,9494,9499,9502],{},[4869,9487,9488],{},[138,9489,9490],{},"+ SWE-Pruner (Claude 4.5)",[4869,9492,9493],{},"41.7 (-18.2%)",[4869,9495,9496],{},[138,9497,9498],{},"72.0 (+1.4%)",[4869,9500,9501],{},"0.701 (-23.1%)",[4869,9503,9504],{},"0.369 (-26.8%)",[4854,9506,9507,9510,9513,9516,9519],{},[4869,9508,9509],{},"Mini SWE Agent (GLM 4.6)",[4869,9511,9512],{},"49.3",[4869,9514,9515],{},"55.4",[4869,9517,9518],{},"0.791",[4869,9520,9521],{},"0.055",[4854,9523,9524,9529,9532,9537,9540],{},[4869,9525,9526],{},[138,9527,9528],{},"+ SWE-Pruner (GLM 4.6)",[4869,9530,9531],{},"36.6 (-25.7%)",[4869,9533,9534],{},[138,9535,9536],{},"56.6 (+1.2%)",[4869,9538,9539],{},"0.488 (-38.3%)",[4869,9541,9542],{},"0.035 (-36.4%)",[112,9544,9545],{},"這組數據揭示了一個深刻的認知科學洞察：在程式碼推理任務中，上下文壓縮不單單是為了「節省運算成本」的妥協手段，它實質上是一種「強化認知專注力」的主動機制 。透過神經略讀器前置過濾掉冗餘的探索雜訊，底層語言模型不再被無關的邏輯分支所迷惑，因而能夠展現出更果斷的推理直覺，直接減少了無效的反覆讀取操作，進而促成了整體互動輪次與成本的斷崖式下降 。",[5454,9547,9548],{"id":9548},"壓縮策略的橫向切片對比分析",[112,9550,9551],{},"為了進一步驗證「為何」是 SWE-Pruner 取得最佳表現，研究團隊在 SWE-Bench 的隨機 50 個樣本子集上，針對目前主流的上下文管理策略進行了橫向對比實證 。",[4848,9553,9554,9570],{},[4851,9555,9556],{},[4854,9557,9558,9561,9564,9567],{},[4857,9559,9560],{},"上下文管理策略",[4857,9562,9563],{},"平均輪數",[4857,9565,9566],{},"任務成功率 (%)",[4857,9568,9569],{},"詞塊總消耗 (M)",[4864,9571,9572,9586,9600,9614,9628,9641],{},[4854,9573,9574,9577,9580,9583],{},[4869,9575,9576],{},"無壓縮基準 (Mini SWE Agent)",[4869,9578,9579],{},"52.3",[4869,9581,9582],{},"62.0",[4869,9584,9585],{},"0.972",[4854,9587,9588,9591,9594,9597],{},[4869,9589,9590],{},"+ LLMLingua2 (詞塊級修剪)",[4869,9592,9593],{},"42.1",[4869,9595,9596],{},"54.0",[4869,9598,9599],{},"0.856",[4854,9601,9602,9605,9608,9611],{},[4869,9603,9604],{},"+ RAG (粗粒度段落檢索)",[4869,9606,9607],{},"40.2",[4869,9609,9610],{},"50.0",[4869,9612,9613],{},"0.771",[4854,9615,9616,9619,9622,9625],{},[4869,9617,9618],{},"+ LLM Summarize (生成式摘要)",[4869,9620,9621],{},"41.3",[4869,9623,9624],{},"56.0",[4869,9626,9627],{},"0.794",[4854,9629,9630,9633,9636,9638],{},[4869,9631,9632],{},"+ LongCodeZip (語法結構壓縮)",[4869,9634,9635],{},"44.3",[4869,9637,9596],{},[4869,9639,9640],{},"0.889",[4854,9642,9643,9648,9651,9656],{},[4869,9644,9645],{},[138,9646,9647],{},"+ SWE-Pruner (任務感知行級修剪)",[4869,9649,9650],{},"41.1",[4869,9652,9653],{},[138,9654,9655],{},"64.0",[4869,9657,9658],{},[138,9659,9660],{},"0.670",[112,9662,9663],{},"對比數據無情地暴露了傳統壓縮理論在軟體工程領域的死穴。LLMLingua2 與 RAG 雖然表面上達成了降低輪數與詞塊的目的，但代價是任務成功率從 62.0% 崩盤至 54.0% 與 50.0% 。這印證了先前的理論假設：詞塊級修剪撕裂了代碼的語法完整性，導致模型產生編譯幻覺；而 RAG 的粗粒度檢索則無可避免地漏掉了關鍵的實作微觀細節 。唯有 SWE-Pruner 成功在「結構保存」與「雜訊過濾」之間找到了完美的帕雷托最佳解（Pareto Optimum），以最低的資源消耗（0.670M）斬獲了全場最高的成功率（64.0%） 。",[112,9665,9666],{},"而在針對程式碼問答的 SWE-QA 測試中，SWE-Pruner 搭載於 OpenHands 框架上，橫跨 Streamlink、Reflex 與 Conan 等三大相異生態的儲存庫，同樣取得了 29% 至 54% 的驚人詞塊縮減，且模型在 LLM-as-a-Judge 的五維度品質評分上未見任何明顯衰退，再次證實了該架構極強的泛化生存能力 。",[5119,9668,9669],{"id":9669},"單輪理解任務下的極限壓縮邊界探索",[112,9671,9672],{},"儘管 SWE-Pruner 的誕生是為了解決多輪智能體的互動瓶頸，但其核心的目標感知與行級過濾機制，在處理傳統的單輪長上下文理解任務時，同樣展現出了降維打擊般的統治力 。研究團隊選擇了 Long Code QA（應對長達 100 萬詞塊的超大代碼庫問答）與 Long Code Completion 兩項極端基準，並設定了 4 倍與 8 倍的嚴格壓縮約束條件進行壓力測試 。",[112,9674,9675],{},"在使用 Qwen2.5-Coder-7B-Instruct 進行評估的 8 倍壓縮極端環境下，SWE-Pruner 的表現令人矚目：\n在 Long Code QA 任務中，當多數基線模型因上下文坍縮而導致準確率大跌時，SWE-Pruner 憑藉精準的目標鎖定，實際達成了 14.84 倍的超高壓縮率，並將準確率死守在 58.71% 的高位。作為強烈對比，依賴純量檢索的 RAG 方法僅能勉強維持 55.86%（且壓縮率不足 7 倍），而強調結構壓縮的 LongCodeZip 準確率更跌至 54.95% 。\n在 Long Code Completion 任務中，SWE-Pruner 在 10.92 倍的實際壓縮比下，依舊取得了 57.58 的編輯相似度（ES）與 31.0 的精確匹配率（EM）。反觀基於資訊熵的 Selective-Context 方法，其 ES 分數在此約束下直接雪崩至 48.67，宣告了該方法在長代碼場景下的實用性破產 。",[4848,9677,9678,9708],{},[4851,9679,9680],{},[4854,9681,9682,9685,9688,9691,9694,9697,9700,9703],{},[4857,9683,9684],{},"測試任務",[4857,9686,9687],{},"壓縮約束",[4857,9689,9690],{},"指標",[4857,9692,9693],{},"Selective-Context",[4857,9695,9696],{},"LLMLingua2",[4857,9698,9699],{},"RAG",[4857,9701,9702],{},"LongCodeZip",[4857,9704,9705],{},[138,9706,9707],{},"SWE-Pruner",[4864,9709,9710,9740,9766,9795],{},[4854,9711,9712,9717,9720,9723,9726,9729,9732,9735],{},[4869,9713,9714],{},[138,9715,9716],{},"Long Code Completion",[4869,9718,9719],{},"4x",[4869,9721,9722],{},"有效壓縮率",[4869,9724,9725],{},"3.27x",[4869,9727,9728],{},"3.32x",[4869,9730,9731],{},"3.29x",[4869,9733,9734],{},"2.77x",[4869,9736,9737],{},[138,9738,9739],{},"5.56x",[4854,9741,9742,9744,9746,9749,9752,9755,9758,9761],{},[4869,9743],{},[4869,9745],{},[4869,9747,9748],{},"ES \u002F EM",[4869,9750,9751],{},"52.48 \u002F 22.0",[4869,9753,9754],{},"49.47 \u002F 15.5",[4869,9756,9757],{},"58.97 \u002F 30.5",[4869,9759,9760],{},"57.77 \u002F 28.0",[4869,9762,9763],{},[138,9764,9765],{},"58.63 \u002F 31.5",[4854,9767,9768,9773,9776,9778,9781,9784,9787,9790],{},[4869,9769,9770],{},[138,9771,9772],{},"Long Code QA",[4869,9774,9775],{},"8x",[4869,9777,9722],{},[4869,9779,9780],{},"3.69x",[4869,9782,9783],{},"3.57x",[4869,9785,9786],{},"3.06x",[4869,9788,9789],{},"3.98x",[4869,9791,9792],{},[138,9793,9794],{},"14.84x",[4854,9796,9797,9799,9801,9804,9807,9810,9813,9816],{},[4869,9798],{},[4869,9800],{},[4869,9802,9803],{},"準確率 (Acc)",[4869,9805,9806],{},"55.36%",[4869,9808,9809],{},"51.79%",[4869,9811,9812],{},"55.86%",[4869,9814,9815],{},"54.95%",[4869,9817,9818],{},[138,9819,9820],{},"58.71%",[112,9822,9823],{},"為了進一步確認此現象並非 Qwen 模型家族的特異功能，團隊更換為架構不同的 Seed-Coder-8B-Instruct 進行交叉驗證 。數據顯示，SWE-Pruner 在 8 倍約束的 Long Code QA 任務中依然斬獲 14.68 倍壓縮與 55.75% 的最高準確率，遙遙領先群雄。這項獨立驗證堅實地證明了：結合任務感知與行級保留的修剪範式，是一種跨越模型底層架構的通用型普適真理 。",[5119,9825,9826],{"id":9826},"系統效率與句法結構保留的底層機制解密",[112,9828,9829],{},"任何被標榜為「中介軟體」的增強模組，若其自身引入的運算開銷超過了其節省的成本，便只是一場理論上的空中樓閣。為了檢驗 SWE-Pruner 的工程落地可行性，研究深入剖析了其系統延遲與代碼結構維護能力。",[5454,9831,9833],{"id":9832},"首字節延遲ttft與運算經濟學","首字節延遲（TTFT）與運算經濟學",[112,9835,9836],{},"在模型推論的生命週期中，首字節延遲（Time To First Token）是決定系統響應靈敏度的核心指標。實驗針對不同長度的輸入序列，測量了 SWE-Pruner 與多個常規生成式大模型的 TTFT 。",[4848,9838,9839,9863],{},[4851,9840,9841],{},[4854,9842,9843,9846,9849,9852,9855,9858],{},[4857,9844,9845],{},"序列長度 (Tokens)",[4857,9847,9848],{},"Qwen3-0.6B",[4857,9850,9851],{},"Qwen3-4B",[4857,9853,9854],{},"Qwen3-14B",[4857,9856,9857],{},"Qwen3-32B",[4857,9859,9860],{},[138,9861,9862],{},"SWE-Pruner (0.6B)",[4864,9864,9865,9887,9909],{},[4854,9866,9867,9870,9873,9876,9879,9882],{},[4869,9868,9869],{},"512",[4869,9871,9872],{},"32.00 ms",[4869,9874,9875],{},"104.93 ms",[4869,9877,9878],{},"143.65 ms",[4869,9880,9881],{},"84.01 ms",[4869,9883,9884],{},[138,9885,9886],{},"42.05 ms",[4854,9888,9889,9892,9895,9898,9901,9904],{},[4869,9890,9891],{},"2048",[4869,9893,9894],{},"29.64 ms",[4869,9896,9897],{},"99.10 ms",[4869,9899,9900],{},"129.97 ms",[4869,9902,9903],{},"274.22 ms",[4869,9905,9906],{},[138,9907,9908],{},"49.05 ms",[4854,9910,9911,9914,9917,9920,9923,9926],{},[4869,9912,9913],{},"8192",[4869,9915,9916],{},"76.73 ms",[4869,9918,9919],{},"241.97 ms",[4869,9921,9922],{},"529.45 ms",[4869,9924,9925],{},"1188.67 ms",[4869,9927,9928],{},[138,9929,9930],{},"102.00 ms",[112,9932,9933],{},"數據展示了 SWE-Pruner 極其優越的次線性（Sublinear）延展特性。在處理高達 8192 個詞塊的龐大檔案時，SWE-Pruner 的耗時僅需 102.00 毫秒，這速度是 14B 模型的 5 倍以上，更是 32B 模型（需 1188.67 毫秒）的 11 倍之快 。從企業運算的經濟學視角切入，呼叫如 Claude 4.5 這樣的高階閉源模型，單次 API 的網路往返與運算往往耗時數秒；SWE-Pruner 區區不到 0.1 秒的「攔截檢查費」，在其創造的 23%~54% 詞塊縮減與高達 20% 的整體推論時間節省面前，其投資報酬率無疑是壓倒性的優勢 。",[5454,9935,9937],{"id":9936},"條件隨機場與-ast-完整性的物理關聯","條件隨機場與 AST 完整性的物理關聯",[112,9939,9940,9941,9943],{},"除了速度，壓縮後的「代碼品質」更是決定生死存亡的關鍵。研究團隊運用 ",[118,9942,5584],{}," 工具對各種方法壓縮後的代碼進行了嚴格的抽象語法樹（AST）正確性測試 。",[112,9945,9946],{},"結果令人咋舌：依賴機率閾值的詞塊級修剪方法 LLMLingua2 與 Selective Context，其 AST 存活率僅剩淒慘的 0.29% 與 12.4% 。這解釋了為何它們在除錯任務中屢屢讓模型崩潰。然而，SWE-Pruner 在基於 Function RAG 的基礎上進行二次深度修剪後，依然維繫了高達 87.3% 的 AST 正確性 。",[112,9948,9949,9950,9952,9953,9956],{},"這種近乎奇蹟般的結構保存能力，正是源自於其架構中條件隨機場（CRF）的物理特徵。CRF 的轉移矩陣在數萬筆黃金標準資料的訓練下，深刻內化了程式語言的拓樸定律——當模型決定保留某一個核心的賦值語句時，CRF 會透過轉移機率的鏈式傳導，自動賦予該語句所屬的函式宣告行、",[118,9951,8098],{}," 判斷式以及關鍵的閉合右括號 ",[118,9954,9955],{},"}"," 極高的連帶保留權重 。正是這種「牽一髮而動全身」的結構共振設計，讓 SWE-Pruner 得以在剔除大量雜訊的同時，完美捍衛程式碼的神聖句法骨架 。",[5119,9958,9960],{"id":9959},"認知軌跡重塑swe-bench-案例深度還原","認知軌跡重塑：SWE-Bench 案例深度還原",[112,9962,9963],{},"為了將抽象的量化指標轉化為具體的認知行為變化，報告擷取了兩個極具代表性的 SWE-Bench 任務軌跡，生動展示了上下文修剪如何徹底重塑智能體的解題行為學。",[5454,9965,9967],{"id":9966},"案例一從災難性溢位到精準狙擊高影響力場景","案例一：從災難性溢位到精準狙擊（高影響力場景）",[112,9969,9970,9971,9974,9975,9978],{},"在處理 ",[118,9972,9973],{},"django__django-10554"," 任務時，目標是修復一個關於 ",[118,9976,9977],{},"Query.clone()"," 方法中缺失深拷貝的幽微邏輯缺陷 。",[112,9980,9981,9982,9984,9985,9988],{},"在沒有修剪機制的 Baseline 設置下，智能體陷入了嚴重的「資訊恐慌」。它瘋狂呼叫了 59 次讀取與 39 次搜尋指令，試圖用 ",[118,9983,6448],{}," 與片段式的 ",[118,9986,9987],{},"sed"," 指令在大海中撈針。這種無目標的廣度優先探索，導致歷史紀錄中塞滿了支離破碎且互不相關的代碼片段。最終，在其經歷了 164 步的痛苦掙扎後，峰值提示詞長度飆破 87,790 詞塊，總消耗超過 700 萬詞塊，系統因超過資源上限而宣告任務失敗 。",[112,9990,9991,9992,9994,9995,9998,9999,10002],{},"相反地，配備了 SWE-Pruner 的智能體展現了宛如資深工程師般的冷靜與俐落。它僅透過 20 次讀取與 10 次搜尋，並巧妙利用 ",[118,9993,6526],{}," 參數指示系統：「篩選出與 Query 類別中 clone 邏輯相關的部分」。藉助修剪器提供的乾淨視野，智能體迅速鎖定了核心檔案 ",[118,9996,9997],{},"query.py"," 中 ",[118,10000,10001],{},"if self.combinator:"," 的判斷分支。最終，它僅用 56 步便成功結案，總詞塊消耗驟降 83.3%（僅 1.17M tokens） 。這揭示了修剪機制不僅是節約工具，更是將「註定失敗的資源耗竭」扭轉為「成功修復」的決定性力量。",[5454,10004,10006],{"id":10005},"案例二降低認知不確定性結構性效率紅利","案例二：降低認知不確定性（結構性效率紅利）",[112,10008,10009,10010,10013],{},"在另一個任務 ",[118,10011,10012],{},"django__django-11740","（添加外鍵依賴追蹤邏輯）中，Baseline 與 Pruner 智能體最終皆成功解題，但其內部的心智狀態卻大相逕庭 。",[112,10015,10016,10017,10020],{},"Baseline 智能體雖然在 42 步內完工，但在這過程中，由於其面對的是大量未經整理的原始碼，模型展現出了極高的「認知不確定性」（Cognitive Uncertainty）。它不僅反覆讀取相同檔案的不同片段，甚至還耗費了額外的步驟去建立如 ",[118,10018,10019],{},"\u002Ftmp\u002Ftest_uuid_to_fk3.py"," 這樣的臨時驗證腳本，試圖透過執行副程式來確認自己對混亂上下文的理解是否正確 。",[112,10022,10023,10024,10027],{},"而 Pruner 智能體雖然多走了幾步（48 步），但因為它每次讀取得到的都是高度聚焦的核心邏輯，模型充滿自信地直接對 ",[118,10025,10026],{},"autodetector.py"," 進行了精準編輯，完全省略了建立防禦性驗證腳本的冗餘行為。最終，其峰值提示詞長度減少了 30.2%，整體詞塊消耗也更為精簡 。這深刻表明：給予模型更少但更純粹的資訊，能有效激發其推理自信，促成更直接且具備經濟效益的干預行為。",[5119,10029,10030],{"id":10030},"智能體提示詞工程與系統整合哲學",[112,10032,10033],{},"SWE-Pruner 能夠順暢運作，其背後隱藏著一套精心設計的提示詞工程（Prompt Engineering）範本。這些範本在系統底層重塑了智能體與環境對話的標準協議 。",[112,10035,10036,10037,10040,10041,10043,10044,10047],{},"在 Mini SWE Agent 的系統設定中，框架強制規定智能體的每次輸出必須嚴格遵守雙段式結構：首先是闡述內部推理邏輯的 ",[118,10038,10039],{},"THOUGHT"," 區塊，接著才是包含 ",[118,10042,6526],{}," 的 ",[118,10045,10046],{},"bash"," 執行區塊 。系統範本中更是清晰地定義了過濾意圖的邊界：\n優秀的提問必須是自包含且具備強烈業務意圖的，例如「某個功能在代碼中是如何實作的？」；而諸如「載入函數」或「讀取 50 到 100 行」這類包含實體檔案資訊或過於模糊的短語，則被嚴格禁止 。",[112,10049,10050],{},"這種設計哲學實作了「意圖」與「操作」的完美解耦：智能體負責思考「我要找什麼邏輯」（Cognitive Chain），而底層的神經略讀器則負責執行「在實體檔案中將這段邏輯完整無損地抽離出來」（Execution Chain）。同時，為了確保資料集訓練出的略讀器具備高度的實戰價值，團隊在品質評估提示詞（Quality Evaluation Prompt）中明確告誡裁判模型：評估修剪成敗的最高指導原則在於「語義與結構的留存」，即使移除了大量代碼，只要剩餘的邏輯具備獨立可讀性與括號匹配性，即應賦予高分 。這些隱含在提示詞深處的工程巧思，共同構建了 SWE-Pruner 無懈可擊的運作閉環。",[5119,10052,10053],{"id":10053},"系統局限性探討與未來研究的戰略方向",[112,10055,10056],{},"儘管 SWE-Pruner 展現了顛覆性的效能躍升，本報告秉持客觀嚴謹之學術立場，深入剖析該系統目前面臨的理論局限與未來優化空間：",[112,10058,10059],{},"首先是跨生態系統的語言泛化挑戰。當前的實作與大規模實證測試，主要集中於 Python 語言構建的開源專案。雖然基於 CRF 的行級修剪機制在數學原理上並不依賴特定語言的保留字或語法樹結構，但若要全面平移至具有極度複雜泛型（Generics）與巨集（Macros）展開機制的強型別語言（如 C++ 或 Rust），神經網絡對底層記憶體管理語義的感知能力仍需透過擴充多語種訓練語料庫來進一步鍛鍊 。",[112,10061,10062],{},"其次是極致低延遲場景的微觀開銷。在非同步除錯或代理後台運作中，100 毫秒的處理延遲微不足道；但若將 SWE-Pruner 直接佈署於開發者整合開發環境（IDE）的實時即打即補全（Type-and-complete）場景中，這微小的延遲仍可能造成使用者的感知頓挫。未來的研究可透過深度的知識蒸餾（Knowledge Distillation）技術，或引入基於信賴度的提早退出（Early-exit）動態推理網路，將 0.6B 的參數規模進一步極致壓縮至 0.1B 等級 。",[112,10064,10065],{},"最後，是與時間序列維度壓縮技術的終極融合。SWE-Pruner 作為一個中介層，其核心任務是解決「空間維度」的資訊爆炸，即單次讀取龐大實體檔案的雜訊過濾。然而，這項技術與專注於「時間維度」壓縮（即折疊智能體過去冗長的思考軌跡）的框架（如 AgentFold 等）並無衝突，反而處於完美的正交互補關係。未來的終極架構，必然是一個融合了「空間精剪」與「時間折疊」的雙核驅動系統，屆時，LLM 程式碼智能體將徹底擺脫記憶容量的物理桎梏，邁向真正的全知與自治 。",[5119,10067,10068],{"id":10068},"總結",[112,10070,10071],{},"本分析報告對 SWE-Pruner 框架進行了全方位、深層次的技術解構與機制探討。在解決大型語言模型處理軟體工程任務時面臨的「上下文之牆」挑戰中，SWE-Pruner 以無可辯駁的數據與嚴密的數學邏輯證明了一個核心真理：單純依賴擴展基底模型的上下文視窗（Context Window），不僅是不具經濟效益的粗暴做法，更是放任認知雜訊氾濫的錯誤方向 。",[112,10073,10074],{},"真正的破局之道，在於賦予智能體如同人類資深開發者般，具備強烈「目標導向」與「選擇性略讀」的過濾直覺。透過巧妙地將 0.6B 輕量級特徵融合神經網絡、基於條件隨機場（CRF）的行級結構感知算法，以及與現有開發工具鏈無縫接軌的意圖提示機制相結合，SWE-Pruner 成功地在複雜多變的軟體庫中殺出一條血路 。",[112,10076,10077],{},"它不僅在極端測試中為模型削減了高達 54% 的 API 計費詞塊開銷，顯著遏制了無效探勘帶來的輪數虛耗，更在極度嚴苛的壓縮壓力下，完美捍衛了程式語言神聖不可侵犯的語法結構骨架 。SWE-Pruner 的誕生，不僅宣告了早期靜態、盲目且破壞性的詞塊級上下文壓縮時代的全面終結，更為未來打造具備極高經濟效益、認知專注力與結構嚴謹性的次世代自主軟體開發智能體，樹立了一座難以逾越的工程與理論標竿。",{"title":311,"searchDepth":312,"depth":312,"links":10079},[10080],{"id":6416,"depth":312,"text":6406,"children":10081},[10082,10083,10084,10085,10086,10087,10088,10089,10090,10091,10092,10093],{"id":6419,"depth":4984,"text":6422},{"id":5713,"depth":4984,"text":5713},{"id":5727,"depth":4984,"text":5727},{"id":6499,"depth":4984,"text":6500},{"id":9396,"depth":4984,"text":9396},{"id":9430,"depth":4984,"text":9430},{"id":9669,"depth":4984,"text":9669},{"id":9826,"depth":4984,"text":9826},{"id":9959,"depth":4984,"text":9960},{"id":10030,"depth":4984,"text":10030},{"id":10053,"depth":4984,"text":10053},{"id":10068,"depth":4984,"text":10068},"2026-02-25","在當代軟體工程的演進歷程中，大型語言模型（Large Language Models, LLMs）的應用已經跨越了單純的程式碼片段補全與靜態語義分析，正式邁入能夠自主導航龐大程式碼庫",{},"\u002Fblog\u002Fzh-tw\u002Farxiv-2601-16746v2",{"title":6406,"description":10095},"arxiv-2601-16746v2","blog\u002Fzh-TW\u002Farxiv-2601-16746v2",[5424,6401,5692],"EEjZKo9sgBqwgnfYO1N5LIg38YnIidezuMLcm7r1NqY",{"id":10104,"title":10105,"author":103,"body":10106,"cover":15734,"coverAlt":10105,"date":15735,"description":15736,"draft":324,"extension":325,"featured":324,"lang":327,"meta":15737,"navigation":326,"path":15738,"seo":15739,"slug":15740,"stem":15741,"tags":15742,"translationKey":15740,"updated":337,"__hash__":15744},"blog\u002Fblog\u002Fzh-TW\u002Farxiv-2512-22471v2.md","Transformer 注意力機制的貝葉斯幾何學：機制與理論深度分析報告",{"type":105,"value":10107,"toc":15701},[10108,10114,10117,10120,10123,10129,10135,10138,10141,10297,10303,10310,10381,10384,10620,10623,10653,10656,10974,10977,11190,11248,11251,11643,11650,11656,11667,11673,11676,12124,12370,12373,12652,12658,12661,12944,13390,13393,13471,13601,13607,13610,13872,13878,13881,13887,13890,14114,14120,14123,14666,14672,14675,14699,14702,15012,15018,15024,15035,15041,15044,15064,15070,15073,15146,15152,15155,15175,15181,15184,15190,15193,15351,15357,15438,15444,15447,15453,15456,15476,15482,15488,15518,15524,15531,15549,15555,15558,15648,15654,15660,15675,15678,15681,15685],[108,10109,10111],{"id":10110},"_1-執行摘要",[138,10112,10113],{},"1. 執行摘要",[112,10115,10116],{},"在當前的人工智慧領域中，Transformer 架構已成為大型語言模型（LLMs）的基石。然而，關於其強大的上下文學習（In-Context Learning）能力背後的確切運作機制，學術界長期以來存在著激烈的爭論。核心問題在於：這些模型究竟是僅僅進行高維度的模式匹配與記憶，還是在演算法層面上真正實現了機率推理？由 Naman Agarwal、Siddhartha R. Dalal 和 Vishal Misra 於 2026 年發表的開創性研究《Transformer 注意力機制的貝葉斯幾何學》（The Bayesian Geometry of Transformer Attention）為此提供了具體的實證解答。",[112,10118,10119],{},"本報告旨在對該研究進行詳盡的深度分析與總結。研究團隊透過構建「貝葉斯風洞」（Bayesian Wind Tunnels）——一種具備解析解且無法通過記憶解決的受控實驗環境——證明了小型 Transformer 模型不僅能夠模擬貝葉斯推理，更能以機器級的精度（誤差低至10^{-3} 到 10^{-4} bits）追蹤確切的貝葉斯後驗分佈。與此形成鮮明對比的是，參數規模相當的多層感知機（MLP）在相同任務中表現出災難性的失敗，這確立了注意力機制在動態推理中的不可或缺性。",[112,10121,10122],{},"進一步的幾何診斷揭示了 Transformer 實現此一過程的三階段機制：首先在底層構建正交的假設框架（Hypothesis Frame），接著在中層透過注意力機制進行順序性的假設消除（Sequential Elimination），最後在頂層將後驗機率精確編碼於低維度的值流形（Value Manifold）上。這些發現不僅解釋了 Transformer 的成功原因，更為提示工程（Prompt Engineering）提供了理論依據——即提示應被視為消除假設空間的證據。",[108,10124,10126],{"id":10125},"_2-理論背景與研究動機",[138,10127,10128],{},"2. 理論背景與研究動機",[5119,10130,10132],{"id":10131},"_21-大型語言模型的驗證危機",[138,10133,10134],{},"2.1 大型語言模型的驗證危機",[112,10136,10137],{},"隨著大型語言模型（如 GPT-4, Claude 等）展現出驚人的推理能力，一個根本性的科學難題隨之浮現：我們無法驗證這些模型是否真的在「推理」。自然語言數據本質上缺乏「基本真理」（Ground Truth）的後驗機率分佈。例如，在一段文本續寫任務中，下一個詞的真實機率分佈是未知的，且往往受到多種潛在因素的影響。此外，現代 LLM 的訓練數據量極其龐大，使得研究人員難以區分模型是在進行真正的即時機率計算，還是僅僅在提取海量訓練數據中記憶的片段。",[112,10139,10140],{},"這種「驗證危機」導致了對模型能力的解釋往往停留在定性描述或擬人化的比喻上，缺乏嚴謹的定量評估。為了突破這一局限，Agarwal 等人提出了「貝葉斯風洞」的概念。這是一種合成的實驗環境，設計上必須滿足三個嚴格條件：",[5193,10142,10143,10149,10291],{},[275,10144,10145,10148],{},[138,10146,10147],{},"解析可解性（Analytic Tractability）："," 在序列的每一步，真實的貝葉斯後驗分佈必須是已知且可精確計算的閉式解（Closed-form）。",[275,10150,10151,10154,10155,10223,10224,10290],{},[138,10152,10153],{},"抗記憶性（Memorization Resistance）："," 假設空間必須大到組合爆炸的程度（例如 ",[360,10156,10158,10178],{"className":10157},[363],[360,10159,10161],{"className":10160},[367],[369,10162,10163],{"xmlns":371},[373,10164,10165,10175],{},[376,10166,10167],{},[387,10168,10169,10172],{},[1185,10170,10171],{},"10",[1185,10173,10174],{},"18",[405,10176,10177],{"encoding":407},"10^{18}",[360,10179,10181],{"className":10180,"ariaHidden":413},[412],[360,10182,10184,10188,10191],{"className":10183},[417],[360,10185],{"className":10186,"style":10187},[421],"height:0.8141em;",[360,10189,1187],{"className":10190},[426],[360,10192,10194,10197],{"className":10193},[426],[360,10195,4052],{"className":10196},[426],[360,10198,10200],{"className":10199},[457],[360,10201,10203],{"className":10202},[461],[360,10204,10206],{"className":10205},[465],[360,10207,10209],{"className":10208,"style":10187},[469],[360,10210,10211,10214],{"style":472},[360,10212],{"className":10213,"style":477},[476],[360,10215,10217],{"className":10216},[481,482,483,484],[360,10218,10220],{"className":10219},[426,484],[360,10221,10174],{"className":10222},[426,484]," 或 ",[360,10225,10227,10246],{"className":10226},[363],[360,10228,10230],{"className":10229},[367],[369,10231,10232],{"xmlns":371},[373,10233,10234,10243],{},[376,10235,10236],{},[387,10237,10238,10240],{},[1185,10239,10171],{},[1185,10241,10242],{},"40",[405,10244,10245],{"encoding":407},"10^{40}",[360,10247,10249],{"className":10248,"ariaHidden":413},[412],[360,10250,10252,10255,10258],{"className":10251},[417],[360,10253],{"className":10254,"style":10187},[421],[360,10256,1187],{"className":10257},[426],[360,10259,10261,10264],{"className":10260},[426],[360,10262,4052],{"className":10263},[426],[360,10265,10267],{"className":10266},[457],[360,10268,10270],{"className":10269},[461],[360,10271,10273],{"className":10272},[465],[360,10274,10276],{"className":10275,"style":10187},[469],[360,10277,10278,10281],{"style":472},[360,10279],{"className":10280,"style":477},[476],[360,10282,10284],{"className":10283},[481,482,483,484],[360,10285,10287],{"className":10286},[426,484],[360,10288,10242],{"className":10289},[426,484]," 種可能性），從而數學上排除模型通過背誦訓練樣本來解決任務的可能性。",[275,10292,10293,10296],{},[138,10294,10295],{},"推理必要性（Inference Necessity）："," 任務必須要求模型根據上下文中的新證據動態更新信念，這正是貝葉斯推理的定義。",[5119,10298,10300],{"id":10299},"_22-交叉熵與貝葉斯推理的數學對偶性",[138,10301,10302],{},"2.2 交叉熵與貝葉斯推理的數學對偶性",[112,10304,10305,10306,10309],{},"本研究的理論基礎建立在深度學習訓練目標與機率論之間的深層聯繫上。研究團隊提出了",[138,10307,10308],{},"定理 1：交叉熵的群體最優解","（Population Optimum of Cross-Entropy）。",[112,10311,10312,10313],{},"設想一個由潛在參數\n",[360,10314,10316,10342],{"className":10315},[363],[360,10317,10319],{"className":10318},[367],[369,10320,10321],{"xmlns":371},[373,10322,10323,10339],{},[376,10324,10325,10327,10330,10333,10335,10337],{},[379,10326,6596],{},[383,10328,10329],{},"∼",[379,10331,10332],{},"π",[383,10334,779],{"stretchy":778},[379,10336,6596],{},[383,10338,793],{"stretchy":778},[405,10340,10341],{"encoding":407},"\\theta \\sim \\pi(\\theta)",[360,10343,10345,10363],{"className":10344,"ariaHidden":413},[412],[360,10346,10348,10351,10354,10357,10360],{"className":10347},[417],[360,10349],{"className":10350,"style":9324},[421],[360,10352,6596],{"className":10353,"style":3944},[426,427],[360,10355],{"className":10356,"style":432},[431],[360,10358,10329],{"className":10359},[436],[360,10361],{"className":10362,"style":432},[431],[360,10364,10366,10369,10372,10375,10378],{"className":10365},[417],[360,10367],{"className":10368,"style":848},[421],[360,10370,10332],{"className":10371,"style":736},[426,427],[360,10373,779],{"className":10374},[860],[360,10376,6596],{"className":10377,"style":3944},[426,427],[360,10379,793],{"className":10380},[887],[112,10382,10383],{},"索引的任務族。對於每個任務，模型觀察到上下文",[112,10385,10386],{},[360,10387,10389,10437],{"className":10388},[363],[360,10390,10392],{"className":10391},[367],[369,10393,10394],{"xmlns":371},[373,10395,10396,10434],{},[376,10397,10398,10400,10402],{},[379,10399,7219],{},[383,10401,796],{},[1206,10403,10404,10424,10432],{},[376,10405,10406,10408,10414,10416,10422],{},[383,10407,779],{"stretchy":778},[628,10409,10410,10412],{},[379,10411,6582],{},[379,10413,1126],{},[383,10415,784],{"separator":413},[628,10417,10418,10420],{},[379,10419,7236],{},[379,10421,1126],{},[383,10423,793],{"stretchy":778},[376,10425,10426,10428,10430],{},[379,10427,1126],{},[383,10429,796],{},[1185,10431,1187],{},[379,10433,1219],{},[405,10435,10436],{"encoding":407},"c={(x_i, y_i)}_{i=1}^k",[360,10438,10440,10458],{"className":10439,"ariaHidden":413},[412],[360,10441,10443,10446,10449,10452,10455],{"className":10442},[417],[360,10444],{"className":10445,"style":3107},[421],[360,10447,7219],{"className":10448},[426,427],[360,10450],{"className":10451,"style":432},[431],[360,10453,796],{"className":10454},[436],[360,10456],{"className":10457,"style":432},[431],[360,10459,10461,10465],{"className":10460},[417],[360,10462],{"className":10463,"style":10464},[421],"height:1.2887em;vertical-align:-0.2997em;",[360,10466,10468,10563],{"className":10467},[426],[360,10469,10471,10474,10514,10517,10520,10560],{"className":10470},[426],[360,10472,779],{"className":10473},[860],[360,10475,10477,10480],{"className":10476},[426],[360,10478,6582],{"className":10479},[426,427],[360,10481,10483],{"className":10482},[457],[360,10484,10486,10506],{"className":10485},[461,713],[360,10487,10489,10503],{"className":10488},[465],[360,10490,10492],{"className":10491,"style":1317},[469],[360,10493,10494,10497],{"style":1320},[360,10495],{"className":10496,"style":477},[476],[360,10498,10500],{"className":10499},[481,482,483,484],[360,10501,1126],{"className":10502},[426,427,484],[360,10504,741],{"className":10505},[740],[360,10507,10509],{"className":10508},[465],[360,10510,10512],{"className":10511,"style":1339},[469],[360,10513],{},[360,10515,784],{"className":10516},[867],[360,10518],{"className":10519,"style":871},[431],[360,10521,10523,10526],{"className":10522},[426],[360,10524,7236],{"className":10525,"style":736},[426,427],[360,10527,10529],{"className":10528},[457],[360,10530,10532,10552],{"className":10531},[461,713],[360,10533,10535,10549],{"className":10534},[465],[360,10536,10538],{"className":10537,"style":1317},[469],[360,10539,10540,10543],{"style":2044},[360,10541],{"className":10542,"style":477},[476],[360,10544,10546],{"className":10545},[481,482,483,484],[360,10547,1126],{"className":10548},[426,427,484],[360,10550,741],{"className":10551},[740],[360,10553,10555],{"className":10554},[465],[360,10556,10558],{"className":10557,"style":1339},[469],[360,10559],{},[360,10561,793],{"className":10562},[887],[360,10564,10566],{"className":10565},[457],[360,10567,10569,10611],{"className":10568},[461,713],[360,10570,10572,10608],{"className":10571},[465],[360,10573,10576,10597],{"className":10574,"style":10575},[469],"height:0.989em;",[360,10577,10579,10582],{"style":10578},"top:-2.4003em;margin-right:0.05em;",[360,10580],{"className":10581,"style":477},[476],[360,10583,10585],{"className":10584},[481,482,483,484],[360,10586,10588,10591,10594],{"className":10587},[426,484],[360,10589,1126],{"className":10590},[426,427,484],[360,10592,796],{"className":10593},[436,484],[360,10595,1187],{"className":10596},[426,484],[360,10598,10599,10602],{"style":1433},[360,10600],{"className":10601,"style":477},[476],[360,10603,10605],{"className":10604},[481,482,483,484],[360,10606,1219],{"className":10607,"style":1645},[426,427,484],[360,10609,741],{"className":10610},[740],[360,10612,10614],{"className":10613},[465],[360,10615,10618],{"className":10616,"style":10617},[469],"height:0.2997em;",[360,10619],{},[112,10621,10622],{},"並需要預測新的標籤",[112,10624,10625],{},[360,10626,10628,10641],{"className":10627},[363],[360,10629,10631],{"className":10630},[367],[369,10632,10633],{"xmlns":371},[373,10634,10635,10639],{},[376,10636,10637],{},[379,10638,7236],{},[405,10640,7236],{"encoding":407},[360,10642,10644],{"className":10643,"ariaHidden":413},[412],[360,10645,10647,10650],{"className":10646},[417],[360,10648],{"className":10649,"style":2189},[421],[360,10651,7236],{"className":10652,"style":736},[426,427],[112,10654,10655],{},"訓練目標是最小化預期交叉熵損失：",[360,10657,10659],{"className":10658},[755],[360,10660,10662,10759],{"className":10661},[363],[360,10663,10665],{"className":10664},[367],[369,10666,10667],{"xmlns":371,"display":764},[373,10668,10669,10756],{},[376,10670,10671,10673,10675,10677,10679,10681,10694,10728,10731,10733,10735,10737,10739,10741,10743,10745,10747,10749,10751,10753],{},[379,10672,8453],{"mathvariant":6570},[383,10674,779],{"stretchy":778},[379,10676,1210],{},[383,10678,793],{"stretchy":778},[383,10680,796],{},[628,10682,10683,10686],{},[379,10684,10685],{"mathvariant":391},"E",[376,10687,10688,10690,10692],{},[379,10689,6596],{},[383,10691,10329],{},[379,10693,10332],{},[628,10695,10696,10698],{},[379,10697,10685],{"mathvariant":391},[376,10699,10700,10702,10704,10706,10708,10710,10712,10714,10716,10718,10720,10722,10724,10726],{},[379,10701,7219],{},[383,10703,784],{"separator":413},[383,10705,779],{"stretchy":778},[379,10707,6582],{},[383,10709,784],{"separator":413},[379,10711,7236],{},[383,10713,793],{"stretchy":778},[383,10715,10329],{},[379,10717,112],{},[383,10719,779],{"stretchy":778},[383,10721,9081],{},[379,10723,6587],{"mathvariant":771},[379,10725,6596],{},[383,10727,793],{"stretchy":778},[383,10729,10730],{"stretchy":778},"[",[383,10732,7367],{},[379,10734,8121],{},[383,10736,775],{},[379,10738,1210],{},[383,10740,779],{"stretchy":778},[379,10742,7236],{},[379,10744,6587],{"mathvariant":771},[379,10746,6582],{},[383,10748,784],{"separator":413},[379,10750,7219],{},[383,10752,793],{"stretchy":778},[383,10754,10755],{"stretchy":778},"]",[405,10757,10758],{"encoding":407},"\\mathcal{L}(q)=\\mathbb{E}_{\\theta \\sim \\pi} \\mathbb{E}_{c, (x,y) \\sim p(\\cdot|\\theta)} [-\\log q(y|x,c)]",[360,10760,10762,10789],{"className":10761,"ariaHidden":413},[412],[360,10763,10765,10768,10771,10774,10777,10780,10783,10786],{"className":10764},[417],[360,10766],{"className":10767,"style":848},[421],[360,10769,8453],{"className":10770},[426,6672],[360,10772,779],{"className":10773},[860],[360,10775,1210],{"className":10776,"style":736},[426,427],[360,10778,793],{"className":10779},[887],[360,10781],{"className":10782,"style":432},[431],[360,10784,796],{"className":10785},[436],[360,10787],{"className":10788,"style":432},[431],[360,10790,10792,10796,10845,10929,10932,10935,10938,10943,10946,10949,10952,10955,10958,10961,10964,10967,10970],{"className":10791},[417],[360,10793],{"className":10794,"style":10795},[421],"height:1.1052em;vertical-align:-0.3552em;",[360,10797,10799,10802],{"className":10798},[426],[360,10800,10685],{"className":10801},[426,453],[360,10803,10805],{"className":10804},[457],[360,10806,10808,10837],{"className":10807},[461,713],[360,10809,10811,10834],{"className":10810},[465],[360,10812,10814],{"className":10813,"style":4928},[469],[360,10815,10816,10819],{"style":1320},[360,10817],{"className":10818,"style":477},[476],[360,10820,10822],{"className":10821},[481,482,483,484],[360,10823,10825,10828,10831],{"className":10824},[426,484],[360,10826,6596],{"className":10827,"style":3944},[426,427,484],[360,10829,10329],{"className":10830},[436,484],[360,10832,10332],{"className":10833,"style":736},[426,427,484],[360,10835,741],{"className":10836},[740],[360,10838,10840],{"className":10839},[465],[360,10841,10843],{"className":10842,"style":1339},[469],[360,10844],{},[360,10846,10848,10851],{"className":10847},[426],[360,10849,10685],{"className":10850},[426,453],[360,10852,10854],{"className":10853},[457],[360,10855,10857,10920],{"className":10856},[461,713],[360,10858,10860,10917],{"className":10859},[465],[360,10861,10863],{"className":10862,"style":8054},[469],[360,10864,10866,10869],{"style":10865},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[360,10867],{"className":10868,"style":477},[476],[360,10870,10872],{"className":10871},[481,482,483,484],[360,10873,10875,10878,10881,10884,10887,10890,10893,10896,10899,10902,10905,10908,10911,10914],{"className":10874},[426,484],[360,10876,7219],{"className":10877},[426,427,484],[360,10879,784],{"className":10880},[867,484],[360,10882,779],{"className":10883},[860,484],[360,10885,6582],{"className":10886},[426,427,484],[360,10888,784],{"className":10889},[867,484],[360,10891,7236],{"className":10892,"style":736},[426,427,484],[360,10894,793],{"className":10895},[887,484],[360,10897,10329],{"className":10898},[436,484],[360,10900,112],{"className":10901},[426,427,484],[360,10903,779],{"className":10904},[860,484],[360,10906,9081],{"className":10907},[426,484],[360,10909,6587],{"className":10910},[426,484],[360,10912,6596],{"className":10913,"style":3944},[426,427,484],[360,10915,793],{"className":10916},[887,484],[360,10918,741],{"className":10919},[740],[360,10921,10923],{"className":10922},[465],[360,10924,10927],{"className":10925,"style":10926},[469],"height:0.3552em;",[360,10928],{},[360,10930,10730],{"className":10931},[860],[360,10933,7367],{"className":10934},[426],[360,10936],{"className":10937,"style":871},[431],[360,10939,8208,10941],{"className":10940},[852],[360,10942,8212],{"style":8211},[360,10944],{"className":10945,"style":871},[431],[360,10947,1210],{"className":10948,"style":736},[426,427],[360,10950,779],{"className":10951},[860],[360,10953,7236],{"className":10954,"style":736},[426,427],[360,10956,6587],{"className":10957},[426],[360,10959,6582],{"className":10960},[426,427],[360,10962,784],{"className":10963},[867],[360,10965],{"className":10966,"style":871},[431],[360,10968,7219],{"className":10969},[426,427],[360,10971,10973],{"className":10972},[887],")]",[112,10975,10976],{},"該定理證明，此目標函數的唯一全局最小值即為貝葉斯後驗預測分佈（Bayesian Posterior Predictive Distribution）：",[360,10978,10980],{"className":10979},[755],[360,10981,10983,11053],{"className":10982},[363],[360,10984,10986],{"className":10985},[367],[369,10987,10988],{"xmlns":371,"display":764},[373,10989,10990,11050],{},[376,10991,10992,10999,11001,11003,11005,11007,11009,11011,11013,11015,11018,11020,11022,11024,11026,11028,11030,11032,11034,11036,11038,11040,11042,11044,11046,11048],{},[387,10993,10994,10996],{},[379,10995,1210],{},[383,10997,10998],{},"∗",[383,11000,779],{"stretchy":778},[379,11002,7236],{},[379,11004,6587],{"mathvariant":771},[379,11006,6582],{},[383,11008,784],{"separator":413},[379,11010,7219],{},[383,11012,793],{"stretchy":778},[383,11014,796],{},[383,11016,11017],{},"∫",[379,11019,112],{},[383,11021,779],{"stretchy":778},[379,11023,7236],{},[379,11025,6587],{"mathvariant":771},[379,11027,6582],{},[383,11029,784],{"separator":413},[379,11031,6596],{},[383,11033,793],{"stretchy":778},[379,11035,112],{},[383,11037,779],{"stretchy":778},[379,11039,6596],{},[379,11041,6587],{"mathvariant":771},[379,11043,7219],{},[383,11045,793],{"stretchy":778},[379,11047,403],{},[379,11049,6596],{},[405,11051,11052],{"encoding":407},"q^*(y|x,c)=\\int p(y|x,\\theta) p(\\theta|c) d\\theta",[360,11054,11056,11125],{"className":11055,"ariaHidden":413},[412],[360,11057,11059,11062,11092,11095,11098,11101,11104,11107,11110,11113,11116,11119,11122],{"className":11058},[417],[360,11060],{"className":11061,"style":848},[421],[360,11063,11065,11068],{"className":11064},[426],[360,11066,1210],{"className":11067,"style":736},[426,427],[360,11069,11071],{"className":11070},[457],[360,11072,11074],{"className":11073},[461],[360,11075,11077],{"className":11076},[465],[360,11078,11081],{"className":11079,"style":11080},[469],"height:0.7387em;",[360,11082,11083,11086],{"style":4335},[360,11084],{"className":11085,"style":477},[476],[360,11087,11089],{"className":11088},[481,482,483,484],[360,11090,10998],{"className":11091},[494,484],[360,11093,779],{"className":11094},[860],[360,11096,7236],{"className":11097,"style":736},[426,427],[360,11099,6587],{"className":11100},[426],[360,11102,6582],{"className":11103},[426,427],[360,11105,784],{"className":11106},[867],[360,11108],{"className":11109,"style":871},[431],[360,11111,7219],{"className":11112},[426,427],[360,11114,793],{"className":11115},[887],[360,11117],{"className":11118,"style":432},[431],[360,11120,796],{"className":11121},[436],[360,11123],{"className":11124,"style":432},[431],[360,11126,11128,11132,11136,11139,11142,11145,11148,11151,11154,11157,11160,11163,11166,11169,11172,11175,11178,11181,11184,11187],{"className":11127},[417],[360,11129],{"className":11130,"style":11131},[421],"height:2.2222em;vertical-align:-0.8622em;",[360,11133,11017],{"className":11134,"style":11135},[852,1393,7127],"margin-right:0.4445em;position:relative;top:-0.0011em;",[360,11137],{"className":11138,"style":871},[431],[360,11140,112],{"className":11141},[426,427],[360,11143,779],{"className":11144},[860],[360,11146,7236],{"className":11147,"style":736},[426,427],[360,11149,6587],{"className":11150},[426],[360,11152,6582],{"className":11153},[426,427],[360,11155,784],{"className":11156},[867],[360,11158],{"className":11159,"style":871},[431],[360,11161,6596],{"className":11162,"style":3944},[426,427],[360,11164,793],{"className":11165},[887],[360,11167,112],{"className":11168},[426,427],[360,11170,779],{"className":11171},[860],[360,11173,6596],{"className":11174,"style":3944},[426,427],[360,11176,6587],{"className":11177},[426],[360,11179,7219],{"className":11180},[426,427],[360,11182,793],{"className":11183},[887],[360,11185,403],{"className":11186},[426,427],[360,11188,6596],{"className":11189,"style":3944},[426,427],[112,11191,11192,11193,11247],{},"其中 ",[360,11194,11196,11220],{"className":11195},[363],[360,11197,11199],{"className":11198},[367],[369,11200,11201],{"xmlns":371},[373,11202,11203,11217],{},[376,11204,11205,11207,11209,11211,11213,11215],{},[379,11206,112],{},[383,11208,779],{"stretchy":778},[379,11210,6596],{},[379,11212,6587],{"mathvariant":771},[379,11214,7219],{},[383,11216,793],{"stretchy":778},[405,11218,11219],{"encoding":407},"p(\\theta|c)",[360,11221,11223],{"className":11222,"ariaHidden":413},[412],[360,11224,11226,11229,11232,11235,11238,11241,11244],{"className":11225},[417],[360,11227],{"className":11228,"style":848},[421],[360,11230,112],{"className":11231},[426,427],[360,11233,779],{"className":11234},[860],[360,11236,6596],{"className":11237,"style":3944},[426,427],[360,11239,6587],{"className":11240},[426],[360,11242,7219],{"className":11243},[426,427],[360,11245,793],{"className":11246},[887],">",[112,11249,11250],{},"是根據貝葉斯法則計算的潛在參數後驗分佈：",[112,11252,11253],{},[360,11254,11256,11344],{"className":11255},[363],[360,11257,11259],{"className":11258},[367],[369,11260,11261],{"xmlns":371},[373,11262,11263,11341],{},[376,11264,11265,11267,11269,11271,11273,11275,11277,11280,11282,11284,11286,11288,11317,11319,11321,11327,11329,11335,11337,11339],{},[379,11266,112],{},[383,11268,779],{"stretchy":778},[379,11270,6596],{},[379,11272,6587],{"mathvariant":771},[379,11274,7219],{},[383,11276,793],{"stretchy":778},[383,11278,11279],{},"∝",[379,11281,10332],{},[383,11283,779],{"stretchy":778},[379,11285,6596],{},[383,11287,793],{"stretchy":778},[628,11289,11290,11293],{},[383,11291,11292],{},"∏",[376,11294,11295,11297,11303,11305,11311,11313,11315],{},[383,11296,779],{"stretchy":778},[628,11298,11299,11301],{},[379,11300,6582],{},[379,11302,1126],{},[383,11304,784],{"separator":413},[628,11306,11307,11309],{},[379,11308,7236],{},[379,11310,1126],{},[383,11312,793],{"stretchy":778},[383,11314,385],{},[379,11316,7219],{},[379,11318,112],{},[383,11320,779],{"stretchy":778},[628,11322,11323,11325],{},[379,11324,7236],{},[379,11326,1126],{},[379,11328,6587],{"mathvariant":771},[628,11330,11331,11333],{},[379,11332,6582],{},[379,11334,1126],{},[383,11336,784],{"separator":413},[379,11338,6596],{},[383,11340,793],{"stretchy":778},[405,11342,11343],{"encoding":407},"p(\\theta|c) \\propto \\pi(\\theta) \\prod_{(x_i, y_i) \\in c} p(y_i|x_i, \\theta)",[360,11345,11347,11380],{"className":11346,"ariaHidden":413},[412],[360,11348,11350,11353,11356,11359,11362,11365,11368,11371,11374,11377],{"className":11349},[417],[360,11351],{"className":11352,"style":848},[421],[360,11354,112],{"className":11355},[426,427],[360,11357,779],{"className":11358},[860],[360,11360,6596],{"className":11361,"style":3944},[426,427],[360,11363,6587],{"className":11364},[426],[360,11366,7219],{"className":11367},[426,427],[360,11369,793],{"className":11370},[887],[360,11372],{"className":11373,"style":432},[431],[360,11375,11279],{"className":11376},[436],[360,11378],{"className":11379,"style":432},[431],[360,11381,11383,11387,11390,11393,11396,11399,11402,11539,11542,11545,11548,11588,11591,11631,11634,11637,11640],{"className":11382},[417],[360,11384],{"className":11385,"style":11386},[421],"height:1.2247em;vertical-align:-0.4747em;",[360,11388,10332],{"className":11389,"style":736},[426,427],[360,11391,779],{"className":11392},[860],[360,11394,6596],{"className":11395,"style":3944},[426,427],[360,11397,793],{"className":11398},[887],[360,11400],{"className":11401,"style":871},[431],[360,11403,11405,11408],{"className":11404},[852],[360,11406,11292],{"className":11407,"style":1395},[852,1393,1394],[360,11409,11411],{"className":11410},[457],[360,11412,11414,11530],{"className":11413},[461,713],[360,11415,11417,11527],{"className":11416},[465],[360,11418,11421],{"className":11419,"style":11420},[469],"height:0.2253em;",[360,11422,11423,11426],{"style":1411},[360,11424],{"className":11425,"style":477},[476],[360,11427,11429],{"className":11428},[481,482,483,484],[360,11430,11432,11435,11475,11478,11518,11521,11524],{"className":11431},[426,484],[360,11433,779],{"className":11434},[860,484],[360,11436,11438,11441],{"className":11437},[426,484],[360,11439,6582],{"className":11440},[426,427,484],[360,11442,11444],{"className":11443},[457],[360,11445,11447,11467],{"className":11446},[461,713],[360,11448,11450,11464],{"className":11449},[465],[360,11451,11453],{"className":11452,"style":1658},[469],[360,11454,11455,11458],{"style":723},[360,11456],{"className":11457,"style":727},[476],[360,11459,11461],{"className":11460},[481,731,732,484],[360,11462,1126],{"className":11463},[426,427,484],[360,11465,741],{"className":11466},[740],[360,11468,11470],{"className":11469},[465],[360,11471,11473],{"className":11472,"style":748},[469],[360,11474],{},[360,11476,784],{"className":11477},[867,484],[360,11479,11481,11484],{"className":11480},[426,484],[360,11482,7236],{"className":11483,"style":736},[426,427,484],[360,11485,11487],{"className":11486},[457],[360,11488,11490,11510],{"className":11489},[461,713],[360,11491,11493,11507],{"className":11492},[465],[360,11494,11496],{"className":11495,"style":1658},[469],[360,11497,11498,11501],{"style":7502},[360,11499],{"className":11500,"style":727},[476],[360,11502,11504],{"className":11503},[481,731,732,484],[360,11505,1126],{"className":11506},[426,427,484],[360,11508,741],{"className":11509},[740],[360,11511,11513],{"className":11512},[465],[360,11514,11516],{"className":11515,"style":748},[469],[360,11517],{},[360,11519,793],{"className":11520},[887,484],[360,11522,385],{"className":11523},[436,484],[360,11525,7219],{"className":11526},[426,427,484],[360,11528,741],{"className":11529},[740],[360,11531,11533],{"className":11532},[465],[360,11534,11537],{"className":11535,"style":11536},[469],"height:0.4747em;",[360,11538],{},[360,11540],{"className":11541,"style":871},[431],[360,11543,112],{"className":11544},[426,427],[360,11546,779],{"className":11547},[860],[360,11549,11551,11554],{"className":11550},[426],[360,11552,7236],{"className":11553,"style":736},[426,427],[360,11555,11557],{"className":11556},[457],[360,11558,11560,11580],{"className":11559},[461,713],[360,11561,11563,11577],{"className":11562},[465],[360,11564,11566],{"className":11565,"style":1317},[469],[360,11567,11568,11571],{"style":2044},[360,11569],{"className":11570,"style":477},[476],[360,11572,11574],{"className":11573},[481,482,483,484],[360,11575,1126],{"className":11576},[426,427,484],[360,11578,741],{"className":11579},[740],[360,11581,11583],{"className":11582},[465],[360,11584,11586],{"className":11585,"style":1339},[469],[360,11587],{},[360,11589,6587],{"className":11590},[426],[360,11592,11594,11597],{"className":11593},[426],[360,11595,6582],{"className":11596},[426,427],[360,11598,11600],{"className":11599},[457],[360,11601,11603,11623],{"className":11602},[461,713],[360,11604,11606,11620],{"className":11605},[465],[360,11607,11609],{"className":11608,"style":1317},[469],[360,11610,11611,11614],{"style":1320},[360,11612],{"className":11613,"style":477},[476],[360,11615,11617],{"className":11616},[481,482,483,484],[360,11618,1126],{"className":11619},[426,427,484],[360,11621,741],{"className":11622},[740],[360,11624,11626],{"className":11625},[465],[360,11627,11629],{"className":11628,"style":1339},[469],[360,11630],{},[360,11632,784],{"className":11633},[867],[360,11635],{"className":11636,"style":871},[431],[360,11638,6596],{"className":11639,"style":3944},[426,427],[360,11641,793],{"className":11642},[887],[112,11644,11645,11646,11649],{},"這一理論結果具有深遠的意義：它表明在無限數據和無限容量的極限下，通過交叉熵訓練的模型",[5450,11647,11648],{},"應該","收斂為一個完美的貝葉斯推理機。然而，這僅僅定義了「目標函數」的性質。關鍵的實證問題在於：在有限的參數、有限的數據和特定的架構（Transformer vs. MLP）下，哪種模型能夠真正實現這一數學理想？這正是本研究試圖回答的核心問題。",[108,11651,11653],{"id":11652},"_3-實驗設計構建貝葉斯風洞",[138,11654,11655],{},"3. 實驗設計：構建貝葉斯風洞",[112,11657,11658,11659,11662,11663,11666],{},"為了全面評估模型的推理能力，研究團隊設計了兩個難度遞增的「風洞」任務：",[138,11660,11661],{},"雙射學習","（Bijection Learning）與",[138,11664,11665],{},"隱馬可夫模型狀態追蹤","（HMM State Tracking）。這兩個任務分別測試了離散假設消除和遞迴信念更新兩種根本不同的推理模式。",[5119,11668,11670],{"id":11669},"_31-風洞一雙射學習bijection-learning",[138,11671,11672],{},"3.1 風洞一：雙射學習（Bijection Learning）",[112,11674,11675],{},"此任務模擬了一個嚴格的邏輯消除過程，類似於數獨或邏輯謎題。",[272,11677,11678,12064,12070],{},[275,11679,11680,11683,11684,11713,11714,11868,11869,11922,11923,11993,11994,153],{},[138,11681,11682],{},"任務定義："," 潛在參數 ",[360,11685,11687,11701],{"className":11686},[363],[360,11688,11690],{"className":11689},[367],[369,11691,11692],{"xmlns":371},[373,11693,11694,11698],{},[376,11695,11696],{},[379,11697,6596],{},[405,11699,11700],{"encoding":407},"\\theta",[360,11702,11704],{"className":11703,"ariaHidden":413},[412],[360,11705,11707,11710],{"className":11706},[417],[360,11708],{"className":11709,"style":9324},[421],[360,11711,6596],{"className":11712,"style":3944},[426,427]," 是一個雙射函數（一對一映射） ",[360,11715,11717,11766],{"className":11716},[363],[360,11718,11720],{"className":11719},[367],[369,11721,11722],{"xmlns":371},[373,11723,11724,11763],{},[376,11725,11726,11728,11730,11733,11735,11737,11740,11742,11744,11746,11749,11751,11753,11755,11757,11759,11761],{},[379,11727,10332],{},[383,11729,4066],{},[383,11731,11732],{"stretchy":778},"{",[1185,11734,1187],{},[383,11736,784],{"separator":413},[383,11738,11739],{},"…",[383,11741,784],{"separator":413},[379,11743,614],{},[383,11745,9955],{"stretchy":778},[383,11747,11748],{},"→",[383,11750,11732],{"stretchy":778},[1185,11752,1187],{},[383,11754,784],{"separator":413},[383,11756,11739],{},[383,11758,784],{"separator":413},[379,11760,614],{},[383,11762,9955],{"stretchy":778},[405,11764,11765],{"encoding":407},"\\pi: \\{1, \\dots, V\\} \\to \\{1, \\dots, V\\}",[360,11767,11769,11787,11832],{"className":11768,"ariaHidden":413},[412],[360,11770,11772,11775,11778,11781,11784],{"className":11771},[417],[360,11773],{"className":11774,"style":3107},[421],[360,11776,10332],{"className":11777,"style":736},[426,427],[360,11779],{"className":11780,"style":432},[431],[360,11782,4066],{"className":11783},[436],[360,11785],{"className":11786,"style":432},[431],[360,11788,11790,11793,11796,11799,11802,11805,11808,11811,11814,11817,11820,11823,11826,11829],{"className":11789},[417],[360,11791],{"className":11792,"style":848},[421],[360,11794,11732],{"className":11795},[860],[360,11797,1187],{"className":11798},[426],[360,11800,784],{"className":11801},[867],[360,11803],{"className":11804,"style":871},[431],[360,11806,11739],{"className":11807},[920],[360,11809],{"className":11810,"style":871},[431],[360,11812,784],{"className":11813},[867],[360,11815],{"className":11816,"style":871},[431],[360,11818,614],{"className":11819,"style":650},[426,427],[360,11821,9955],{"className":11822},[887],[360,11824],{"className":11825,"style":432},[431],[360,11827,11748],{"className":11828},[436],[360,11830],{"className":11831,"style":432},[431],[360,11833,11835,11838,11841,11844,11847,11850,11853,11856,11859,11862,11865],{"className":11834},[417],[360,11836],{"className":11837,"style":848},[421],[360,11839,11732],{"className":11840},[860],[360,11842,1187],{"className":11843},[426],[360,11845,784],{"className":11846},[867],[360,11848],{"className":11849,"style":871},[431],[360,11851,11739],{"className":11852},[920],[360,11854],{"className":11855,"style":871},[431],[360,11857,784],{"className":11858},[867],[360,11860],{"className":11861,"style":871},[431],[360,11863,614],{"className":11864,"style":650},[426,427],[360,11866,9955],{"className":11867},[887],"。模型在觀察到 ",[360,11870,11872,11890],{"className":11871},[363],[360,11873,11875],{"className":11874},[367],[369,11876,11877],{"xmlns":371},[373,11878,11879,11887],{},[376,11880,11881,11883,11885],{},[379,11882,1219],{},[383,11884,7367],{},[1185,11886,1187],{},[405,11888,11889],{"encoding":407},"k-1",[360,11891,11893,11912],{"className":11892,"ariaHidden":413},[412],[360,11894,11896,11900,11903,11906,11909],{"className":11895},[417],[360,11897],{"className":11898,"style":11899},[421],"height:0.7778em;vertical-align:-0.0833em;",[360,11901,1219],{"className":11902,"style":1645},[426,427],[360,11904],{"className":11905,"style":650},[431],[360,11907,7367],{"className":11908},[494],[360,11910],{"className":11911,"style":650},[431],[360,11913,11915,11919],{"className":11914},[417],[360,11916],{"className":11917,"style":11918},[421],"height:0.6444em;",[360,11920,1187],{"className":11921},[426]," 個輸入-輸出對後，必須預測下一個新輸入 ",[360,11924,11926,11944],{"className":11925},[363],[360,11927,11929],{"className":11928},[367],[369,11930,11931],{"xmlns":371},[373,11932,11933,11941],{},[376,11934,11935],{},[628,11936,11937,11939],{},[379,11938,6582],{},[379,11940,1219],{},[405,11942,11943],{"encoding":407},"x_k",[360,11945,11947],{"className":11946,"ariaHidden":413},[412],[360,11948,11950,11953],{"className":11949},[417],[360,11951],{"className":11952,"style":1298},[421],[360,11954,11956,11959],{"className":11955},[426],[360,11957,6582],{"className":11958},[426,427],[360,11960,11962],{"className":11961},[457],[360,11963,11965,11985],{"className":11964},[461,713],[360,11966,11968,11982],{"className":11967},[465],[360,11969,11971],{"className":11970,"style":4928},[469],[360,11972,11973,11976],{"style":1320},[360,11974],{"className":11975,"style":477},[476],[360,11977,11979],{"className":11978},[481,482,483,484],[360,11980,1219],{"className":11981,"style":1645},[426,427,484],[360,11983,741],{"className":11984},[740],[360,11986,11988],{"className":11987},[465],[360,11989,11991],{"className":11990,"style":1339},[469],[360,11992],{}," 的對應輸出 ",[360,11995,11997,12015],{"className":11996},[363],[360,11998,12000],{"className":11999},[367],[369,12001,12002],{"xmlns":371},[373,12003,12004,12012],{},[376,12005,12006],{},[628,12007,12008,12010],{},[379,12009,7236],{},[379,12011,1219],{},[405,12013,12014],{"encoding":407},"y_k",[360,12016,12018],{"className":12017,"ariaHidden":413},[412],[360,12019,12021,12024],{"className":12020},[417],[360,12022],{"className":12023,"style":2189},[421],[360,12025,12027,12030],{"className":12026},[426],[360,12028,7236],{"className":12029,"style":736},[426,427],[360,12031,12033],{"className":12032},[457],[360,12034,12036,12056],{"className":12035},[461,713],[360,12037,12039,12053],{"className":12038},[465],[360,12040,12042],{"className":12041,"style":4928},[469],[360,12043,12044,12047],{"style":2044},[360,12045],{"className":12046,"style":477},[476],[360,12048,12050],{"className":12049},[481,482,483,484],[360,12051,1219],{"className":12052,"style":1645},[426,427,484],[360,12054,741],{"className":12055},[740],[360,12057,12059],{"className":12058},[465],[360,12060,12062],{"className":12061,"style":1339},[469],[360,12063],{},[275,12065,12066,12069],{},[138,12067,12068],{},"貝葉斯邏輯："," 由於映射是雙射的，每一個觀察到的輸出都排除了該輸出對應其他輸入的可能性。對於一個全新的輸入，其可能的輸出必須均勻分佈在尚未出現過的輸出集合中。",[275,12071,12072,12073,12123],{},"解析後驗： 若已觀察到 ",[360,12074,12076,12093],{"className":12075},[363],[360,12077,12079],{"className":12078},[367],[369,12080,12081],{"xmlns":371},[373,12082,12083,12091],{},[376,12084,12085,12087,12089],{},[379,12086,1219],{},[383,12088,7367],{},[1185,12090,1187],{},[405,12092,11889],{"encoding":407},[360,12094,12096,12114],{"className":12095,"ariaHidden":413},[412],[360,12097,12099,12102,12105,12108,12111],{"className":12098},[417],[360,12100],{"className":12101,"style":11899},[421],[360,12103,1219],{"className":12104,"style":1645},[426,427],[360,12106],{"className":12107,"style":650},[431],[360,12109,7367],{"className":12110},[494],[360,12112],{"className":12113,"style":650},[431],[360,12115,12117,12120],{"className":12116},[417],[360,12118],{"className":12119,"style":11918},[421],[360,12121,1187],{"className":12122},[426]," 個不同的輸出，則下一個輸出的真實機率分佈為均勻分佈，其熵（Entropy）為：",[360,12125,12127],{"className":12126},[755],[360,12128,12130,12191],{"className":12129},[363],[360,12131,12133],{"className":12132},[367],[369,12134,12135],{"xmlns":371,"display":764},[373,12136,12137,12188],{},[376,12138,12139,12156,12158,12160,12162,12164,12174,12176,12178,12180,12182,12184,12186],{},[628,12140,12141,12144],{},[379,12142,12143],{},"H",[376,12145,12146,12148,12150,12152,12154],{},[379,12147,8480],{},[379,12149,5055],{},[379,12151,7236],{},[379,12153,7227],{},[379,12155,6563],{},[383,12157,779],{"stretchy":778},[379,12159,1219],{},[383,12161,793],{"stretchy":778},[383,12163,796],{},[628,12165,12166,12172],{},[376,12167,12168,12170],{},[379,12169,8121],{},[383,12171,775],{},[1185,12173,4119],{},[383,12175,779],{"stretchy":778},[379,12177,614],{},[383,12179,7367],{},[379,12181,1219],{},[383,12183,3976],{},[1185,12185,1187],{},[383,12187,793],{"stretchy":778},[405,12189,12190],{"encoding":407},"H_{Bayes}(k)= \\log_2(V - k + 1)",[360,12192,12194,12274,12340,12358],{"className":12193,"ariaHidden":413},[412],[360,12195,12197,12201,12256,12259,12262,12265,12268,12271],{"className":12196},[417],[360,12198],{"className":12199,"style":12200},[421],"height:1.0361em;vertical-align:-0.2861em;",[360,12202,12204,12208],{"className":12203},[426],[360,12205,12143],{"className":12206,"style":12207},[426,427],"margin-right:0.0813em;",[360,12209,12211],{"className":12210},[457],[360,12212,12214,12248],{"className":12213},[461,713],[360,12215,12217,12245],{"className":12216},[465],[360,12218,12220],{"className":12219,"style":8873},[469],[360,12221,12223,12226],{"style":12222},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[360,12224],{"className":12225,"style":477},[476],[360,12227,12229],{"className":12228},[481,482,483,484],[360,12230,12232,12235,12238,12241],{"className":12231},[426,484],[360,12233,8480],{"className":12234,"style":8659},[426,427,484],[360,12236,5055],{"className":12237},[426,427,484],[360,12239,7236],{"className":12240,"style":736},[426,427,484],[360,12242,12244],{"className":12243},[426,427,484],"es",[360,12246,741],{"className":12247},[740],[360,12249,12251],{"className":12250},[465],[360,12252,12254],{"className":12253,"style":2063},[469],[360,12255],{},[360,12257,779],{"className":12258},[860],[360,12260,1219],{"className":12261,"style":1645},[426,427],[360,12263,793],{"className":12264},[887],[360,12266],{"className":12267,"style":432},[431],[360,12269,796],{"className":12270},[436],[360,12272],{"className":12273,"style":432},[431],[360,12275,12277,12280,12325,12328,12331,12334,12337],{"className":12276},[417],[360,12278],{"className":12279,"style":848},[421],[360,12281,12283,12288],{"className":12282},[852],[360,12284,8208,12286],{"className":12285},[852],[360,12287,8212],{"style":8211},[360,12289,12291],{"className":12290},[457],[360,12292,12294,12316],{"className":12293},[461,713],[360,12295,12297,12313],{"className":12296},[465],[360,12298,12301],{"className":12299,"style":12300},[469],"height:0.207em;",[360,12302,12304,12307],{"style":12303},"top:-2.4559em;margin-right:0.05em;",[360,12305],{"className":12306,"style":477},[476],[360,12308,12310],{"className":12309},[481,482,483,484],[360,12311,4119],{"className":12312},[426,484],[360,12314,741],{"className":12315},[740],[360,12317,12319],{"className":12318},[465],[360,12320,12323],{"className":12321,"style":12322},[469],"height:0.2441em;",[360,12324],{},[360,12326,779],{"className":12327},[860],[360,12329,614],{"className":12330,"style":650},[426,427],[360,12332],{"className":12333,"style":650},[431],[360,12335,7367],{"className":12336},[494],[360,12338],{"className":12339,"style":650},[431],[360,12341,12343,12346,12349,12352,12355],{"className":12342},[417],[360,12344],{"className":12345,"style":11899},[421],[360,12347,1219],{"className":12348,"style":1645},[426,427],[360,12350],{"className":12351,"style":650},[431],[360,12353,3976],{"className":12354},[494],[360,12356],{"className":12357,"style":650},[431],[360,12359,12361,12364,12367],{"className":12360},[417],[360,12362],{"className":12363,"style":848},[421],[360,12365,1187],{"className":12366},[426],[360,12368,793],{"className":12369},[887],[112,12371,12372],{},"隨著序列推進，可能的假設數量減少，熵呈現階梯狀下降。",[272,12374,12375],{},[275,12376,12377,12380,12381,12433,12434,12587,12588,12651],{},[138,12378,12379],{},"複雜度控制","： 設定詞彙表大小 ",[360,12382,12384,12403],{"className":12383},[363],[360,12385,12387],{"className":12386},[367],[369,12388,12389],{"xmlns":371},[373,12390,12391,12400],{},[376,12392,12393,12395,12397],{},[379,12394,614],{},[383,12396,796],{},[1185,12398,12399],{},"20",[405,12401,12402],{"encoding":407},"V=20",[360,12404,12406,12424],{"className":12405,"ariaHidden":413},[412],[360,12407,12409,12412,12415,12418,12421],{"className":12408},[417],[360,12410],{"className":12411,"style":5226},[421],[360,12413,614],{"className":12414,"style":650},[426,427],[360,12416],{"className":12417,"style":432},[431],[360,12419,796],{"className":12420},[436],[360,12422],{"className":12423,"style":432},[431],[360,12425,12427,12430],{"className":12426},[417],[360,12428],{"className":12429,"style":11918},[421],[360,12431,12399],{"className":12432},[426],"，可能的雙射函數總數為 ",[360,12435,12437,12489],{"className":12436},[363],[360,12438,12440],{"className":12439},[367],[369,12441,12442],{"xmlns":371},[373,12443,12444,12486],{},[376,12445,12446,12448,12450,12453,12455,12457,12459,12461,12463,12465,12468,12470,12472,12474,12476,12478,12480],{},[1185,12447,12399],{},[803,12449,805],{},[431,12451],{"linebreak":12452},"newline",[379,12454,5055],{},[379,12456,112],{},[379,12458,112],{},[379,12460,7224],{},[379,12462,1162],{},[379,12464,6582],{},[1185,12466,12467],{},"2.4",[431,12469],{"linebreak":12452},[379,12471,6804],{},[379,12473,1126],{},[379,12475,524],{},[379,12477,7227],{},[379,12479,6563],{},[387,12481,12482,12484],{},[1185,12483,10171],{},[1185,12485,10174],{},[405,12487,12488],{"encoding":407},"20\\! \\\\approx 2.4 \\\\times 10^{18}",[360,12490,12492,12504,12507,12533,12536],{"className":12491,"ariaHidden":413},[412],[360,12493,12495,12498,12501],{"className":12494},[417],[360,12496],{"className":12497,"style":11918},[421],[360,12499,12399],{"className":12500},[426],[360,12502],{"className":12503,"style":913},[431],[360,12505],{"className":12506},[431,12452],[360,12508,12510,12514,12517,12521,12524,12527,12530],{"className":12509},[417],[360,12511],{"className":12512,"style":12513},[421],"height:0.8389em;vertical-align:-0.1944em;",[360,12515,5055],{"className":12516},[426,427],[360,12518,12520],{"className":12519},[426,427],"pp",[360,12522,7224],{"className":12523,"style":3944},[426,427],[360,12525,1162],{"className":12526},[426,427],[360,12528,6582],{"className":12529},[426,427],[360,12531,12467],{"className":12532},[426],[360,12534],{"className":12535},[431,12452],[360,12537,12539,12542,12545,12549,12552,12555],{"className":12538},[417],[360,12540],{"className":12541,"style":10187},[421],[360,12543,6804],{"className":12544},[426,427],[360,12546,12548],{"className":12547},[426,427],"im",[360,12550,12244],{"className":12551},[426,427],[360,12553,1187],{"className":12554},[426],[360,12556,12558,12561],{"className":12557},[426],[360,12559,4052],{"className":12560},[426],[360,12562,12564],{"className":12563},[457],[360,12565,12567],{"className":12566},[461],[360,12568,12570],{"className":12569},[465],[360,12571,12573],{"className":12572,"style":10187},[469],[360,12574,12575,12578],{"style":472},[360,12576],{"className":12577,"style":477},[476],[360,12579,12581],{"className":12580},[481,482,483,484],[360,12582,12584],{"className":12583},[426,484],[360,12585,10174],{"className":12586},[426,484],"。由於訓練樣本僅有 ",[360,12589,12591,12610],{"className":12590},[363],[360,12592,12594],{"className":12593},[367],[369,12595,12596],{"xmlns":371},[373,12597,12598,12607],{},[376,12599,12600],{},[387,12601,12602,12604],{},[1185,12603,10171],{},[1185,12605,12606],{},"5",[405,12608,12609],{"encoding":407},"10^5",[360,12611,12613],{"className":12612,"ariaHidden":413},[412],[360,12614,12616,12619,12622],{"className":12615},[417],[360,12617],{"className":12618,"style":10187},[421],[360,12620,1187],{"className":12621},[426],[360,12623,12625,12628],{"className":12624},[426],[360,12626,4052],{"className":12627},[426],[360,12629,12631],{"className":12630},[457],[360,12632,12634],{"className":12633},[461],[360,12635,12637],{"className":12636},[465],[360,12638,12640],{"className":12639,"style":10187},[469],[360,12641,12642,12645],{"style":472},[360,12643],{"className":12644,"style":477},[476],[360,12646,12648],{"className":12647},[481,482,483,484],[360,12649,12606],{"className":12650},[426,484]," 筆，模型在訓練中幾乎不可能見過測試集中的雙射函數。因此，模型無法「背誦」答案，必須學會「排除法」這一演算法。",[5119,12653,12655],{"id":12654},"_32-風洞二隱馬可夫模型狀態追蹤hmm-state-tracking",[138,12656,12657],{},"3.2 風洞二：隱馬可夫模型狀態追蹤（HMM State Tracking）",[112,12659,12660],{},"此任務測試模型處理隨機性、噪聲以及遞迴狀態更新的能力，要求模型在內部激活中實現「前向演算法」（Forward Algorithm）。",[272,12662,12663,12941],{},[275,12664,12665,12667,12668,12671,12672,12700,12701,12730,12731,12869,12870,12940],{},[138,12666,11682],{}," 每個序列由一個",[5450,12669,12670],{},"全新","的隱馬可夫模型生成。序列開頭包含一個「標頭」（Header），編碼了該特定 HMM 的轉移矩陣（",[360,12673,12675,12688],{"className":12674},[363],[360,12676,12678],{"className":12677},[367],[369,12679,12680],{"xmlns":371},[373,12681,12682,12686],{},[376,12683,12684],{},[379,12685,6791],{},[405,12687,6791],{"encoding":407},[360,12689,12691],{"className":12690,"ariaHidden":413},[412],[360,12692,12694,12697],{"className":12693},[417],[360,12695],{"className":12696,"style":5226},[421],[360,12698,6791],{"className":12699,"style":6960},[426,427],"）和發射矩陣（",[360,12702,12704,12717],{"className":12703},[363],[360,12705,12707],{"className":12706},[367],[369,12708,12709],{"xmlns":371},[373,12710,12711,12715],{},[376,12712,12713],{},[379,12714,10685],{},[405,12716,10685],{"encoding":407},[360,12718,12720],{"className":12719,"ariaHidden":413},[412],[360,12721,12723,12726],{"className":12722},[417],[360,12724],{"className":12725,"style":5226},[421],[360,12727,10685],{"className":12728,"style":12729},[426,427],"margin-right:0.0576em;","）。隨後，模型觀察一連串的觀測值 ",[360,12732,12734,12764],{"className":12733},[363],[360,12735,12737],{"className":12736},[367],[369,12738,12739],{"xmlns":371},[373,12740,12741,12761],{},[376,12742,12743,12749,12751,12757,12759],{},[628,12744,12745,12747],{},[379,12746,1162],{},[1185,12748,1187],{},[383,12750,784],{"separator":413},[628,12752,12753,12755],{},[379,12754,1162],{},[1185,12756,4119],{},[383,12758,784],{"separator":413},[383,12760,11739],{},[405,12762,12763],{"encoding":407},"o_1, o_2, \\dots",[360,12765,12767],{"className":12766,"ariaHidden":413},[412],[360,12768,12770,12773,12814,12817,12820,12860,12863,12866],{"className":12769},[417],[360,12771],{"className":12772,"style":2189},[421],[360,12774,12776,12779],{"className":12775},[426],[360,12777,1162],{"className":12778},[426,427],[360,12780,12782],{"className":12781},[457],[360,12783,12785,12806],{"className":12784},[461,713],[360,12786,12788,12803],{"className":12787},[465],[360,12789,12792],{"className":12790,"style":12791},[469],"height:0.3011em;",[360,12793,12794,12797],{"style":1320},[360,12795],{"className":12796,"style":477},[476],[360,12798,12800],{"className":12799},[481,482,483,484],[360,12801,1187],{"className":12802},[426,484],[360,12804,741],{"className":12805},[740],[360,12807,12809],{"className":12808},[465],[360,12810,12812],{"className":12811,"style":1339},[469],[360,12813],{},[360,12815,784],{"className":12816},[867],[360,12818],{"className":12819,"style":871},[431],[360,12821,12823,12826],{"className":12822},[426],[360,12824,1162],{"className":12825},[426,427],[360,12827,12829],{"className":12828},[457],[360,12830,12832,12852],{"className":12831},[461,713],[360,12833,12835,12849],{"className":12834},[465],[360,12836,12838],{"className":12837,"style":12791},[469],[360,12839,12840,12843],{"style":1320},[360,12841],{"className":12842,"style":477},[476],[360,12844,12846],{"className":12845},[481,482,483,484],[360,12847,4119],{"className":12848},[426,484],[360,12850,741],{"className":12851},[740],[360,12853,12855],{"className":12854},[465],[360,12856,12858],{"className":12857,"style":1339},[469],[360,12859],{},[360,12861,784],{"className":12862},[867],[360,12864],{"className":12865,"style":871},[431],[360,12867,11739],{"className":12868},[920],"，並需預測當前隱藏狀態 ",[360,12871,12873,12891],{"className":12872},[363],[360,12874,12876],{"className":12875},[367],[369,12877,12878],{"xmlns":371},[373,12879,12880,12888],{},[376,12881,12882],{},[628,12883,12884,12886],{},[379,12885,6563],{},[379,12887,6804],{},[405,12889,12890],{"encoding":407},"s_t",[360,12892,12894],{"className":12893,"ariaHidden":413},[412],[360,12895,12897,12900],{"className":12896},[417],[360,12898],{"className":12899,"style":1298},[421],[360,12901,12903,12906],{"className":12902},[426],[360,12904,6563],{"className":12905},[426,427],[360,12907,12909],{"className":12908},[457],[360,12910,12912,12932],{"className":12911},[461,713],[360,12913,12915,12929],{"className":12914},[465],[360,12916,12918],{"className":12917,"style":7161},[469],[360,12919,12920,12923],{"style":1320},[360,12921],{"className":12922,"style":477},[476],[360,12924,12926],{"className":12925},[481,482,483,484],[360,12927,6804],{"className":12928},[426,427,484],[360,12930,741],{"className":12931},[740],[360,12933,12935],{"className":12934},[465],[360,12936,12938],{"className":12937,"style":1339},[469],[360,12939],{}," 的後驗機率分佈。",[275,12942,12943],{},"貝葉斯邏輯： 真實的貝葉斯後驗由遞迴的前向演算法給出：",[360,12945,12947],{"className":12946},[755],[360,12948,12950,13041],{"className":12949},[363],[360,12951,12953],{"className":12952},[367],[369,12954,12955],{"xmlns":371,"display":764},[373,12956,12957,13038],{},[376,12958,12959,12966,12968,12970,12972,12974,12976,12978,12984,12986,12988,12990,13000,13002,13004,13006,13008,13014,13016,13028,13030,13036],{},[628,12960,12961,12964],{},[379,12962,12963],{},"α",[379,12965,6804],{},[383,12967,779],{"stretchy":778},[379,12969,6563],{},[383,12971,793],{"stretchy":778},[383,12973,11279],{},[379,12975,10685],{},[383,12977,779],{"stretchy":778},[628,12979,12980,12982],{},[379,12981,1162],{},[379,12983,6804],{},[379,12985,6587],{"mathvariant":771},[379,12987,6563],{},[383,12989,793],{"stretchy":778},[2547,12991,12992,12994],{},[383,12993,1176],{},[387,12995,12996,12998],{},[379,12997,6563],{},[383,12999,8151],{"mathvariant":771,"lspace":4042,"rspace":4042},[379,13001,6791],{},[383,13003,779],{"stretchy":778},[379,13005,6563],{},[379,13007,6587],{"mathvariant":771},[387,13009,13010,13012],{},[379,13011,6563],{},[383,13013,8151],{"mathvariant":771,"lspace":4042,"rspace":4042},[383,13015,793],{"stretchy":778},[628,13017,13018,13020],{},[379,13019,12963],{},[376,13021,13022,13024,13026],{},[379,13023,6804],{},[383,13025,7367],{},[1185,13027,1187],{},[383,13029,779],{"stretchy":778},[387,13031,13032,13034],{},[379,13033,6563],{},[383,13035,8151],{"mathvariant":771,"lspace":4042,"rspace":4042},[383,13037,793],{"stretchy":778},[405,13039,13040],{"encoding":407},"\\alpha_t(s) \\propto E(o_t|s) \\sum_{s'} T(s|s') \\alpha_{t-1}(s')",[360,13042,13044,13110],{"className":13043,"ariaHidden":413},[412],[360,13045,13047,13050,13092,13095,13098,13101,13104,13107],{"className":13046},[417],[360,13048],{"className":13049,"style":848},[421],[360,13051,13053,13057],{"className":13052},[426],[360,13054,12963],{"className":13055,"style":13056},[426,427],"margin-right:0.0037em;",[360,13058,13060],{"className":13059},[457],[360,13061,13063,13084],{"className":13062},[461,713],[360,13064,13066,13081],{"className":13065},[465],[360,13067,13069],{"className":13068,"style":7161},[469],[360,13070,13072,13075],{"style":13071},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[360,13073],{"className":13074,"style":477},[476],[360,13076,13078],{"className":13077},[481,482,483,484],[360,13079,6804],{"className":13080},[426,427,484],[360,13082,741],{"className":13083},[740],[360,13085,13087],{"className":13086},[465],[360,13088,13090],{"className":13089,"style":1339},[469],[360,13091],{},[360,13093,779],{"className":13094},[860],[360,13096,6563],{"className":13097},[426,427],[360,13099,793],{"className":13100},[887],[360,13102],{"className":13103,"style":432},[431],[360,13105,11279],{"className":13106},[436],[360,13108],{"className":13109,"style":432},[431],[360,13111,13113,13117,13120,13123,13163,13166,13169,13172,13175,13253,13256,13259,13262,13265,13268,13300,13303,13352,13355,13387],{"className":13112},[417],[360,13114],{"className":13115,"style":13116},[421],"height:2.344em;vertical-align:-1.294em;",[360,13118,10685],{"className":13119,"style":12729},[426,427],[360,13121,779],{"className":13122},[860],[360,13124,13126,13129],{"className":13125},[426],[360,13127,1162],{"className":13128},[426,427],[360,13130,13132],{"className":13131},[457],[360,13133,13135,13155],{"className":13134},[461,713],[360,13136,13138,13152],{"className":13137},[465],[360,13139,13141],{"className":13140,"style":7161},[469],[360,13142,13143,13146],{"style":1320},[360,13144],{"className":13145,"style":477},[476],[360,13147,13149],{"className":13148},[481,482,483,484],[360,13150,6804],{"className":13151},[426,427,484],[360,13153,741],{"className":13154},[740],[360,13156,13158],{"className":13157},[465],[360,13159,13161],{"className":13160,"style":1339},[469],[360,13162],{},[360,13164,6587],{"className":13165},[426],[360,13167,6563],{"className":13168},[426,427],[360,13170,793],{"className":13171},[887],[360,13173],{"className":13174,"style":871},[431],[360,13176,13178],{"className":13177},[852,7042],[360,13179,13181,13244],{"className":13180},[461,713],[360,13182,13184,13241],{"className":13183},[465],[360,13185,13187,13231],{"className":13186,"style":7052},[469],[360,13188,13190,13193],{"style":13189},"top:-1.856em;margin-left:0em;",[360,13191],{"className":13192,"style":7059},[476],[360,13194,13196],{"className":13195},[481,482,483,484],[360,13197,13199],{"className":13198},[426,484],[360,13200,13202,13205],{"className":13201},[426,484],[360,13203,6563],{"className":13204},[426,427,484],[360,13206,13208],{"className":13207},[457],[360,13209,13211],{"className":13210},[461],[360,13212,13214],{"className":13213},[465],[360,13215,13217],{"className":13216,"style":8293},[469],[360,13218,13219,13222],{"style":8296},[360,13220],{"className":13221,"style":727},[476],[360,13223,13225],{"className":13224},[481,731,732,484],[360,13226,13228],{"className":13227},[426,484],[360,13229,8151],{"className":13230},[426,484],[360,13232,13233,13236],{"style":7118},[360,13234],{"className":13235,"style":7059},[476],[360,13237,13238],{},[360,13239,1176],{"className":13240},[852,1393,7127],[360,13242,741],{"className":13243},[740],[360,13245,13247],{"className":13246},[465],[360,13248,13251],{"className":13249,"style":13250},[469],"height:1.294em;",[360,13252],{},[360,13254],{"className":13255,"style":871},[431],[360,13257,6791],{"className":13258,"style":6960},[426,427],[360,13260,779],{"className":13261},[860],[360,13263,6563],{"className":13264},[426,427],[360,13266,6587],{"className":13267},[426],[360,13269,13271,13274],{"className":13270},[426],[360,13272,6563],{"className":13273},[426,427],[360,13275,13277],{"className":13276},[457],[360,13278,13280],{"className":13279},[461],[360,13281,13283],{"className":13282},[465],[360,13284,13286],{"className":13285,"style":8412},[469],[360,13287,13288,13291],{"style":4335},[360,13289],{"className":13290,"style":477},[476],[360,13292,13294],{"className":13293},[481,482,483,484],[360,13295,13297],{"className":13296},[426,484],[360,13298,8151],{"className":13299},[426,484],[360,13301,793],{"className":13302},[887],[360,13304,13306,13309],{"className":13305},[426],[360,13307,12963],{"className":13308,"style":13056},[426,427],[360,13310,13312],{"className":13311},[457],[360,13313,13315,13344],{"className":13314},[461,713],[360,13316,13318,13341],{"className":13317},[465],[360,13319,13321],{"className":13320,"style":12791},[469],[360,13322,13323,13326],{"style":13071},[360,13324],{"className":13325,"style":477},[476],[360,13327,13329],{"className":13328},[481,482,483,484],[360,13330,13332,13335,13338],{"className":13331},[426,484],[360,13333,6804],{"className":13334},[426,427,484],[360,13336,7367],{"className":13337},[494,484],[360,13339,1187],{"className":13340},[426,484],[360,13342,741],{"className":13343},[740],[360,13345,13347],{"className":13346},[465],[360,13348,13350],{"className":13349,"style":8915},[469],[360,13351],{},[360,13353,779],{"className":13354},[860],[360,13356,13358,13361],{"className":13357},[426],[360,13359,6563],{"className":13360},[426,427],[360,13362,13364],{"className":13363},[457],[360,13365,13367],{"className":13366},[461],[360,13368,13370],{"className":13369},[465],[360,13371,13373],{"className":13372,"style":8412},[469],[360,13374,13375,13378],{"style":4335},[360,13376],{"className":13377,"style":477},[476],[360,13379,13381],{"className":13380},[481,482,483,484],[360,13382,13384],{"className":13383},[426,484],[360,13385,8151],{"className":13386},[426,484],[360,13388,793],{"className":13389},[887],[112,13391,13392],{},"這要求模型執行複雜的操作：",[5193,13394,13395,13459,13465],{},[275,13396,13397,13400,13401,13429,13430,13458],{},[138,13398,13399],{},"解析上下文："," 從標頭中讀取並存儲 ",[360,13402,13404,13417],{"className":13403},[363],[360,13405,13407],{"className":13406},[367],[369,13408,13409],{"xmlns":371},[373,13410,13411,13415],{},[376,13412,13413],{},[379,13414,6791],{},[405,13416,6791],{"encoding":407},[360,13418,13420],{"className":13419,"ariaHidden":413},[412],[360,13421,13423,13426],{"className":13422},[417],[360,13424],{"className":13425,"style":5226},[421],[360,13427,6791],{"className":13428,"style":6960},[426,427]," 和 ",[360,13431,13433,13446],{"className":13432},[363],[360,13434,13436],{"className":13435},[367],[369,13437,13438],{"xmlns":371},[373,13439,13440,13444],{},[376,13441,13442],{},[379,13443,10685],{},[405,13445,10685],{"encoding":407},[360,13447,13449],{"className":13448,"ariaHidden":413},[412],[360,13450,13452,13455],{"className":13451},[417],[360,13453],{"className":13454,"style":5226},[421],[360,13456,10685],{"className":13457,"style":12729},[426,427]," 矩陣。",[275,13460,13461,13464],{},[138,13462,13463],{},"狀態維護："," 在殘差流（Residual Stream）中維護一個信念狀態（Belief State）。",[275,13466,13467,13470],{},[138,13468,13469],{},"遞迴更新："," 每一步根據新的觀測值和轉移邏輯更新信念。",[272,13472,13473],{},[275,13474,13475,13478,13479,13429,13507,13535,13536,13600],{},[138,13476,13477],{},"複雜度控制："," ",[360,13480,13482,13495],{"className":13481},[363],[360,13483,13485],{"className":13484},[367],[369,13486,13487],{"xmlns":371},[373,13488,13489,13493],{},[376,13490,13491],{},[379,13492,6791],{},[405,13494,6791],{"encoding":407},[360,13496,13498],{"className":13497,"ariaHidden":413},[412],[360,13499,13501,13504],{"className":13500},[417],[360,13502],{"className":13503,"style":5226},[421],[360,13505,6791],{"className":13506,"style":6960},[426,427],[360,13508,13510,13523],{"className":13509},[363],[360,13511,13513],{"className":13512},[367],[369,13514,13515],{"xmlns":371},[373,13516,13517,13521],{},[376,13518,13519],{},[379,13520,10685],{},[405,13522,10685],{"encoding":407},[360,13524,13526],{"className":13525,"ariaHidden":413},[412],[360,13527,13529,13532],{"className":13528},[417],[360,13530],{"className":13531,"style":5226},[421],[360,13533,10685],{"className":13534,"style":12729},[426,427]," 的參數從 Dirichlet 分佈中採樣。可能的 HMM 空間在連續參數下是無限的，即使離散化後也超過 ",[360,13537,13539,13556],{"className":13538},[363],[360,13540,13542],{"className":13541},[367],[369,13543,13544],{"xmlns":371},[373,13545,13546,13554],{},[376,13547,13548],{},[387,13549,13550,13552],{},[1185,13551,10171],{},[1185,13553,10242],{},[405,13555,10245],{"encoding":407},[360,13557,13559],{"className":13558,"ariaHidden":413},[412],[360,13560,13562,13565,13568],{"className":13561},[417],[360,13563],{"className":13564,"style":10187},[421],[360,13566,1187],{"className":13567},[426],[360,13569,13571,13574],{"className":13570},[426],[360,13572,4052],{"className":13573},[426],[360,13575,13577],{"className":13576},[457],[360,13578,13580],{"className":13579},[461],[360,13581,13583],{"className":13582},[465],[360,13584,13586],{"className":13585,"style":10187},[469],[360,13587,13588,13591],{"style":472},[360,13589],{"className":13590,"style":477},[476],[360,13592,13594],{"className":13593},[481,482,483,484],[360,13595,13597],{"className":13596},[426,484],[360,13598,10242],{"className":13599},[426,484],"。這迫使模型必須成為一個「元學習者」（Meta-Learner），即學會執行前向演算法，而非記憶特定的 HMM。",[5119,13602,13604],{"id":13603},"_33-模型架構與基準對照",[138,13605,13606],{},"3.3 模型架構與基準對照",[112,13608,13609],{},"為了隔離 Transformer 架構（特別是注意力機制）的貢獻，研究使用了嚴格控制的對照組。",[272,13611,13612,13866],{},[275,13613,13614,13617,13618],{},[138,13615,13616],{},"Transformer 模型："," 採用小型但標準的架構（約 267-268 萬參數）。\n",[272,13619,13620,13742,13863],{},[275,13621,13622,13623,153],{},"雙射任務：6 層，6 個注意力頭，模型維度 ",[360,13624,13626,13659],{"className":13625},[363],[360,13627,13629],{"className":13628},[367],[369,13630,13631],{"xmlns":371},[373,13632,13633,13656],{},[376,13634,13635,13651,13653],{},[628,13636,13637,13639],{},[379,13638,403],{},[376,13640,13641,13643,13645,13647,13649],{},[379,13642,524],{},[379,13644,1162],{},[379,13646,403],{},[379,13648,7227],{},[379,13650,9065],{},[383,13652,796],{},[1185,13654,13655],{},"192",[405,13657,13658],{"encoding":407},"d_{model}=192",[360,13660,13662,13733],{"className":13661,"ariaHidden":413},[412],[360,13663,13665,13669,13724,13727,13730],{"className":13664},[417],[360,13666],{"className":13667,"style":13668},[421],"height:0.8444em;vertical-align:-0.15em;",[360,13670,13672,13675],{"className":13671},[426],[360,13673,403],{"className":13674},[426,427],[360,13676,13678],{"className":13677},[457],[360,13679,13681,13716],{"className":13680},[461,713],[360,13682,13684,13713],{"className":13683},[465],[360,13685,13687],{"className":13686,"style":4928},[469],[360,13688,13689,13692],{"style":1320},[360,13690],{"className":13691,"style":477},[476],[360,13693,13695],{"className":13694},[481,482,483,484],[360,13696,13698,13701,13704,13707,13710],{"className":13697},[426,484],[360,13699,524],{"className":13700},[426,427,484],[360,13702,1162],{"className":13703},[426,427,484],[360,13705,403],{"className":13706},[426,427,484],[360,13708,7227],{"className":13709},[426,427,484],[360,13711,9065],{"className":13712,"style":9185},[426,427,484],[360,13714,741],{"className":13715},[740],[360,13717,13719],{"className":13718},[465],[360,13720,13722],{"className":13721,"style":1339},[469],[360,13723],{},[360,13725],{"className":13726,"style":432},[431],[360,13728,796],{"className":13729},[436],[360,13731],{"className":13732,"style":432},[431],[360,13734,13736,13739],{"className":13735},[417],[360,13737],{"className":13738,"style":11918},[421],[360,13740,13655],{"className":13741},[426],[275,13743,13744,13745,153],{},"HMM 任務：9 層，8 個注意力頭，模型維度 ",[360,13746,13748,13781],{"className":13747},[363],[360,13749,13751],{"className":13750},[367],[369,13752,13753],{"xmlns":371},[373,13754,13755,13778],{},[376,13756,13757,13773,13775],{},[628,13758,13759,13761],{},[379,13760,403],{},[376,13762,13763,13765,13767,13769,13771],{},[379,13764,524],{},[379,13766,1162],{},[379,13768,403],{},[379,13770,7227],{},[379,13772,9065],{},[383,13774,796],{},[1185,13776,13777],{},"256",[405,13779,13780],{"encoding":407},"d_{model}=256",[360,13782,13784,13854],{"className":13783,"ariaHidden":413},[412],[360,13785,13787,13790,13845,13848,13851],{"className":13786},[417],[360,13788],{"className":13789,"style":13668},[421],[360,13791,13793,13796],{"className":13792},[426],[360,13794,403],{"className":13795},[426,427],[360,13797,13799],{"className":13798},[457],[360,13800,13802,13837],{"className":13801},[461,713],[360,13803,13805,13834],{"className":13804},[465],[360,13806,13808],{"className":13807,"style":4928},[469],[360,13809,13810,13813],{"style":1320},[360,13811],{"className":13812,"style":477},[476],[360,13814,13816],{"className":13815},[481,482,483,484],[360,13817,13819,13822,13825,13828,13831],{"className":13818},[426,484],[360,13820,524],{"className":13821},[426,427,484],[360,13823,1162],{"className":13824},[426,427,484],[360,13826,403],{"className":13827},[426,427,484],[360,13829,7227],{"className":13830},[426,427,484],[360,13832,9065],{"className":13833,"style":9185},[426,427,484],[360,13835,741],{"className":13836},[740],[360,13838,13840],{"className":13839},[465],[360,13841,13843],{"className":13842,"style":1339},[469],[360,13844],{},[360,13846],{"className":13847,"style":432},[431],[360,13849,796],{"className":13850},[436],[360,13852],{"className":13853,"style":432},[431],[360,13855,13857,13860],{"className":13856},[417],[360,13858],{"className":13859,"style":11918},[421],[360,13861,13777],{"className":13862},[426],[275,13864,13865],{},"配置：包含學習的位置嵌入、Pre-norm 殘差塊和標準的多頭自注意力機制。",[275,13867,13868,13871],{},[138,13869,13870],{},"MLP 基準模型："," 為了測試注意力機制的必要性，研究訓練了容量匹配（Capacity-matched）的多層感知機（MLP）。這些 MLP 擁有與 Transformer 幾乎相同的參數數量（誤差在 1% 以內），採用 18-20 層的深層結構，並配備了殘差連接和層歸一化（LayerNorm）。這確保了任何性能差異都源於架構本身（是否存在注意力機制），而非參數規模或訓練技巧的差異。",[108,13873,13875],{"id":13874},"_4-實證結果行為層面的深度分析",[138,13876,13877],{},"4. 實證結果：行為層面的深度分析",[112,13879,13880],{},"實驗結果提供了決定性的證據，表明 Transformer 在演算法層面上與貝葉斯推理具有驚人的一致性，而傳統 MLP 則完全無法勝任此類任務。",[5119,13882,13884],{"id":13883},"_41-雙射任務中的精確熵追蹤",[138,13885,13886],{},"4.1 雙射任務中的精確熵追蹤",[112,13888,13889],{},"在雙射學習任務中，Transformer 展現了近乎完美的假設消除能力。",[272,13891,13892,14006,14108],{},[275,13893,13894,13897,13898,14005],{},[138,13895,13896],{},"階梯狀熵曲線："," 隨著模型接收到更多的輸入-輸出對，剩餘的可能性空間縮小，預測的不確定性（熵）應呈現離散的階梯狀下降。實驗數據顯示，Transformer 的預測熵曲線與理論上的 ",[360,13899,13901,13935],{"className":13900},[363],[360,13902,13904],{"className":13903},[367],[369,13905,13906],{"xmlns":371},[373,13907,13908,13932],{},[376,13909,13910,13926,13928,13930],{},[628,13911,13912,13914],{},[379,13913,12143],{},[376,13915,13916,13918,13920,13922,13924],{},[379,13917,8480],{},[379,13919,5055],{},[379,13921,7236],{},[379,13923,7227],{},[379,13925,6563],{},[383,13927,779],{"stretchy":778},[379,13929,1219],{},[383,13931,793],{"stretchy":778},[405,13933,13934],{"encoding":407},"H_{Bayes}(k)",[360,13936,13938],{"className":13937,"ariaHidden":413},[412],[360,13939,13941,13944,13996,13999,14002],{"className":13940},[417],[360,13942],{"className":13943,"style":12200},[421],[360,13945,13947,13950],{"className":13946},[426],[360,13948,12143],{"className":13949,"style":12207},[426,427],[360,13951,13953],{"className":13952},[457],[360,13954,13956,13988],{"className":13955},[461,713],[360,13957,13959,13985],{"className":13958},[465],[360,13960,13962],{"className":13961,"style":8873},[469],[360,13963,13964,13967],{"style":12222},[360,13965],{"className":13966,"style":477},[476],[360,13968,13970],{"className":13969},[481,482,483,484],[360,13971,13973,13976,13979,13982],{"className":13972},[426,484],[360,13974,8480],{"className":13975,"style":8659},[426,427,484],[360,13977,5055],{"className":13978},[426,427,484],[360,13980,7236],{"className":13981,"style":736},[426,427,484],[360,13983,12244],{"className":13984},[426,427,484],[360,13986,741],{"className":13987},[740],[360,13989,13991],{"className":13990},[465],[360,13992,13994],{"className":13993,"style":2063},[469],[360,13995],{},[360,13997,779],{"className":13998},[860],[360,14000,1219],{"className":14001,"style":1645},[426,427],[360,14003,793],{"className":14004},[887]," 曲線幾乎完全重疊（參見原論文 Figure 1）。",[275,14007,14008,14011,14012,14107],{},[138,14009,14010],{},"微小的誤差："," 定量分析顯示，Transformer 的平均絕對熵誤差（MAE）僅為 ",[360,14013,14015,14041],{"className":14014},[363],[360,14016,14018],{"className":14017},[367],[369,14019,14020],{"xmlns":371},[373,14021,14022,14038],{},[376,14023,14024,14026,14028],{},[1185,14025,6508],{},[383,14027,400],{},[387,14029,14030,14032],{},[1185,14031,10171],{},[376,14033,14034,14036],{},[383,14035,7367],{},[1185,14037,6508],{},[405,14039,14040],{"encoding":407},"3 \\times 10^{-3}",[360,14042,14044,14063],{"className":14043,"ariaHidden":413},[412],[360,14045,14047,14051,14054,14057,14060],{"className":14046},[417],[360,14048],{"className":14049,"style":14050},[421],"height:0.7278em;vertical-align:-0.0833em;",[360,14052,6508],{"className":14053},[426],[360,14055],{"className":14056,"style":650},[431],[360,14058,400],{"className":14059},[494],[360,14061],{"className":14062,"style":650},[431],[360,14064,14066,14069,14072],{"className":14065},[417],[360,14067],{"className":14068,"style":10187},[421],[360,14070,1187],{"className":14071},[426],[360,14073,14075,14078],{"className":14074},[426],[360,14076,4052],{"className":14077},[426],[360,14079,14081],{"className":14080},[457],[360,14082,14084],{"className":14083},[461],[360,14085,14087],{"className":14086},[465],[360,14088,14090],{"className":14089,"style":10187},[469],[360,14091,14092,14095],{"style":472},[360,14093],{"className":14094,"style":477},[476],[360,14096,14098],{"className":14097},[481,482,483,484],[360,14099,14101,14104],{"className":14100},[426,484],[360,14102,7367],{"className":14103},[426,484],[360,14105,6508],{"className":14106},[426,484]," bits。這一誤差級別極低，幾乎等同於單精度浮點數運算引入的數值噪聲，意味著模型已經達到了機器精度的極限。",[275,14109,14110,14113],{},[138,14111,14112],{},"序列級別的動態性："," 更重要的是，模型並非僅僅擬合了平均趨勢。在單個序列的層面上（原論文 Figure 2），當模型遇到一個已知的輸入重複出現時，其預測熵會瞬間坍縮至零；而當遇到新輸入時，則準確回到消除曲線。這種「鋸齒狀」的動態行為證明了模型正在進行逐個序列的即時推理，而非簡單的統計平均。",[5119,14115,14117],{"id":14116},"_42-hmm-任務中的遞迴演算法學習",[138,14118,14119],{},"4.2 HMM 任務中的遞迴演算法學習",[112,14121,14122],{},"HMM 任務的結果進一步證實了模型處理隨機性和遞迴計算的能力。",[272,14124,14125,14279,14608],{},[275,14126,14127,14130,14131,14182,14183,14278],{},[138,14128,14129],{},"前向演算法的模擬："," 在訓練長度（",[360,14132,14134,14152],{"className":14133},[363],[360,14135,14137],{"className":14136},[367],[369,14138,14139],{"xmlns":371},[373,14140,14141,14149],{},[376,14142,14143,14145,14147],{},[379,14144,513],{},[383,14146,796],{},[1185,14148,12399],{},[405,14150,14151],{"encoding":407},"K=20",[360,14153,14155,14173],{"className":14154,"ariaHidden":413},[412],[360,14156,14158,14161,14164,14167,14170],{"className":14157},[417],[360,14159],{"className":14160,"style":5226},[421],[360,14162,513],{"className":14163,"style":545},[426,427],[360,14165],{"className":14166,"style":432},[431],[360,14168,796],{"className":14169},[436],[360,14171],{"className":14172,"style":432},[431],[360,14174,14176,14179],{"className":14175},[417],[360,14177],{"className":14178,"style":11918},[421],[360,14180,12399],{"className":14181},[426],"）內，Transformer 的熵誤差低至 ",[360,14184,14186,14213],{"className":14185},[363],[360,14187,14189],{"className":14188},[367],[369,14190,14191],{"xmlns":371},[373,14192,14193,14210],{},[376,14194,14195,14198,14200],{},[1185,14196,14197],{},"7.5",[383,14199,400],{},[387,14201,14202,14204],{},[1185,14203,10171],{},[376,14205,14206,14208],{},[383,14207,7367],{},[1185,14209,12606],{},[405,14211,14212],{"encoding":407},"7.5 \\times 10^{-5}",[360,14214,14216,14234],{"className":14215,"ariaHidden":413},[412],[360,14217,14219,14222,14225,14228,14231],{"className":14218},[417],[360,14220],{"className":14221,"style":14050},[421],[360,14223,14197],{"className":14224},[426],[360,14226],{"className":14227,"style":650},[431],[360,14229,400],{"className":14230},[494],[360,14232],{"className":14233,"style":650},[431],[360,14235,14237,14240,14243],{"className":14236},[417],[360,14238],{"className":14239,"style":10187},[421],[360,14241,1187],{"className":14242},[426],[360,14244,14246,14249],{"className":14245},[426],[360,14247,4052],{"className":14248},[426],[360,14250,14252],{"className":14251},[457],[360,14253,14255],{"className":14254},[461],[360,14256,14258],{"className":14257},[465],[360,14259,14261],{"className":14260,"style":10187},[469],[360,14262,14263,14266],{"style":472},[360,14264],{"className":14265,"style":477},[476],[360,14267,14269],{"className":14268},[481,482,483,484],[360,14270,14272,14275],{"className":14271},[426,484],[360,14273,7367],{"className":14274},[426,484],[360,14276,12606],{"className":14277},[426,484]," bits。這意味著模型內部實現的計算過程與數學上的前向演算法在功能上是等價的。",[275,14280,14281,14284,14285,13429,14337,14389,14390,14442,14443,14493,14494,14556,14557,14607],{},[138,14282,14283],{},"長度泛化（Length Generalization）："," 這是區分「推理」與「記憶」的黃金標準。研究人員將測試序列長度延伸至 ",[360,14286,14288,14307],{"className":14287},[363],[360,14289,14291],{"className":14290},[367],[369,14292,14293],{"xmlns":371},[373,14294,14295,14304],{},[376,14296,14297,14299,14301],{},[379,14298,513],{},[383,14300,796],{},[1185,14302,14303],{},"30",[405,14305,14306],{"encoding":407},"K=30",[360,14308,14310,14328],{"className":14309,"ariaHidden":413},[412],[360,14311,14313,14316,14319,14322,14325],{"className":14312},[417],[360,14314],{"className":14315,"style":5226},[421],[360,14317,513],{"className":14318,"style":545},[426,427],[360,14320],{"className":14321,"style":432},[431],[360,14323,796],{"className":14324},[436],[360,14326],{"className":14327,"style":432},[431],[360,14329,14331,14334],{"className":14330},[417],[360,14332],{"className":14333,"style":11918},[421],[360,14335,14303],{"className":14336},[426],[360,14338,14340,14359],{"className":14339},[363],[360,14341,14343],{"className":14342},[367],[369,14344,14345],{"xmlns":371},[373,14346,14347,14356],{},[376,14348,14349,14351,14353],{},[379,14350,513],{},[383,14352,796],{},[1185,14354,14355],{},"50",[405,14357,14358],{"encoding":407},"K=50",[360,14360,14362,14380],{"className":14361,"ariaHidden":413},[412],[360,14363,14365,14368,14371,14374,14377],{"className":14364},[417],[360,14366],{"className":14367,"style":5226},[421],[360,14369,513],{"className":14370,"style":545},[426,427],[360,14372],{"className":14373,"style":432},[431],[360,14375,796],{"className":14376},[436],[360,14378],{"className":14379,"style":432},[431],[360,14381,14383,14386],{"className":14382},[417],[360,14384],{"className":14385,"style":11918},[421],[360,14387,14355],{"className":14388},[426],"（即訓練長度的 1.5 倍和 2.5 倍）。結果顯示，Transformer 在超過訓練邊界（",[360,14391,14393,14411],{"className":14392},[363],[360,14394,14396],{"className":14395},[367],[369,14397,14398],{"xmlns":371},[373,14399,14400,14408],{},[376,14401,14402,14404,14406],{},[379,14403,6804],{},[383,14405,796],{},[1185,14407,12399],{},[405,14409,14410],{"encoding":407},"t=20",[360,14412,14414,14433],{"className":14413,"ariaHidden":413},[412],[360,14415,14417,14421,14424,14427,14430],{"className":14416},[417],[360,14418],{"className":14419,"style":14420},[421],"height:0.6151em;",[360,14422,6804],{"className":14423},[426,427],[360,14425],{"className":14426,"style":432},[431],[360,14428,796],{"className":14429},[436],[360,14431],{"className":14432,"style":432},[431],[360,14434,14436,14439],{"className":14435},[417],[360,14437],{"className":14438,"style":11918},[421],[360,14440,12399],{"className":14441},[426],"）後，誤差僅呈現平滑且緩慢的增長（",[360,14444,14446,14463],{"className":14445},[363],[360,14447,14449],{"className":14448},[367],[369,14450,14451],{"xmlns":371},[373,14452,14453,14461],{},[376,14454,14455,14457,14459],{},[379,14456,513],{},[383,14458,796],{},[1185,14460,14355],{},[405,14462,14358],{"encoding":407},[360,14464,14466,14484],{"className":14465,"ariaHidden":413},[412],[360,14467,14469,14472,14475,14478,14481],{"className":14468},[417],[360,14470],{"className":14471,"style":5226},[421],[360,14473,513],{"className":14474,"style":545},[426,427],[360,14476],{"className":14477,"style":432},[431],[360,14479,796],{"className":14480},[436],[360,14482],{"className":14483,"style":432},[431],[360,14485,14487,14490],{"className":14486},[417],[360,14488],{"className":14489,"style":11918},[421],[360,14491,14355],{"className":14492},[426]," 時 MAE ",[360,14495,14497,14526],{"className":14496},[363],[360,14498,14500],{"className":14499},[367],[369,14501,14502],{"xmlns":371},[373,14503,14504,14523],{},[376,14505,14506,14508,14510,14512,14514,14516,14518,14520],{},[431,14507],{"linebreak":12452},[379,14509,5055],{},[379,14511,112],{},[379,14513,112],{},[379,14515,7224],{},[379,14517,1162],{},[379,14519,6582],{},[1185,14521,14522],{},"0.02",[405,14524,14525],{"encoding":407},"\\\\approx 0.02",[360,14527,14529,14532],{"className":14528,"ariaHidden":413},[412],[360,14530],{"className":14531},[431,12452],[360,14533,14535,14538,14541,14544,14547,14550,14553],{"className":14534},[417],[360,14536],{"className":14537,"style":12513},[421],[360,14539,5055],{"className":14540},[426,427],[360,14542,12520],{"className":14543},[426,427],[360,14545,7224],{"className":14546,"style":3944},[426,427],[360,14548,1162],{"className":14549},[426,427],[360,14551,6582],{"className":14552},[426,427],[360,14554,14522],{"className":14555},[426]," bits），且在 ",[360,14558,14560,14577],{"className":14559},[363],[360,14561,14563],{"className":14562},[367],[369,14564,14565],{"xmlns":371},[373,14566,14567,14575],{},[376,14568,14569,14571,14573],{},[379,14570,6804],{},[383,14572,796],{},[1185,14574,12399],{},[405,14576,14410],{"encoding":407},[360,14578,14580,14598],{"className":14579,"ariaHidden":413},[412],[360,14581,14583,14586,14589,14592,14595],{"className":14582},[417],[360,14584],{"className":14585,"style":14420},[421],[360,14587,6804],{"className":14588},[426,427],[360,14590],{"className":14591,"style":432},[431],[360,14593,796],{"className":14594},[436],[360,14596],{"className":14597,"style":432},[431],[360,14599,14601,14604],{"className":14600},[417],[360,14602],{"className":14603,"style":11918},[421],[360,14605,12399],{"className":14606},[426]," 處沒有出現任何突變或斷裂。這證明模型學到的是一個與位置無關的通用遞迴算子（Recursive Operator），而非死記硬背了長度為 20 的特定模式。",[275,14609,14610,14613,14614,14665],{},[138,14611,14612],{},"語義不變性（Semantic Invariance）："," 研究人員進行了一項巧妙的測試：隨機置換 HMM 隱藏狀態的標籤（例如將「狀態 1」與「狀態 3」互換），並重新生成數據進行測試。結果顯示，模型的校準誤差在置換前後幾乎完全一致（分佈在 ",[360,14615,14617,14635],{"className":14616},[363],[360,14618,14620],{"className":14619},[367],[369,14621,14622],{"xmlns":371},[373,14623,14624,14632],{},[376,14625,14626,14628,14630],{},[379,14627,7236],{},[383,14629,796],{},[379,14631,6582],{},[405,14633,14634],{"encoding":407},"y=x",[360,14636,14638,14656],{"className":14637,"ariaHidden":413},[412],[360,14639,14641,14644,14647,14650,14653],{"className":14640},[417],[360,14642],{"className":14643,"style":2189},[421],[360,14645,7236],{"className":14646,"style":736},[426,427],[360,14648],{"className":14649,"style":432},[431],[360,14651,796],{"className":14652},[436],[360,14654],{"className":14655,"style":432},[431],[360,14657,14659,14662],{"className":14658},[417],[360,14660],{"className":14661,"style":3107},[421],[360,14663,6582],{"className":14664},[426,427]," 對角線上，如 Figure 8 所示）。這表明模型理解的是 HMM 的拓撲結構和轉移邏輯，而非依賴於特定的狀態 ID。",[5119,14667,14669],{"id":14668},"_43-mlp-的架構性崩潰",[138,14670,14671],{},"4.3 MLP 的架構性崩潰",[112,14673,14674],{},"與 Transformer 的卓越表現形成強烈對比的是，容量匹配的 MLP 在兩個風洞任務中均遭遇了災難性的失敗。",[272,14676,14677,14683,14689],{},[275,14678,14679,14682],{},[138,14680,14681],{},"無法進行假設消除："," 在雙射任務中，MLP 的 MAE 高達約 1.85 bits，比 Transformer 差了近三個數量級。其熵曲線幾乎保持平坦，顯示它只能學習到輸出的邊際分佈，完全無法根據上下文進行動態的假設剔除。",[275,14684,14685,14688],{},[138,14686,14687],{},"無法進行狀態追蹤："," 在 HMM 任務中，MLP 的誤差穩定在 0.4 bits 左右，且不隨序列長度變化。這表明它無法維護或更新信念狀態，只能進行靜態的猜測。",[275,14690,14691,14694,14695,14698],{},[138,14692,14693],{},"根本原因分析："," 這一失敗並非源於參數不足或訓練不當，而是源於架構缺陷。MLP 缺乏",[138,14696,14697],{},"內容定址（Content-Addressable）的路由機制","。在貝葉斯推理中，模型需要根據當前的證據（Query）去檢索相關的假設或參數（Key）。Transformer 的注意力機制天然支持這種操作，而 MLP 只能依賴固定的權重矩陣，無法動態地「查找」上下文中的信息。",[112,14700,14701],{},"為了直觀展示兩種架構的巨大差異，下表總結了關鍵的誤差數據：",[4848,14703,14704,14723],{},[4851,14705,14706],{},[4854,14707,14708,14711,14714,14717,14720],{},[4857,14709,14710],{"align":5929},"任務 (Task)",[4857,14712,14713],{"align":5929},"測試條件 (Condition)",[4857,14715,14716],{"align":5929},"Transformer MAE (bits)",[4857,14718,14719],{"align":5929},"MLP MAE (bits)",[4857,14721,14722],{"align":5929},"性能差距 (Degradation Factor)",[4864,14724,14725,14797,14869,14941],{},[4854,14726,14727,14731,14784,14789,14792],{},[4869,14728,14729],{"align":5929},[138,14730,11661],{},[4869,14732,14733,14734,793],{"align":5929},"訓練長度(",[360,14735,14737,14754],{"className":14736},[363],[360,14738,14740],{"className":14739},[367],[369,14741,14742],{"xmlns":371},[373,14743,14744,14752],{},[376,14745,14746,14748,14750],{},[379,14747,513],{},[383,14749,796],{},[1185,14751,12399],{},[405,14753,14151],{"encoding":407},[360,14755,14757,14775],{"className":14756,"ariaHidden":413},[412],[360,14758,14760,14763,14766,14769,14772],{"className":14759},[417],[360,14761],{"className":14762,"style":5226},[421],[360,14764,513],{"className":14765,"style":545},[426,427],[360,14767],{"className":14768,"style":432},[431],[360,14770,796],{"className":14771},[436],[360,14773],{"className":14774,"style":432},[431],[360,14776,14778,14781],{"className":14777},[417],[360,14779],{"className":14780,"style":11918},[421],[360,14782,12399],{"className":14783},[426],[4869,14785,14786],{"align":5929},[138,14787,14788],{},"0.0030",[4869,14790,14791],{"align":5929},"1.8540",[4869,14793,14794],{"align":5929},[138,14795,14796],{},"~618倍",[4854,14798,14799,14804,14856,14861,14864],{},[4869,14800,14801],{"align":5929},[138,14802,14803],{},"HMM 狀態追蹤",[4869,14805,14733,14806,793],{"align":5929},[360,14807,14809,14826],{"className":14808},[363],[360,14810,14812],{"className":14811},[367],[369,14813,14814],{"xmlns":371},[373,14815,14816,14824],{},[376,14817,14818,14820,14822],{},[379,14819,513],{},[383,14821,796],{},[1185,14823,12399],{},[405,14825,14151],{"encoding":407},[360,14827,14829,14847],{"className":14828,"ariaHidden":413},[412],[360,14830,14832,14835,14838,14841,14844],{"className":14831},[417],[360,14833],{"className":14834,"style":5226},[421],[360,14836,513],{"className":14837,"style":545},[426,427],[360,14839],{"className":14840,"style":432},[431],[360,14842,796],{"className":14843},[436],[360,14845],{"className":14846,"style":432},[431],[360,14848,14850,14853],{"className":14849},[417],[360,14851],{"className":14852,"style":11918},[421],[360,14854,12399],{"className":14855},[426],[4869,14857,14858],{"align":5929},[138,14859,14860],{},"0.000075",[4869,14862,14863],{"align":5929},"0.4090",[4869,14865,14866],{"align":5929},[138,14867,14868],{},"~5,467倍",[4854,14870,14871,14875,14928,14933,14936],{},[4869,14872,14873],{"align":5929},[138,14874,14803],{},[4869,14876,14877,14878,793],{"align":5929},"泛化長度(",[360,14879,14881,14898],{"className":14880},[363],[360,14882,14884],{"className":14883},[367],[369,14885,14886],{"xmlns":371},[373,14887,14888,14896],{},[376,14889,14890,14892,14894],{},[379,14891,513],{},[383,14893,796],{},[1185,14895,14303],{},[405,14897,14306],{"encoding":407},[360,14899,14901,14919],{"className":14900,"ariaHidden":413},[412],[360,14902,14904,14907,14910,14913,14916],{"className":14903},[417],[360,14905],{"className":14906,"style":5226},[421],[360,14908,513],{"className":14909,"style":545},[426,427],[360,14911],{"className":14912,"style":432},[431],[360,14914,796],{"className":14915},[436],[360,14917],{"className":14918,"style":432},[431],[360,14920,14922,14925],{"className":14921},[417],[360,14923],{"className":14924,"style":11918},[421],[360,14926,14303],{"className":14927},[426],[4869,14929,14930],{"align":5929},[138,14931,14932],{},"0.0125",[4869,14934,14935],{"align":5929},"0.4000",[4869,14937,14938],{"align":5929},[138,14939,14940],{},"32倍",[4854,14942,14943,14947,14999,15004,15007],{},[4869,14944,14945],{"align":5929},[138,14946,14803],{},[4869,14948,14877,14949,793],{"align":5929},[360,14950,14952,14969],{"className":14951},[363],[360,14953,14955],{"className":14954},[367],[369,14956,14957],{"xmlns":371},[373,14958,14959,14967],{},[376,14960,14961,14963,14965],{},[379,14962,513],{},[383,14964,796],{},[1185,14966,14355],{},[405,14968,14358],{"encoding":407},[360,14970,14972,14990],{"className":14971,"ariaHidden":413},[412],[360,14973,14975,14978,14981,14984,14987],{"className":14974},[417],[360,14976],{"className":14977,"style":5226},[421],[360,14979,513],{"className":14980,"style":545},[426,427],[360,14982],{"className":14983,"style":432},[431],[360,14985,796],{"className":14986},[436],[360,14988],{"className":14989,"style":432},[431],[360,14991,14993,14996],{"className":14992},[417],[360,14994],{"className":14995,"style":11918},[421],[360,14997,14355],{"className":14998},[426],[4869,15000,15001],{"align":5929},[138,15002,15003],{},"0.0288",[4869,15005,15006],{"align":5929},"0.4020",[4869,15008,15009],{"align":5929},[138,15010,15011],{},"14倍",[112,15013,15014,15015,153],{},"這組數據強有力地證明：在需要複雜推理的任務中，注意力機制並非錦上添花，而是",[138,15016,15017],{},"幾何上的必要條件",[108,15019,15021],{"id":15020},"_5-機制解釋注意力機制的貝葉斯幾何學",[138,15022,15023],{},"5. 機制解釋：注意力機制的貝葉斯幾何學",[112,15025,15026,15027,15030,15031,15034],{},"既然證實了 Transformer ",[5450,15028,15029],{},"能夠","執行貝葉斯推理，接下來的關鍵問題是：它是",[5450,15032,15033],{},"如何","做到的？Agarwal 等人利用幾何診斷工具，分析了模型內部的向量表徵，揭示了一個跨越層級的三階段運作機制。這一機制將貝葉斯推理的抽象數學運算轉化為具體的幾何操作。",[5119,15036,15038],{"id":15037},"_51-第一階段基礎綁定與假設框架layer-0",[138,15039,15040],{},"5.1 第一階段：基礎綁定與假設框架（Layer 0）",[112,15042,15043],{},"推理的起點在於構建一個能夠表示所有可能假設的幾何空間，這發生在網絡的最底層。",[272,15045,15046,15052,15058],{},[275,15047,15048,15051],{},[138,15049,15050],{},"正交鍵基底（Orthogonal Key Basis）："," 在 Layer 0，模型學會將輸入 Token 的鍵向量（Keys）組織成一個近似正交的基底。幾何分析顯示，不同輸入 Token 對應的 Key 向量之間的餘弦相似度接近於零（參見 Figure 14）。這意味著每個輸入都被映射到高維空間中的一個獨立軸或子空間上，形成了一個坐標系。",[275,15053,15054,15057],{},[138,15055,15056],{},"假設頭（Hypothesis Head）的關鍵性："," 消融研究（Ablation Study）發現，Layer 0 中存在一個特定的注意力頭，其重要性遠超其他頭。移除這個「假設頭」會導致整體誤差激增。這個頭的功能是將原始輸入「綁定」到殘差流中的特定幾何位置，從而定義了後續推理的「假設空間」（Hypothesis Space）。如果沒有這個框架，後續層級就無法對假設進行操作。",[275,15059,15060,15063],{},[138,15061,15062],{},"早期穩定性："," 訓練動態顯示，這個幾何框架在訓練早期就已形成並保持穩定。模型首先學會「如何表示問題」，然後才學會「如何精確計算」。",[5119,15065,15067],{"id":15066},"_52-第二階段順序消除與信息路由middle-layers",[138,15068,15069],{},"5.2 第二階段：順序消除與信息路由（Middle Layers）",[112,15071,15072],{},"一旦假設框架建立，中間層級開始執行核心的貝葉斯更新——即根據證據剔除不可能的假設。",[272,15074,15075,15081,15087],{},[275,15076,15077,15080],{},[138,15078,15079],{},"注意力作為路由（Attention as Routing）："," 注意力機制在此充當了動態路由器的角色。當前位置的查詢向量（Query）會與上下文中的鍵向量（Key）進行比對。",[275,15082,15083,15086],{},[138,15084,15085],{},"漸進式銳化（Progressive Sharpening）："," 在淺層，注意力分佈可能較為分散。但隨著層數加深，Query 與 Key 之間的對齊變得越來越尖銳（Sharpened）。在雙射任務的 Layer 5 中，Query 向量幾乎只與那些與當前證據一致的「可行假設」的 Key 產生高相似度（參見 Figure 15）。",[275,15088,15089,15092,15093,15145],{},[138,15090,15091],{},"貝葉斯「與」運算（Bayesian AND-ing）："," 這種幾何上的聚焦效應對應於貝葉斯更新中的乘法性質。如果一個假設在第 ",[360,15094,15096,15114],{"className":15095},[363],[360,15097,15099],{"className":15098},[367],[369,15100,15101],{"xmlns":371},[373,15102,15103,15111],{},[376,15104,15105,15107,15109],{},[379,15106,6804],{},[383,15108,7367],{},[1185,15110,1187],{},[405,15112,15113],{"encoding":407},"t-1",[360,15115,15117,15136],{"className":15116,"ariaHidden":413},[412],[360,15118,15120,15124,15127,15130,15133],{"className":15119},[417],[360,15121],{"className":15122,"style":15123},[421],"height:0.6984em;vertical-align:-0.0833em;",[360,15125,6804],{"className":15126},[426,427],[360,15128],{"className":15129,"style":650},[431],[360,15131,7367],{"className":15132},[494],[360,15134],{"className":15135,"style":650},[431],[360,15137,15139,15142],{"className":15138},[417],[360,15140],{"className":15141,"style":11918},[421],[360,15143,1187],{"className":15144},[426]," 步被證據排除，後續層級的注意力機制就會停止從該假設路由信息。前饋網絡（FFN）則負責處理這些路由過來的信息，執行數值上的後驗計算，並更新殘差流中的信念狀態。這一過程展示了 Transformer 如何利用深度（Depth）來實現序列化的邏輯消除。",[5119,15147,15149],{"id":15148},"_53-第三階段流形展開與精度精煉late-layers",[138,15150,15151],{},"5.3 第三階段：流形展開與精度精煉（Late Layers）",[112,15153,15154],{},"最後的階段涉及將粗略的信念狀態轉化為精確的機率輸出。",[272,15156,15157,15163,15169],{},[275,15158,15159,15162],{},[138,15160,15161],{},"值流形（Value Manifold）："," 研究人員對模型末層的「值向量」（Value Vectors）進行主成分分析（PCA），發現了一個驚人的幾何結構。這些向量並非隨機分佈，而是排列在一個低維流形上（在雙射任務中是一條一維曲線）。",[275,15164,15165,15168],{},[138,15166,15167],{},"熵參數化："," 這條曲線被後驗熵完美地參數化。也就是說，模型將其「不確定性」直接編碼為流形上的位置。高熵（不確定）的狀態聚集在曲線的一端，低熵（確定）的狀態則位於另一端（參見 Figure 16）。",[275,15170,15171,15174],{},[138,15172,15173],{},"框架-精度分離（Frame-Precision Dissociation）："," 一個重要的發現是，「路由」（注意力圖）的穩定早於「流形」（數值精度）的完善。模型先學會了「去哪裡找信息」（幾何路由），然後在訓練後期才慢慢學會「如何精確表達機率」（流形展開）。這種分離解釋了為什麼模型在訓練後期能顯著降低誤差——它是在一個固定的幾何框架內微調數值精度。",[108,15176,15178],{"id":15177},"_6-深入數據分析與消融研究",[138,15179,15180],{},"6. 深入數據分析與消融研究",[112,15182,15183],{},"為了驗證上述機制的必要性與穩健性，報告提供了詳細的消融實驗數據。這些數據揭示了 Transformer 內部各組件的功能特異性。",[5119,15185,15187],{"id":15186},"_61-層級與組件的消融影響",[138,15188,15189],{},"6.1 層級與組件的消融影響",[112,15191,15192],{},"研究人員逐一移除了模型的不同部分，觀察誤差（MAE）的變化。",[4848,15194,15195,15211],{},[4851,15196,15197],{},[4854,15198,15199,15202,15205,15208],{},[4857,15200,15201],{"align":5929},"消融類型 (Ablation Type)",[4857,15203,15204],{"align":5929},"被移除組件 (Component Removed)",[4857,15206,15207],{"align":5929},"對 MAE 的影響",[4857,15209,15210],{"align":5929},"洞察 (Insight)",[4864,15212,15213,15257,15275,15290],{},[4854,15214,15215,15220,15251,15254],{},[4869,15216,15217],{"align":5929},[138,15218,15219],{},"逐層消融",[4869,15221,15222,15223,793],{"align":5929},"任意單一層 (Layer ",[360,15224,15226,15239],{"className":15225},[363],[360,15227,15229],{"className":15228},[367],[369,15230,15231],{"xmlns":371},[373,15232,15233,15237],{},[376,15234,15235],{},[379,15236,1126],{},[405,15238,1126],{"encoding":407},[360,15240,15242],{"className":15241,"ariaHidden":413},[412],[360,15243,15245,15248],{"className":15244},[417],[360,15246],{"className":15247,"style":1138},[421],[360,15249,1126],{"className":15250},[426,427],[4869,15252,15253],{"align":5929},"誤差增加 >10倍",[4869,15255,15256],{"align":5929},"證明計算是深度組合的（Compositional），沒有哪一層是冗餘的。每一層都執行了一個必要的消除步驟。",[4854,15258,15259,15264,15267,15272],{},[4869,15260,15261],{"align":5929},[138,15262,15263],{},"逐頭消融",[4869,15265,15266],{"align":5929},"Layer 0 「假設頭」",[4869,15268,15269],{"align":5929},[138,15270,15271],{},"系統性崩潰",[4869,15273,15274],{"align":5929},"此頭負責構建幾何基底。移除它等於摧毀了推理的坐標系，導致後續所有計算失效。",[4854,15276,15277,15281,15284,15287],{},[4869,15278,15279],{"align":5929},[138,15280,15263],{},[4869,15282,15283],{"align":5929},"後層注意力頭",[4869,15285,15286],{"align":5929},"中等影響",[4869,15288,15289],{"align":5929},"後層頭部主要負責精煉值流形，具有一定的並行冗餘性。",[4854,15291,15292,15297,15300,15344],{},[4869,15293,15294],{"align":5929},[138,15295,15296],{},"架構變體",[4869,15298,15299],{"align":5929},"頂層兩層的注意力",[4869,15301,15302,15303,15343],{"align":5929},"訓練長度內影響小，但在 ",[360,15304,15306,15325],{"className":15305},[363],[360,15307,15309],{"className":15308},[367],[369,15310,15311],{"xmlns":371},[373,15312,15313,15322],{},[376,15314,15315,15317,15320],{},[379,15316,513],{},[803,15318,15319],{}," ",[1185,15321,12399],{},[405,15323,15324],{"encoding":407},"K\\>20",[360,15326,15328],{"className":15327,"ariaHidden":413},[412],[360,15329,15331,15334,15337,15340],{"className":15330},[417],[360,15332],{"className":15333,"style":5226},[421],[360,15335,513],{"className":15336,"style":545},[426,427],[360,15338],{"className":15339,"style":650},[431],[360,15341,12399],{"className":15342},[426]," 時崩潰",[4869,15345,15346,15347,15350],{"align":5929},"頂層注意力對於",[138,15348,15349],{},"長度泛化","至關重要。它負責在長序列中維持穩定的信念更新，防止誤差累積。",[5119,15352,15354],{"id":15353},"_62-多種子穩健性分析",[138,15355,15356],{},"6.2 多種子穩健性分析",[112,15358,15359,15360,15437],{},"為了確保這些發現不是隨機初始化的產物，研究在 5 個不同的隨機種子下重複了 HMM 實驗。結果顯示（參見原論文 Figure 11），不同種子之間的誤差曲線在 ",[360,15361,15363,15389],{"className":15362},[363],[360,15364,15366],{"className":15365},[367],[369,15367,15368],{"xmlns":371},[373,15369,15370,15386],{},[376,15371,15372,15374,15376,15378,15380,15382,15384],{},[379,15373,513],{},[383,15375,796],{},[1185,15377,12399],{},[383,15379,784],{"separator":413},[1185,15381,14303],{},[383,15383,784],{"separator":413},[1185,15385,14355],{},[405,15387,15388],{"encoding":407},"K=20, 30, 50",[360,15390,15392,15410],{"className":15391,"ariaHidden":413},[412],[360,15393,15395,15398,15401,15404,15407],{"className":15394},[417],[360,15396],{"className":15397,"style":5226},[421],[360,15399,513],{"className":15400,"style":545},[426,427],[360,15402],{"className":15403,"style":432},[431],[360,15405,796],{"className":15406},[436],[360,15408],{"className":15409,"style":432},[431],[360,15411,15413,15416,15419,15422,15425,15428,15431,15434],{"className":15412},[417],[360,15414],{"className":15415,"style":12513},[421],[360,15417,12399],{"className":15418},[426],[360,15420,784],{"className":15421},[867],[360,15423],{"className":15424,"style":871},[431],[360,15426,14303],{"className":15427},[426],[360,15429,784],{"className":15430},[867],[360,15432],{"className":15433,"style":871},[431],[360,15435,14355],{"className":15436},[426]," 時幾乎完全重合。種子間的變異性遠小於 Transformer 與 MLP 之間的性能差距，這證實了 Transformer 收斂到貝葉斯後驗是一個穩健的架構特性，而非運氣。",[108,15439,15441],{"id":15440},"_7-對大型語言模型與提示工程的啟示",[138,15442,15443],{},"7. 對大型語言模型與提示工程的啟示",[112,15445,15446],{},"雖然本研究是在小型受控環境中進行的，但其發現對於理解數千億參數的 LLM 及其應用（如提示工程）具有深遠的第二階與第三階啟示。",[5119,15448,15450],{"id":15449},"_71-啟示一幻覺的幾何解釋",[138,15451,15452],{},"7.1 啟示一：「幻覺」的幾何解釋",[112,15454,15455],{},"本研究發現的「值流形」為 LLM 的幻覺（Hallucination）現象提供了一個嚴謹的幾何解釋。",[272,15457,15458,15464,15470],{},[275,15459,15460,15463],{},[138,15461,15462],{},"機制："," 模型將其信心編碼為高維流形上的坐標。當模型處於高熵狀態（即不確定）時，其內部向量位於流形的「高不確定性」區域。",[275,15465,15466,15469],{},[138,15467,15468],{},"失效模式："," 然而，語言生成的最後一步是強制性的採樣（Sampling）。即使內部狀態準確地反映了「我不知道」（高熵），模型仍被迫選擇一個 Token 輸出。如果中間層的路由未能成功消除錯誤假設（即「不完全消除」，Incomplete Elimination），殘差流中的向量可能停留在流形的模糊地帶，導致模型自信地輸出錯誤答案。",[275,15471,15472,15475],{},[138,15473,15474],{},"結論："," 幻覺往往不是模型「發瘋」，而是模型在幾何上正確地表示了不確定性，但被解碼過程強制坍縮的結果。",[5119,15477,15479],{"id":15478},"_72-啟示二提示工程即證據供給",[138,15480,15481],{},"7.2 啟示二：提示工程即「證據供給」",[112,15483,15484,15485,153],{},"基於貝葉斯視角，提示工程（Prompt Engineering）的本質不再是「下指令」，而是",[138,15486,15487],{},"注入證據以切割假設空間",[272,15489,15490,15496,15502,15512],{},[275,15491,15492,15495],{},[138,15493,15494],{},"假設空間初始化："," 當輸入一個提示時，Layer 0 立即建立了一個巨大的假設空間（例如：「這是一個代碼任務？還是詩歌創作？是翻譯？」）。",[275,15497,15498,15501],{},[138,15499,15500],{},"證據消除："," 提示中的每一個詞、每一個範例都在充當「證據」，用於消除與之不符的子空間。例如，提供代碼範例可以迅速消除「詩歌」和「翻譯」的假設子空間，引導注意力機制聚焦於「編程」軌道。",[275,15503,15504,15507,15508,15511],{},[138,15505,15506],{},"邊界範例的重要性："," 正如機器人交換（Robot Exchange）分析所指出的，提供位於",[138,15509,15510],{},"決策邊界","（Boundaries）的範例比提供典型範例更有效。邊界範例能更銳利地切割流形，幫助注意力機制區分相似的潛在假設。",[275,15513,15514,15517],{},[138,15515,15516],{},"領域干擾："," 如果提示中混合了多個領域（例如要求「用寫詩的風格寫代碼」），模型被迫在殘差流中同時維護多個互斥的假設軌道，這可能導致路由干擾（Interference），降低推理精度。清晰、單一領域的提示能讓幾何路由更乾淨地收斂。",[5119,15519,15521],{"id":15520},"_73-啟示三深度即時間depth-as-time",[138,15522,15523],{},"7.3 啟示三：深度即時間（Depth as Time）",[112,15525,15526,15527,15530],{},"研究發現移除任何一層都會導致誤差激增，這強調了",[138,15528,15529],{},"深度","在推理中的不可替代性。",[272,15532,15533,15539],{},[275,15534,15535,15538],{},[138,15536,15537],{},"序列化推理："," 貝葉斯推理本質上是序列化的：你必須先根據證據 A 更新信念，才能處理證據 B。在 Transformer 中，層數（Depth）代表了計算步驟或「時間」。",[275,15540,15541,15544,15545,15548],{},[138,15542,15543],{},"思維鏈（Chain of Thought）："," 這解釋了為什麼「思維鏈」技術如此有效。通過讓模型生成中間步驟，我們實際上是在利用",[138,15546,15547],{},"序列維度（Token位置）來擴展模型的虛擬深度","。這給予模型更多的幾何變換次數來執行必要的貝葉斯消除步驟，從而解決那些單次前向傳播（其物理深度有限）無法解決的複雜推理問題。",[108,15550,15552],{"id":15551},"_8-局限性與未來展望",[138,15553,15554],{},"8. 局限性與未來展望",[112,15556,15557],{},"儘管這項研究具有里程碑意義，但作者也誠實地指出了其局限性，這些局限性為未來的研究指明了方向。",[272,15559,15560,15617,15623],{},[275,15561,15562,15565,15566,15616],{},[138,15563,15564],{},"規模限制："," 實驗使用的是小型模型（約 300 萬參數）和小詞彙表（",[360,15567,15569,15586],{"className":15568},[363],[360,15570,15572],{"className":15571},[367],[369,15573,15574],{"xmlns":371},[373,15575,15576,15584],{},[376,15577,15578,15580,15582],{},[379,15579,614],{},[383,15581,796],{},[1185,15583,12399],{},[405,15585,12402],{"encoding":407},[360,15587,15589,15607],{"className":15588,"ariaHidden":413},[412],[360,15590,15592,15595,15598,15601,15604],{"className":15591},[417],[360,15593],{"className":15594,"style":5226},[421],[360,15596,614],{"className":15597,"style":650},[426,427],[360,15599],{"className":15600,"style":432},[431],[360,15602,796],{"className":15603},[436],[360,15605],{"className":15606,"style":432},[431],[360,15608,15610,15613],{"className":15609},[417],[360,15611],{"className":15612,"style":11918},[421],[360,15614,12399],{"className":15615},[426],"）。雖然這足以證明原理，但真實語言模型的潛在空間維度要高得多，且假設空間的結構更為複雜（非結構化）。",[275,15618,15619,15622],{},[138,15620,15621],{},"任務簡化："," 風洞任務（雙射與 HMM）具有清晰的結構和邊界。真實世界的推理往往涉及模糊的定義和隱含的規則。",[275,15624,15625,15628],{},[138,15626,15627],{},"未來方向：",[5193,15629,15630,15636,15642],{},[275,15631,15632,15635],{},[138,15633,15634],{},"擴展風洞："," 開發更複雜的潛在變量模型（如卡爾曼濾波、因果圖模型）的風洞，以測試模型處理連續變量和因果推理的能力。",[275,15637,15638,15641],{},[138,15639,15640],{},"大模型診斷："," 將本研究開發的幾何診斷工具（正交性檢測、流形 PCA）應用於預訓練的 LLM（如 LLaMA 或 GPT），檢查在自然語言訓練下是否湧現出類似的貝葉斯幾何結構。",[275,15643,15644,15647],{},[138,15645,15646],{},"架構探索："," 測試其他架構（如 Mamba、RWKV 等狀態空間模型）是否也能形成類似的貝葉斯流形，這將有助於理解不同架構的歸納偏置（Inductive Bias)。",[108,15649,15651],{"id":15650},"_9-結論",[138,15652,15653],{},"9. 結論",[112,15655,15656,15657],{},"《Transformer 注意力機制的貝葉斯幾何學》一文為我們理解現代 AI 提供了一個強大的理論與實證框架。它超越了以往模糊的「神經網絡是黑盒子」的認知，確立了一個具體的科學事實：",[138,15658,15659],{},"Transformer 是貝葉斯推理的幾何實現引擎。",[112,15661,15662,15663,15666,15667,15670,15671,15674],{},"Agarwal、Dalal 和 Misra 的研究證明，Transformer 之所以能從海量數據中湧現出推理能力，是因為其架構中的",[138,15664,15665],{},"殘差流","（作為信念載體）、",[138,15668,15669],{},"注意力機制","（作為內容定址的路由器）和",[138,15672,15673],{},"前饋網絡","（作為數值更新器）恰好對應了貝葉斯推理的三個核心步驟。它們共同作用，在模型的高維表示空間中雕刻出一個精確的幾何流形，用於整合證據並量化不確定性。",[112,15676,15677],{},"與之相對，MLP 的失敗提醒我們，架構設計至關重要。沒有正確的幾何歸納偏置（如注意力路由），單靠堆砌參數和數據無法產生真正的動態推理能力。",[112,15679,15680],{},"對於 AI 研究者與工程師而言，這份報告不僅證實了 Transformer 的優越性，更提供了一種全新的思維模型：我們不只是在訓練模型預測下一個詞，我們是在構建一個能夠容納並操作機率分佈的幾何機器。理解這一幾何學，將是我們從「模仿智能」邁向「真實推理」的關鍵一步。",[5454,15682,15683],{"id":6208},[138,15684,6211],{},[5193,15686,15687,15694],{},[275,15688,15689,15690],{},"How to Prompt LLM Coding Agents as the Bayesian Inference Engines they are, accessed January 15, 2026, ",[5055,15691,15692],{"href":15692,"rel":15693},"https:\u002F\u002Fwww.robotexchange.io\u002Ft\u002Fhow-to-prompt-llm-coding-agents-as-the-bayesian-inference-engines-they-are\u002F4177",[5059],[275,15695,15696,15697],{},"The Bayesian Geometry of Transformer Attention - arXiv, accessed January 15, 2026, ",[5055,15698,15699],{"href":15699,"rel":15700},"https:\u002F\u002Farxiv.org\u002Fhtml\u002F2512.22471v2",[5059],{"title":311,"searchDepth":312,"depth":312,"links":15702},[15703,15704,15708,15713,15718,15723,15727,15732,15733],{"id":10110,"depth":312,"text":10113},{"id":10125,"depth":312,"text":10128,"children":15705},[15706,15707],{"id":10131,"depth":4984,"text":10134},{"id":10299,"depth":4984,"text":10302},{"id":11652,"depth":312,"text":11655,"children":15709},[15710,15711,15712],{"id":11669,"depth":4984,"text":11672},{"id":12654,"depth":4984,"text":12657},{"id":13603,"depth":4984,"text":13606},{"id":13874,"depth":312,"text":13877,"children":15714},[15715,15716,15717],{"id":13883,"depth":4984,"text":13886},{"id":14116,"depth":4984,"text":14119},{"id":14668,"depth":4984,"text":14671},{"id":15020,"depth":312,"text":15023,"children":15719},[15720,15721,15722],{"id":15037,"depth":4984,"text":15040},{"id":15066,"depth":4984,"text":15069},{"id":15148,"depth":4984,"text":15151},{"id":15177,"depth":312,"text":15180,"children":15724},[15725,15726],{"id":15186,"depth":4984,"text":15189},{"id":15353,"depth":4984,"text":15356},{"id":15440,"depth":312,"text":15443,"children":15728},[15729,15730,15731],{"id":15449,"depth":4984,"text":15452},{"id":15478,"depth":4984,"text":15481},{"id":15520,"depth":4984,"text":15523},{"id":15551,"depth":312,"text":15554},{"id":15650,"depth":312,"text":15653},"\u002Fimages\u002Fblog\u002F2512.22471v2.webp","2026-01-15","Naman Agarwal等人2026年開創性研究證實Transformer透過注意力機制精確實現貝葉斯推理，揭示其上下文學習的幾何本質並為提示工程奠定理論基礎。",{},"\u002Fblog\u002Fzh-tw\u002Farxiv-2512-22471v2",{"title":10105,"description":15736},"arxiv-2512-22471v2","blog\u002Fzh-TW\u002Farxiv-2512-22471v2",[5424,5032,15743],"bayesian","G_DGUjWKK_hCDisM4vLGWf13eighqDB2SOshjvUv17w",{"id":15746,"title":15747,"author":103,"body":15748,"cover":16599,"coverAlt":15747,"date":16600,"description":16601,"draft":324,"extension":325,"featured":324,"lang":327,"meta":16602,"navigation":326,"path":16603,"seo":16604,"slug":16605,"stem":16606,"tags":16607,"translationKey":16605,"updated":337,"__hash__":16610},"blog\u002Fblog\u002Fzh-TW\u002Fintroducing-nested-learning.md","嵌套學習：深度學習架構的幻象",{"type":105,"value":15749,"toc":16593},[15750,15757,15760,15773,15775,15779,15782,15787,15802,15807,15831,15833,15837,15842,15850,15855,15875,15880,15914,15916,15920,15925,15951,15953,15957,15962,15970,15975,16001,16006,16023,16028,16054,16056,16060,16064,16123,16125,16129,16133,16185,16187,16191,16196,16260,16262,16266,16270,16285,16290,16348,16350,16354,16359,16367,16372,16408,16413,16421,16427,16429,16433,16436,16546,16548,16551],[5119,15751,15753],{"id":15752},"httpsresearchgoogleblogintroducing-nested-learning-a-new-ml-paradigm-for-continual-learning",[5055,15754,15755],{"href":15755,"rel":15756},"https:\u002F\u002Fresearch.google\u002Fblog\u002Fintroducing-nested-learning-a-new-ml-paradigm-for-continual-learning\u002F",[5059],[108,15758,15747],{"id":15759},"嵌套學習深度學習架構的幻象",[112,15761,15762,15765,15766,15769,15772],{},[138,15763,15764],{},"作者","：Ali Behrouz、Meisam Razaviyayn、Peilin Zhong、Vahab Mirrokni",[15767,15768],"br",{},[138,15770,15771],{},"機構","：Google Research",[5479,15774],{},[5119,15776,15778],{"id":15777},"全文翻譯與章節重點總結","📖 全文翻譯與章節重點總結",[5454,15780,15781],{"id":5464},"📌 摘要",[112,15783,15784,4023],{},[138,15785,15786],{},"核心觀點",[272,15788,15789,15796,15799],{},[275,15790,15791,15792,15795],{},"提出了 ",[138,15793,15794],{},"嵌套學習"," 這一新的學習典範，將機器學習模型表示為一組嵌套的、多層次的優化問題，每一層都有自己的「上下文流」。",[275,15797,15798],{},"從 NL 的角度看，現有深度學習方法通過壓縮自身的上下文流來從數據中學習，而大型模型中的上下文學習自然湧現。",[275,15800,15801],{},"NL 提出了一種哲學思想，即設計具有更多「層次」、更富表達力的學習演算法，從而實現更高階的上下文學習，並可能解鎖有效的持續學習能力。",[112,15803,15804,4023],{},[138,15805,15806],{},"主要貢獻",[5193,15808,15809,15815,15821],{},[275,15810,15811,15814],{},[138,15812,15813],{},"可表達的最佳化器","：展示了 Adam、帶動量的 SGD 等基於梯度的最佳化器實際上是試圖壓縮梯度資訊的關聯記憶體模組，並在此基礎上提出了具有更深記憶或更強大學習規則的「更富表達力」的最佳化器。",[275,15816,15817,15820],{},[138,15818,15819],{},"自修改學習模組","：利用 NL 對學習演算法的洞察，提出了一種通過學習自己的更新演算法來學習如何修改自身的序列模型。",[275,15822,15823,15826,15827,15830],{},[138,15824,15825],{},"連續體記憶系統","：提出了一種新的記憶系統表述，推廣了傳統的「長\u002F短期記憶」觀點。結合自修改序列模型和連續體記憶系統，提出了名為 ",[138,15828,15829],{},"Hope"," 的持續學習模組，在語言建模、知識整合、少樣本泛化、持續學習和長上下文推理任務中顯示出良好效果。",[5479,15832],{},[5454,15834,15836],{"id":15835},"_1️⃣-引言","1️⃣ 引言",[112,15838,15839,4023],{},[138,15840,15841],{},"核心問題",[272,15843,15844,15847],{},[275,15845,15846],{},"儘管深度學習（尤其是大型語言模型）取得了巨大成功，但它們在初始部署後基本上是靜態的，無法在上下文之外持續獲取新能力，只能通過上下文學習進行快速適應。",[275,15848,15849],{},"當前模型（尤其是 LLMs）的知識僅限於其上下文視窗內的即時上下文或預訓練結束時儲存在 MLP 中的「長期過去」知識，類似於「順行性遺忘症」，無法形成新的長期記憶。",[112,15851,15852,4023],{},[138,15853,15854],{},"神經科學動機",[272,15856,15857,15863,15869],{},[275,15858,15859,15862],{},[138,15860,15861],{},"人腦的持續學習能力"," 歸功於神經可塑性和神經振盪（腦波）。",[275,15864,15865,15868],{},[138,15866,15867],{},"多時間尺度處理系統","：腦波（如 Gamma、Beta、Theta）以不同頻率協調神經計算和資訊處理，對應不同的認知功能和記憶鞏固時間尺度。",[275,15870,15871,15874],{},[138,15872,15873],{},"大腦的均勻可重用結構","：大腦結構是均勻且可重用的（如半球切除術後仍能正常運作），記憶是跨多個區域分佈的神經迴路，而非孤立於特定區域。",[112,15876,15877,4023],{},[138,15878,15879],{},"貢獻與路線圖",[5193,15881,15882,15888,15894,15901,15907],{},[275,15883,15884,15885,153],{},"介紹 ",[138,15886,15887],{},"嵌套學習典範",[275,15889,15890,15891,153],{},"論證 ",[138,15892,15893],{},"最佳化器和架構都是學習模組",[275,15895,15896,15897,15900],{},"重新審視 ",[138,15898,15899],{},"持續學習、上下文學習、預訓練和學習"," 等常見概念。",[275,15902,15903,15904,153],{},"提出 ",[138,15905,15906],{},"連續體記憶系統、自引用 Titans 和 Hope 架構",[275,15908,15909,15910,15913],{},"提供 ",[138,15911,15912],{},"實驗評估","，支持 NL 觀點和提出的元件的有效性。",[5479,15915],{},[5454,15917,15919],{"id":15918},"_2️⃣-預備知識","2️⃣ 預備知識",[112,15921,15922,4023],{},[138,15923,15924],{},"回顧概念",[272,15926,15927,15933,15939,15945],{},[275,15928,15929,15932],{},[138,15930,15931],{},"梯度下降"," 及其變體的公式化表述。",[275,15934,15935,15938],{},[138,15936,15937],{},"元學習","：將模型設計自動化建模為一個兩級優化過程。",[275,15940,15941,15944],{},[138,15942,15943],{},"快速權重程式設計器 (FWPs)","：一種循環神經網路，其記憶是矩陣值的，作為短期記憶。",[275,15946,15947,15950],{},[138,15948,15949],{},"上下文學習 (ICL)","：模型根據給定上下文自適應和學習的能力。本文使用最廣義的定義。",[5479,15952],{},[5454,15954,15956],{"id":15955},"_3️⃣-嵌套學習-nested-learning","3️⃣ 嵌套學習 (Nested Learning)",[112,15958,15959,4023],{},[138,15960,15961],{},"核心思想",[272,15963,15964,15967],{},[275,15965,15966],{},"將機器學習模型（包括其訓練過程）表示為一個相互關聯的、多層次的、嵌套的優化問題系統。",[275,15968,15969],{},"每個元件（架構、最佳化器）都有自己的上下文流和優化目標，它們在不同層次以不同頻率更新。",[112,15971,15972,4023],{},[138,15973,15974],{},"關鍵定義",[5193,15976,15977,15983,15989,15995],{},[275,15978,15979,15982],{},[138,15980,15981],{},"關聯記憶體","：一個將一組鍵映射到一組值的算子。學習是獲取有效記憶體（映射）的過程。",[275,15984,15985,15988],{},[138,15986,15987],{},"更新頻率","：定義每個元件（參數或非參數）的更新頻率。",[275,15990,15991,15994],{},[138,15992,15993],{},"嵌套系統","：一個有序的多層次系統，每層包含一組用梯度下降優化的（關聯記憶體）問題。",[275,15996,15997,16000],{},[138,15998,15999],{},"神經學習模組","：架構和優化過程共同確定的模型的聯合表示。",[112,16002,16003,4023],{},[138,16004,16005],{},"層次間的知識轉移方式",[272,16007,16008,16011,16014,16017,16020],{},[275,16009,16010],{},"參數化直接連接（如 FWPs）",[275,16012,16013],{},"非參數化直接連接（如注意力）",[275,16015,16016],{},"通過反向傳播的知識轉移",[275,16018,16019],{},"通過初始化（如 MAML）",[275,16021,16022],{},"通過生成（權重或上下文生成，如超網路、優化過程）",[112,16024,16025,4023],{},[138,16026,16027],{},"重要洞見",[272,16029,16030,16036,16042,16048],{},[275,16031,16032,16035],{},[138,16033,16034],{},"訓練一個線性層","：通過反向傳播訓練一個線性層可以被視為一個基於「驚奇」的記憶體，它試圖將數據樣本映射到其預測誤差。",[275,16037,16038,16041],{},[138,16039,16040],{},"帶動量的梯度下降","：可以視為一個兩級關聯記憶體（優化過程），其中內層學習將梯度壓縮到其參數中。",[275,16043,16044,16047],{},[138,16045,16046],{},"架構分解","：像線性注意力這樣的架構可以分解為嵌套的優化問題（例如，投影層在低頻優化，記憶體矩陣在高頻優化）。",[275,16049,16050,16053],{},[138,16051,16052],{},"堆疊層次","：嵌套學習允許計算模型由多個（多層）層次組成，以不同的抽象層次和更新頻率學習和處理數據。",[5479,16055],{},[5454,16057,16059],{"id":16058},"_4️⃣-最佳化器作為學習模組","4️⃣ 最佳化器作為學習模組",[112,16061,16062,4023],{},[138,16063,15786],{},[272,16065,16066,16076,16082,16088,16111,16117],{},[275,16067,16068,16071,16072,16075],{},[138,16069,16070],{},"反向傳播作為關聯記憶體","：使用反向傳播訓練神經網路可以看作一個壓縮過程，其中每一層都在儲存其輸入與對應的局部誤差信號（驚奇）之間的映射。這是一個 ",[138,16073,16074],{},"自我指涉過程","，因為用於記憶體的值是由記憶體本身生成的。",[275,16077,16078,16081],{},[138,16079,16080],{},"基於動量的最佳化器作為關聯記憶體","：動量項可以看作一個旨在將過去梯度壓縮到其參數中的關聯記憶體。它是一個兩級優化過程。",[275,16083,16084,16087],{},[138,16085,16086],{},"流行的最佳化器如 Adam、AdaGrad"," 都可以被重新表述為關聯記憶體，它們試圖將梯度映射到其某種全域屬性（如方差）。",[275,16089,16090,16093,16094],{},[138,16091,16092],{},"為持續學習設計更具表達力的最佳化器","：現有動量設計（如指數移動平均）容量有限，對長遠過去梯度記憶不佳。NL 視角啟發我們設計：\n",[272,16095,16096,16099,16102,16105,16108],{},[275,16097,16098],{},"更具表達力的關聯（如添加預處理）。",[275,16100,16101],{},"更具表達力的目標（如用 L2 回歸損失代替點積相似度，得到 Delta Momentum）。",[275,16103,16104],{},"更具表達力的記憶體結構（如用 MLP 代替線性層作為動量記憶體，得到 Deep Momentum GD）。",[275,16106,16107],{},"更高階的特徵映射。",[275,16109,16110],{},"非線性輸出（如 Muon 最佳化器中的牛頓-舒爾茨正交化）。",[275,16112,16113,16116],{},[138,16114,16115],{},"Delta 梯度下降 (DGD)","：用 L2 回歸損失代替點積相似度作為內部目標，得到的更新規則不僅依賴於當前數據，還依賴於權重的先前狀態，從而能捕獲數據樣本間的依賴關係。",[275,16118,16119,16122],{},[138,16120,16121],{},"廣義梯度下降 (GGD)","：將梯度下降定義為任何旨在壓縮數據樣本並將其映射到一組自生成值的自我指涉關聯記憶體。",[5479,16124],{},[5454,16126,16128],{"id":16127},"_5️⃣-現有架構作為神經學習模組","5️⃣ 現有架構作為神經學習模組",[112,16130,16131,4023],{},[138,16132,15786],{},[272,16134,16135,16141,16147,16179],{},[275,16136,16137,16140],{},[138,16138,16139],{},"現代序列模型（Transformer, RNNs）都可以被視為關聯記憶體","，它們旨在根據某種內部目標（注意力偏差）和最佳化演算法學習從鍵到值的映射。",[275,16142,16143,16146],{},[138,16144,16145],{},"從 NL 視角看，所有現代架構都是均勻的","：它們都由一組前饋網路（線性或非線性 MLP 塊）組成，每個網路根據自己的上下文流和優化問題在不同層次和時間尺度上進行訓練。我們觀察到的異質性源於只看到了這些優化問題的最終解，造成了「深度學習架構的幻象」。",[275,16148,16149,16152,16153],{},[138,16150,16151],{},"具體示例","：\n",[272,16154,16155,16161,16167,16173],{},[275,16156,16157,16160],{},[138,16158,16159],{},"Softmax 注意力","：是某個 L2 回歸目標的非參數解（Nadaraya-Watson 估計器）。",[275,16162,16163,16166],{},[138,16164,16165],{},"基於 Hebbian 規則的 RNNs（如線性注意力）","：是使用點積相似度作為內部目標，用梯度下降優化的關聯記憶體。",[275,16168,16169,16172],{},[138,16170,16171],{},"基於 Delta 規則的 RNNs（如 DeltaNet）","：是使用 MSE（L2 回歸）作為內部目標，用梯度下降優化的關聯記憶體。",[275,16174,16175,16178],{},[138,16176,16177],{},"其他學習規則","：如 Oja‘s rule, Omega rule 等，也都是特定內部目標和最佳化演算法的實例。",[275,16180,16181,16184],{},[138,16182,16183],{},"門控機制的作用","：在現代序列模型中，門控線性層的輸出可以被視為 MLP 塊的持久記憶體，或者當記憶體初始狀態未元學習時，作為記憶體模組的初始化。",[5479,16186],{},[5454,16188,16190],{"id":16189},"_6️⃣-要點總結與概念重審","6️⃣ 要點總結與概念重審",[112,16192,16193,4023],{},[138,16194,16195],{},"對常見術語的 NL 視角解讀",[272,16197,16198,16204,16210,16216,16221,16227,16233,16239,16248,16254],{},[275,16199,16200,16203],{},[138,16201,16202],{},"記憶體與學習","：記憶體是由輸入引起的神經更新，學習是獲取有用記憶體的過程。記憶體不是孤立系統，而是分佈在所有參數中。任何由梯度下降（或其他演算法）引起的更新都是記憶體的一種形式。",[275,16205,16206,16209],{},[138,16207,16208],{},"模型參數","：模型的參數不僅包括在預訓練級別優化的那些。NL 表示中出現的所有參數（如動量項、RNN 的隱藏狀態）都對其性能和表達能力有貢獻。",[275,16211,16212,16215],{},[138,16213,16214],{},"每個神經元的更多計算","：堆疊層次可以增加每個低頻參數的計算深度（如 Muon 最佳化器中的正交化步驟）。",[275,16217,16218,16220],{},[138,16219,15949],{},"：從 NL 視角看，每個塊或層次都有自己的上下文流，因此任何對該上下文的神經更新或適應都被視為一種 ICL。Transformer 的 ICL 是非參數 ICL 的一個特例，而 RNN 的記憶體執行的是參數化 ICL。",[275,16222,16223,16226],{},[138,16224,16225],{},"測試時訓練\u002F記憶體","：是參數化 ICL 的實例，當上下文移除時，獲取的上下文知識也隨之消失。",[275,16228,16229,16232],{},[138,16230,16231],{},"預訓練與測試時間","：預訓練只是 ICL 的一個可能實例，其上下文是整個預訓練數據。訓練和測試時間的區分是由於缺乏從最高頻率層（如 Transformer 的上下文）到低頻層（如預訓練）的知識轉移過程。",[275,16234,16235,16238],{},[138,16236,16237],{},"持續學習","：從 NL 視角看，任何機器學習模型都在進行持續學習。知識可能無法持久或轉移到更持久的層次，主要是由於層次間缺乏知識轉移。",[275,16240,16241,16244,16245,153],{},[138,16242,16243],{},"現有架構骨幹與混合模型","：從 NL 視角看，所有現代架構都是均勻的。混合架構可以看作是傳統的 Transformer 模型，只是為某些 MLP 塊增加了新的計算層次。",[138,16246,16247],{},"循環模型正在取代 MLP 塊",[275,16249,16250,16253],{},[138,16251,16252],{},"神經學習模組作為互聯繫統","：架構為最佳化器生成上下文（即梯度）。因此，梯度的適當記憶體管理依賴於架構的選擇。未來需要設計架構特定的最佳化器，以使這個互聯繫統完美和諧地工作。",[275,16255,16256,16259],{},[138,16257,16258],{},"最佳化器 vs. 學習到的最佳化器","：兩者是同一概念的不同實例，只是頻率和上下文流不同。",[5479,16261],{},[5454,16263,16265],{"id":16264},"_7️⃣-連續體多時間尺度記憶系統","7️⃣ 連續體多時間尺度記憶系統",[112,16267,16268,4023],{},[138,16269,15786],{},[272,16271,16272,16279,16282],{},[275,16273,16274,16275,16278],{},"推廣傳統的「長\u002F短期記憶體」觀點，提出 ",[138,16276,16277],{},"連續體記憶系統 (CMS)","：將記憶體視為一個具有更新頻率譜的分佈式互聯繫統。",[275,16280,16281],{},"高頻神經元負責快速適應但儲存記憶體時間短，低頻神經元負責更持久的知識。",[275,16283,16284],{},"與 LSM 相比，這種多頻率設計形成了一個記憶體循環過程，知識在被遺忘後可以部分恢復。",[112,16286,16287,4023],{},[138,16288,16289],{},"CMS 設計",[272,16291,16292,16295,16320,16326,16332,16342],{},[275,16293,16294],{},"由一系列 MLP 塊鏈組成，每個塊與一個更新頻率（塊大小）相關聯。塊參數根據其頻率定期更新（例如，每 C 步更新一次）。",[275,16296,16297,16152,16300],{},[138,16298,16299],{},"變體",[5193,16301,16302,16308,16314],{},[275,16303,16304,16307],{},[138,16305,16306],{},"嵌套 CMS","：高層級 MLP 塊的初始狀態由低層級元學習，允許高階 ICL。",[275,16309,16310,16313],{},[138,16311,16312],{},"順序 CMS","：所有 MLP 塊的初始狀態都在最低頻率層通過反向傳播元學習。",[275,16315,16316,16319],{},[138,16317,16318],{},"獨立（頭-wise）CMS","：獨立的塊具有不同的上下文長度，然後通過聚合函數組合。",[275,16321,16322,16325],{},[138,16323,16324],{},"CMS 有助於持續學習","：當一個塊更新並遺忘舊知識時，這些知識仍可能儲存在其他低頻塊中，並且通過反向傳播的知識轉移可以循環回來，形成一個時間維度的循環，從而很難遺忘重要知識。",[275,16327,16328,16331],{},[138,16329,16330],{},"CMS 的效率","：每次只更新一小部分參數（根據其頻率），並且其設計解鎖了高層級的序列並行化。",[275,16333,16334,16337,16338,16341],{},[138,16335,16336],{},"CMS 在最佳化器中的應用","：作為概念驗證，設計了 ",[138,16339,16340],{},"多尺度動量 Muon (M3) 最佳化器","，結合了 Adam、Muon 和 CMS 的思想，具有多個動量項以更好地壓縮長期梯度資訊。",[275,16343,16344,16347],{},[138,16345,16346],{},"臨時層次堆疊：用預訓練模型初始化 CMS","：可以利用預訓練模型的權重來初始化 CMS 塊的參數，從而幫助模型更快地適應其層次的上下文流。",[5479,16349],{},[5454,16351,16353],{"id":16352},"_8️⃣-hope具有連續體記憶體的自指涉學習模組","8️⃣ Hope：具有連續體記憶體的自指涉學習模組",[112,16355,16356,4023],{},[138,16357,16358],{},"設計動機",[272,16360,16361,16364],{},[275,16362,16363],{},"Transformer 是兩層級元件（投影和 MLP 在第一層，注意力是第二層的非參數解），計算深度有限，且參數在上下文中是靜態的，缺乏自我修改能力。",[275,16365,16366],{},"需要一種允許所有元件在上下文中自適應，並能學習自我修改的模型。",[112,16368,16369,4023],{},[138,16370,16371],{},"Hope 架構組成",[5193,16373,16374,16399],{},[275,16375,16376,16152,16379],{},[138,16377,16378],{},"自修改深度 Titans",[272,16380,16381,16384,16390,16393,16396],{},[275,16382,16383],{},"一個完全自適應的記憶體系統，其中所有元件（生成鍵、值、查詢、學習率、衰減率的映射）都能在上下文中適應。",[275,16385,16386,16389],{},[138,16387,16388],{},"關鍵創新","：模型學習根據上下文生成自己的潛在值（目標值），從而修改自身。",[275,16391,16392],{},"使用 Delta 梯度下降 (DGD) 作為內部優化演算法，因為令牌空間中的元素高度相關。",[275,16394,16395],{},"所有記憶體（映射函數）的架構都是任意的，本文使用 2 層 MLP。",[275,16397,16398],{},"採用分塊訓練演算法實現快速並行化訓練。",[275,16400,16401,16152,16403],{},[138,16402,16277],{},[272,16404,16405],{},[275,16406,16407],{},"為模型提供更持久的記憶體儲存。",[112,16409,16410,4023],{},[138,16411,16412],{},"Hope 的前向傳播",[272,16414,16415,16418],{},[275,16416,16417],{},"輸入首先通過自修改 Titans 模組進行處理，其輸出再通過 CMS 鏈。",[275,16419,16420],{},"輸出是這兩個系統協同工作的結果。",[112,16422,16423,16426],{},[138,16424,16425],{},"Hope-Attention 變體","：將自修改 Titans 替換為 softmax 全域注意力。",[5479,16428],{},[5454,16430,16432],{"id":16431},"_9️⃣-實驗評估","9️⃣ 實驗評估",[112,16434,16435],{},"實驗評估了 NL 觀點及其元件（如 CMS、Hope）在多個任務上的有效性：",[5193,16437,16438,16463,16482,16492,16505,16515,16528],{},[275,16439,16440,16443],{},[138,16441,16442],{},"Hope：持續學習與長上下文理解",[272,16444,16445,16451,16457],{},[275,16446,16447,16450],{},[138,16448,16449],{},"類增量學習","（CLINC, Banking, DBpedia）：Hope 在持續學習任務上優於 ICL、EWC、InCA 等基線，表明 CMS 設計對增強持續學習能力有效。",[275,16452,16453,16456],{},[138,16454,16455],{},"層級對上下文學習的影響","：在 LongHealth、QASPER、MK-NIAH 任務上，更多記憶層級和適當的更新頻率能提升模型的長上下文理解和 ICL 能力。",[275,16458,16459,16462],{},[138,16460,16461],{},"上下文學習新語言 (CTNL)","：Hope 在持續翻譯新語言任務上顯著優於 ICL，能有效緩解災難性遺忘。",[275,16464,16465,16468],{},[138,16466,16467],{},"Hope：長上下文理解",[272,16469,16470,16476],{},[275,16471,16472,16475],{},[138,16473,16474],{},"大海撈針 (NIAH) 任務","：Hope 在各種設置（單針、多鍵、多查詢、多值）和長度下，性能優於其他注意力無關模型（RWKV-7, Comba, Titans）及 Transformer。Hope-Attention 也優於普通 Transformer。",[275,16477,16478,16481],{},[138,16479,16480],{},"BABILong 基準測試","：Hope 在長達 10M 的上下文長度上保持良好性能，優於其他小型微調模型和大型零樣本模型。",[275,16483,16484,16487],{},[138,16485,16486],{},"Hope：語言建模與常識推理",[272,16488,16489],{},[275,16490,16491],{},"在 Wikitext、LAMBADA、PIQA 等多個基準測試上，Hope 在平均性能上優於所有基線模型（包括 Transformer 和各種 RNNs），並且在參數規模擴大時表現出更高的性能增益。",[275,16493,16494,16497],{},[138,16495,16496],{},"Hope：上下文召回任務與 MAD 合成基準測試",[272,16498,16499,16502],{},[275,16500,16501],{},"在多個上下文召回數據集上，Hope 優於所有注意力無關模型，縮小了與 Transformer 的差距。",[275,16503,16504],{},"在 MAD 合成基準測試上，Hope 在壓縮、記憶、複製等任務上優於所有基線，包括 Transformer。",[275,16506,16507,16510],{},[138,16508,16509],{},"語言識別任務",[272,16511,16512],{},[275,16513,16514],{},"在形式語言識別任務上，Hope 取得了與 LSTM、SRWM 等非線性循環模型相當的完美分數，同時保持了可並行訓練的優勢。",[275,16516,16517,16520],{},[138,16518,16519],{},"Hope：消融研究與擴展",[272,16521,16522,16525],{},[275,16523,16524],{},"消融研究表明，Hope 的所有元件（DGD、動量、權重衰減、CMS、內部投影）都對其性能有積極貢獻。",[275,16526,16527],{},"上下文使用越多，模型困惑度越低，表明其具有強大的記憶體管理能力。",[275,16529,16530,16532],{},[138,16531,15813],{},[272,16533,16534,16540],{},[275,16535,16536,16539],{},[138,16537,16538],{},"ImageNet","：在 ImageNet-21K 上訓練 ViT，M3 最佳化器比 AdamW 和 Muon 獲得了更低的訓練\u002F測試損失。",[275,16541,16542,16545],{},[138,16543,16544],{},"大模型效率","：M3 由於使用多個動量（記憶體），訓練速度相對 Muon 較慢，但與 AdaMuon 效率相當。",[5479,16547],{},[5454,16549,16550],{"id":5100},"🔚 結論",[272,16552,16553,16560,16567,16570,16577,16587],{},[275,16554,16555,16556,16559],{},"本文提出了 ",[138,16557,16558],{},"嵌套學習 (NL)","，這是一個新的學習典範，將現代機器學習系統建模為相互關聯的、多層次的優化問題。",[275,16561,16562,16563,16566],{},"從這個角度看，",[138,16564,16565],{},"架構和最佳化器都是嵌套關聯記憶體系統的實例","，它們壓縮自己的上下文（令牌、梯度或高層信號）。",[275,16568,16569],{},"NL 將預訓練、上下文學習和持續學習重新建構為同一底層機制的表現：在不同層次和時間尺度上學習和重用上下文。",[275,16571,16572,16573,16576],{},"基於 NL 的觀點，推導了 ",[138,16574,16575],{},"廣義的基於梯度的更新","（如 DGD、Delta Momentum、M3），並將現代序列架構重新解釋為嵌套關聯記憶體。",[275,16578,16579,16580,16582,16583,16586],{},"為了增強記憶體處理，引入了 ",[138,16581,16277],{},"。基於自修改 Titans 和 CMS 的 ",[138,16584,16585],{},"Hope 架構"," 提高了持續學習和長上下文推理能力，同時作為通用骨幹具有競爭力。",[275,16588,16589,16592],{},[138,16590,16591],{},"災難性遺忘解決了嗎？"," 從 NL 視角看，災難性遺忘是壓縮的自然結果，因為網路的有限容量迫使模型遺忘以保留新資訊的容量。NL 是一個路線圖，它表明在持續學習、長上下文推理、現代最佳化器和自修改模型方面的進展將來自於更好地利用「層次」這一額外設計軸，而不是來自更深的靜態網路。",{"title":311,"searchDepth":312,"depth":312,"links":16594},[16595,16596],{"id":15752,"depth":4984,"text":15755},{"id":15759,"depth":312,"text":15747,"children":16597},[16598],{"id":15777,"depth":4984,"text":15778},"\u002Fimages\u002Fblog\u002FIntroducing_Nested_Learning.webp","2025-12-22","提出了 嵌套學習 這一新的學習典範，將機器學習模型表示為一組嵌套的、多層次的優化問題，每一層都有自己的「上下文流」。從 NL 的角度看，現有深度學習方法通過壓縮自身的上下文流來從數據中學習，而大型模型中的上下文學習自然湧現。NL 提出了一種哲學思想，即設計具有更多「層次」、更富表達力的學習演算法，從而實現更高階的上下文學習，並可能解鎖有效的持續學習能力。",{},"\u002Fblog\u002Fzh-tw\u002Fintroducing-nested-learning",{"title":15747,"description":16601},"introducing-nested-learning","blog\u002Fzh-TW\u002Fintroducing-nested-learning",[5424,16608,16609],"nested-learning","memory","UnpmmILVDKQxNZFHYwWWtgxLsgOmngK3kB9I1u4qAtk",{"id":16612,"title":16613,"author":103,"body":16614,"cover":18200,"coverAlt":16613,"date":18201,"description":18202,"draft":324,"extension":325,"featured":324,"lang":327,"meta":18203,"navigation":326,"path":18204,"seo":18205,"slug":18206,"stem":18207,"tags":18208,"translationKey":18206,"updated":337,"__hash__":18211},"blog\u002Fblog\u002Fzh-TW\u002Foffset-geometric-contact.md","Offset Geometric Contact 論文彙整",{"type":105,"value":16615,"toc":18187},[16616,16621,16628,16631,16634,16638,16662,16666,16690,16694,16728,16732,16922,17531,17614,18041,18045,18073,18077,18140,18144,18172,18176,18184],[16617,16618],"lite-youtube",{"videoid":16619,"playlabel":16620},"xxyniqSLJik","[SIGGRAPH 2025] Offset Geometric Contact",[108,16622,16624],{"id":16623},"httpsgraphicscsutaheduresearchprojectsogc",[5055,16625,16626],{"href":16626,"rel":16627},"https:\u002F\u002Fgraphics.cs.utah.edu\u002Fresearch\u002Fprojects\u002Fogc\u002F",[5059],[108,16629,16613],{"id":16630},"offset-geometric-contact-論文彙整",[112,16632,16633],{},"以下是針對論文《Offset Geometric Contact》的條列式彙整，基於提供的文本內容整理。",[5119,16635,16637],{"id":16636},"_1-論文基本資訊","1. 論文基本資訊",[272,16639,16640,16646,16651,16657],{},[275,16641,16642,16645],{},[138,16643,16644],{},"標題","：Offset Geometric Contact (OGC)",[275,16647,16648,16650],{},[138,16649,15764],{},"：Anka He Chen (University of Utah, NVIDIA), Jerry Hsu (University of Utah), Ziheng Liu (University of Utah), Miles Macklin (NVIDIA), Yin Yang (University of Utah), Cem Yuksel (University of Utah)",[275,16652,16653,16656],{},[138,16654,16655],{},"出處","：ACM Trans. Graph. 44, 4 (August 2025), 21 pages",[275,16658,16659,16661],{},[138,16660,15841],{},"：解決共維物體（codimensional objects）的無穿透模擬（penetration-free simulation）問題，同時降低計算成本。",[5119,16663,16665],{"id":16664},"_2-研究背景與問題","2. 研究背景與問題",[272,16667,16668,16674],{},[275,16669,16670,16673],{},[138,16671,16672],{},"無穿透模擬的重要性","：在圖形應用中至關重要，穿透會導致模擬失真或失敗，尤其在共維模型中更難解決。",[275,16675,16676,16152,16679],{},[138,16677,16678],{},"現有方法的侷限",[272,16680,16681,16684,16687],{},[275,16682,16683],{},"IPC（Incremental Potential Contact）方法雖能保證無穿透，但計算成本高（比非保證方法慢數個數量級），且計算時間不穩定。",[275,16685,16686],{},"IPC使用連續碰撞檢測（CCD），導致全域同步問題，阻礙GPU並行化，且接觸力可能不正交於表面，引起拉伸偽影（stretching artifacts）。",[275,16688,16689],{},"小接觸半徑導致數值剛度（stiffness）問題，收斂困難。",[5119,16691,16693],{"id":16692},"_3-主要貢獻","3. 主要貢獻",[272,16695,16696],{},[275,16697,16698,16701,16702],{},[138,16699,16700],{},"提出OGC方法","：一種新穎的接觸模型，透過幾何偏移實現無穿透模擬，具有以下優勢：\n",[272,16703,16704,16710,16716,16722],{},[275,16705,16706,16709],{},[138,16707,16708],{},"正交接觸力","：確保接觸力始終垂直於表面，避免拉伸偽影。",[275,16711,16712,16715],{},[138,16713,16714],{},"大接觸半徑","：允許使用較大接觸半徑，降低接觸剛度，改善收斂。",[275,16717,16718,16721],{},[138,16719,16720],{},"無CCD需求","：透過局部頂點位移邊界保證無穿透，避免昂貴的CCD。",[275,16723,16724,16727],{},[138,16725,16726],{},"高效並行","：僅依賴局部操作，無需全域同步，適合GPU實現，效能提升兩個數量級以上。",[5119,16729,16731],{"id":16730},"_4-方法概述","4. 方法概述",[272,16733,16734,16747,16843,16853],{},[275,16735,16736,16152,16739],{},[138,16737,16738],{},"偏移幾何建構",[272,16740,16741,16744],{},[275,16742,16743],{},"基於多面體高斯映射（Polyhedral Gauss Map），將每個面沿法線方向偏移，建構體積塊（blocks）作為偏移幾何的基本單元。",[275,16745,16746],{},"包括頂點塊、邊塊和麵塊，確保局部獨佔性和覆蓋性。",[275,16748,16749,16152,16752],{},[138,16750,16751],{},"接觸面集定義",[272,16753,16754],{},[275,16755,16756,16757,16842],{},"使用 ",[360,16758,16760,16785],{"className":16759},[363],[360,16761,16763],{"className":16762},[367],[369,16764,16765],{"xmlns":371},[373,16766,16767,16782],{},[376,16768,16769],{},[628,16770,16771,16773],{},[379,16772,6571],{"mathvariant":6570},[376,16774,16775,16777,16780],{},[379,16776,3920],{"mathvariant":771},[379,16778,16779],{"mathvariant":771},"G",[379,16781,6590],{"mathvariant":771},[405,16783,16784],{"encoding":407},"\\mathcal{F}_{\\mathrm{OGC}}",[360,16786,16788],{"className":16787,"ariaHidden":413},[412],[360,16789,16791,16794],{"className":16790},[417],[360,16792],{"className":16793,"style":9140},[421],[360,16795,16797,16800],{"className":16796},[426],[360,16798,6571],{"className":16799,"style":6673},[426,6672],[360,16801,16803],{"className":16802},[457],[360,16804,16806,16834],{"className":16805},[461,713],[360,16807,16809,16831],{"className":16808},[465],[360,16810,16812],{"className":16811,"style":8873},[469],[360,16813,16815,16818],{"style":16814},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[360,16816],{"className":16817,"style":477},[476],[360,16819,16821],{"className":16820},[481,482,483,484],[360,16822,16824],{"className":16823},[426,484],[360,16825,16827],{"className":16826},[426,484],[360,16828,16830],{"className":16829},[426,856,484],"OGC",[360,16832,741],{"className":16833},[740],[360,16835,16837],{"className":16836},[465],[360,16838,16840],{"className":16839,"style":1339},[469],[360,16841],{},"選擇與頂點接觸的面，保證接觸力正交。",[275,16844,16845,16152,16848],{},[138,16846,16847],{},"穿透深度計算",[272,16849,16850],{},[275,16851,16852],{},"透過距離函數計算穿透深度，用於接觸能量公式。",[275,16854,16855,16152,16858],{},[138,16856,16857],{},"激活函數",[272,16859,16860],{},[275,16861,15903,16862,16921],{},[360,16863,16865,16883],{"className":16864},[363],[360,16866,16868],{"className":16867},[367],[369,16869,16870],{"xmlns":371},[373,16871,16872,16880],{},[376,16873,16874],{},[387,16875,16876,16878],{},[379,16877,6590],{},[1185,16879,4119],{},[405,16881,16882],{"encoding":407},"C^2",[360,16884,16886],{"className":16885,"ariaHidden":413},[412],[360,16887,16889,16892],{"className":16888},[417],[360,16890],{"className":16891,"style":10187},[421],[360,16893,16895,16898],{"className":16894},[426],[360,16896,6590],{"className":16897,"style":545},[426,427],[360,16899,16901],{"className":16900},[457],[360,16902,16904],{"className":16903},[461],[360,16905,16907],{"className":16906},[465],[360,16908,16910],{"className":16909,"style":10187},[469],[360,16911,16912,16915],{"style":472},[360,16913],{"className":16914,"style":477},[476],[360,16916,16918],{"className":16917},[481,482,483,484],[360,16919,4119],{"className":16920},[426,484]," 連續的兩階段激活函數，結合二次函數和對數函數的優點，提高收斂速度。",[360,16923,16925],{"className":16924},[755],[360,16926,16928,17067],{"className":16927},[363],[360,16929,16931],{"className":16930},[367],[369,16932,16933],{"xmlns":371,"display":764},[373,16934,16935,17064],{},[376,16936,16937,16939,16941,16943,16945,16947,16949,16951],{},[379,16938,8212],{},[383,16940,779],{"stretchy":778},[379,16942,403],{},[383,16944,784],{"separator":413},[379,16946,7224],{},[383,16948,793],{"stretchy":778},[383,16950,796],{},[376,16952,16953,16955],{},[383,16954,11732],{"fence":413},[4038,16956,16959,17012],{"rowspacing":16957,"columnalign":16958,"columnspacing":4069},"0.36em","left left",[4044,16960,16961,16991],{},[4047,16962,16963],{},[4050,16964,16965],{"scriptlevel":4052,"displaystyle":778},[376,16966,16967,16977,16979,16981,16983,16985],{},[811,16968,16969,16975],{},[628,16970,16971,16973],{},[379,16972,1219],{},[379,16974,7219],{},[1185,16976,4119],{},[383,16978,779],{"stretchy":778},[379,16980,7224],{},[383,16982,7367],{},[379,16984,403],{},[387,16986,16987,16989],{},[383,16988,793],{"stretchy":778},[1185,16990,4119],{},[4047,16992,16993],{},[4050,16994,16995],{"scriptlevel":4052,"displaystyle":778},[376,16996,16997,17000,17003,17006,17008,17010],{},[803,16998,16999],{},"if ",[379,17001,17002],{},"τ",[383,17004,17005],{},"≤",[379,17007,403],{},[383,17009,17005],{},[379,17011,7224],{},[4044,17013,17014,17045],{},[4047,17015,17016],{},[4050,17017,17018],{"scriptlevel":4052,"displaystyle":778},[376,17019,17020,17022,17030,17032,17034,17036,17038,17040,17042],{},[383,17021,7367],{},[1206,17023,17024,17026,17028],{},[379,17025,1219],{},[379,17027,7219],{},[383,17029,8151],{"mathvariant":771,"lspace":4042,"rspace":4042},[379,17031,8121],{},[383,17033,775],{},[383,17035,779],{"stretchy":778},[379,17037,403],{},[383,17039,793],{"stretchy":778},[383,17041,3976],{},[379,17043,17044],{},"b",[4047,17046,17047],{},[4050,17048,17049],{"scriptlevel":4052,"displaystyle":778},[376,17050,17051,17053,17055,17058,17060,17062],{},[803,17052,16999],{},[1185,17054,4052],{},[383,17056,17057],{},"\u003C",[379,17059,403],{},[383,17061,17057],{},[379,17063,17002],{},[405,17065,17066],{"encoding":407},"g(d, r) = \\begin{cases}\n      \\frac{k_c}{2}(r - d)^2 & \\text{if } \\tau \\leq d \\leq r \\\\\n      -k_c' \\log(d) + b & \\text{if } 0 \u003C d \u003C \\tau\n      \\end{cases}",[360,17068,17070,17106],{"className":17069,"ariaHidden":413},[412],[360,17071,17073,17076,17079,17082,17085,17088,17091,17094,17097,17100,17103],{"className":17072},[417],[360,17074],{"className":17075,"style":848},[421],[360,17077,8212],{"className":17078,"style":736},[426,427],[360,17080,779],{"className":17081},[860],[360,17083,403],{"className":17084},[426,427],[360,17086,784],{"className":17087},[867],[360,17089],{"className":17090,"style":871},[431],[360,17092,7224],{"className":17093,"style":3944},[426,427],[360,17095,793],{"className":17096},[887],[360,17098],{"className":17099,"style":432},[431],[360,17101,796],{"className":17102},[436],[360,17104],{"className":17105,"style":432},[431],[360,17107,17109,17113],{"className":17108},[417],[360,17110],{"className":17111,"style":17112},[421],"height:3em;vertical-align:-1.25em;",[360,17114,17116,17123,17528],{"className":17115},[920],[360,17117,17119],{"className":17118,"style":925},[860,924],[360,17120,11732],{"className":17121},[929,17122],"size4",[360,17124,17126],{"className":17125},[426],[360,17127,17129,17417,17422],{"className":17128},[4038],[360,17130,17132],{"className":17131},[4221],[360,17133,17135,17408],{"className":17134},[461,713],[360,17136,17138,17405],{"className":17137},[465],[360,17139,17141,17307],{"className":17140,"style":2075},[469],[360,17142,17144,17148],{"style":17143},"top:-3.69em;",[360,17145],{"className":17146,"style":17147},[476],"height:3.008em;",[360,17149,17151,17260,17263,17266,17269,17272,17275,17278],{"className":17150},[426],[360,17152,17154,17157,17257],{"className":17153},[426],[360,17155],{"className":17156},[860,936],[360,17158,17160],{"className":17159},[811],[360,17161,17163,17248],{"className":17162},[461,713],[360,17164,17166,17245],{"className":17165},[465],[360,17167,17170,17185,17193],{"className":17168,"style":17169},[469],"height:0.8962em;",[360,17171,17173,17176],{"style":17172},"top:-2.655em;",[360,17174],{"className":17175,"style":956},[476],[360,17177,17179],{"className":17178},[481,482,483,484],[360,17180,17182],{"className":17181},[426,484],[360,17183,4119],{"className":17184},[426,484],[360,17186,17187,17190],{"style":1026},[360,17188],{"className":17189,"style":956},[476],[360,17191],{"className":17192,"style":1034},[1033],[360,17194,17196,17199],{"style":17195},"top:-3.4101em;",[360,17197],{"className":17198,"style":956},[476],[360,17200,17202],{"className":17201},[481,482,483,484],[360,17203,17205],{"className":17204},[426,484],[360,17206,17208,17211],{"className":17207},[426,484],[360,17209,1219],{"className":17210,"style":1645},[426,427,484],[360,17212,17214],{"className":17213},[457],[360,17215,17217,17237],{"className":17216},[461,713],[360,17218,17220,17234],{"className":17219},[465],[360,17221,17223],{"className":17222,"style":720},[469],[360,17224,17225,17228],{"style":1661},[360,17226],{"className":17227,"style":727},[476],[360,17229,17231],{"className":17230},[481,731,732,484],[360,17232,7219],{"className":17233},[426,427,484],[360,17235,741],{"className":17236},[740],[360,17238,17240],{"className":17239},[465],[360,17241,17243],{"className":17242,"style":748},[469],[360,17244],{},[360,17246,741],{"className":17247},[740],[360,17249,17251],{"className":17250},[465],[360,17252,17255],{"className":17253,"style":17254},[469],"height:0.345em;",[360,17256],{},[360,17258],{"className":17259},[887,936],[360,17261,779],{"className":17262},[860],[360,17264,7224],{"className":17265,"style":3944},[426,427],[360,17267],{"className":17268,"style":650},[431],[360,17270,7367],{"className":17271},[494],[360,17273],{"className":17274,"style":650},[431],[360,17276,403],{"className":17277},[426,427],[360,17279,17281,17284],{"className":17280},[887],[360,17282,793],{"className":17283},[887],[360,17285,17287],{"className":17286},[457],[360,17288,17290],{"className":17289},[461],[360,17291,17293],{"className":17292},[465],[360,17294,17296],{"className":17295,"style":10187},[469],[360,17297,17298,17301],{"style":472},[360,17299],{"className":17300,"style":477},[476],[360,17302,17304],{"className":17303},[481,482,483,484],[360,17305,4119],{"className":17306},[426,484],[360,17308,17310,17313],{"style":17309},"top:-2.25em;",[360,17311],{"className":17312,"style":17147},[476],[360,17314,17316,17319,17376,17379,17384,17387,17390,17393,17396,17399,17402],{"className":17315},[426],[360,17317,7367],{"className":17318},[426],[360,17320,17322,17325],{"className":17321},[426],[360,17323,1219],{"className":17324,"style":1645},[426,427],[360,17326,17328],{"className":17327},[457],[360,17329,17331,17367],{"className":17330},[461,713],[360,17332,17334,17364],{"className":17333},[465],[360,17335,17338,17350],{"className":17336,"style":17337},[469],"height:0.7519em;",[360,17339,17341,17344],{"style":17340},"top:-2.453em;margin-left:-0.0315em;margin-right:0.05em;",[360,17342],{"className":17343,"style":477},[476],[360,17345,17347],{"className":17346},[481,482,483,484],[360,17348,7219],{"className":17349},[426,427,484],[360,17351,17352,17355],{"style":472},[360,17353],{"className":17354,"style":477},[476],[360,17356,17358],{"className":17357},[481,482,483,484],[360,17359,17361],{"className":17360},[426,484],[360,17362,8151],{"className":17363},[426,484],[360,17365,741],{"className":17366},[740],[360,17368,17370],{"className":17369},[465],[360,17371,17374],{"className":17372,"style":17373},[469],"height:0.247em;",[360,17375],{},[360,17377],{"className":17378,"style":871},[431],[360,17380,8208,17382],{"className":17381},[852],[360,17383,8212],{"style":8211},[360,17385,779],{"className":17386},[860],[360,17388,403],{"className":17389},[426,427],[360,17391,793],{"className":17392},[887],[360,17394],{"className":17395,"style":650},[431],[360,17397,3976],{"className":17398},[494],[360,17400],{"className":17401,"style":650},[431],[360,17403,17044],{"className":17404},[426,427],[360,17406,741],{"className":17407},[740],[360,17409,17411],{"className":17410},[465],[360,17412,17415],{"className":17413,"style":17414},[469],"height:1.19em;",[360,17416],{},[360,17418],{"className":17419,"style":17421},[17420],"arraycolsep","width:1em;",[360,17423,17425],{"className":17424},[4221],[360,17426,17428,17520],{"className":17427},[461,713],[360,17429,17431,17517],{"className":17430},[465],[360,17432,17434,17476],{"className":17433,"style":2075},[469],[360,17435,17436,17439],{"style":17143},[360,17437],{"className":17438,"style":17147},[476],[360,17440,17442,17448,17452,17455,17458,17461,17464,17467,17470,17473],{"className":17441},[426],[360,17443,17445],{"className":17444},[426,4177],[360,17446,16999],{"className":17447},[426],[360,17449,17002],{"className":17450,"style":17451},[426,427],"margin-right:0.1132em;",[360,17453],{"className":17454,"style":432},[431],[360,17456,17005],{"className":17457},[436],[360,17459],{"className":17460,"style":432},[431],[360,17462,403],{"className":17463},[426,427],[360,17465],{"className":17466,"style":432},[431],[360,17468,17005],{"className":17469},[436],[360,17471],{"className":17472,"style":432},[431],[360,17474,7224],{"className":17475,"style":3944},[426,427],[360,17477,17478,17481],{"style":17309},[360,17479],{"className":17480,"style":17147},[476],[360,17482,17484,17490,17493,17496,17499,17502,17505,17508,17511,17514],{"className":17483},[426],[360,17485,17487],{"className":17486},[426,4177],[360,17488,16999],{"className":17489},[426],[360,17491,4052],{"className":17492},[426],[360,17494],{"className":17495,"style":432},[431],[360,17497,17057],{"className":17498},[436],[360,17500],{"className":17501,"style":432},[431],[360,17503,403],{"className":17504},[426,427],[360,17506],{"className":17507,"style":432},[431],[360,17509,17057],{"className":17510},[436],[360,17512],{"className":17513,"style":432},[431],[360,17515,17002],{"className":17516,"style":17451},[426,427],[360,17518,741],{"className":17519},[740],[360,17521,17523],{"className":17522},[465],[360,17524,17526],{"className":17525,"style":17414},[469],[360,17527],{},[360,17529],{"className":17530},[887,936],[272,17532,17533],{},[275,17534,17535,16152,17538],{},[138,17536,17537],{},"無穿透保證",[272,17539,17540],{},[275,17541,17542,17543,17613],{},"為每個頂點計算保守位移邊界 ",[360,17544,17546,17564],{"className":17545},[363],[360,17547,17549],{"className":17548},[367],[369,17550,17551],{"xmlns":371},[373,17552,17553,17561],{},[376,17554,17555],{},[628,17556,17557,17559],{},[379,17558,17044],{},[379,17560,634],{},[405,17562,17563],{"encoding":407},"b_v",[360,17565,17567],{"className":17566,"ariaHidden":413},[412],[360,17568,17570,17573],{"className":17569},[417],[360,17571],{"className":17572,"style":13668},[421],[360,17574,17576,17579],{"className":17575},[426],[360,17577,17044],{"className":17578},[426,427],[360,17580,17582],{"className":17581},[457],[360,17583,17585,17605],{"className":17584},[461,713],[360,17586,17588,17602],{"className":17587},[465],[360,17589,17591],{"className":17590,"style":7469},[469],[360,17592,17593,17596],{"style":1320},[360,17594],{"className":17595,"style":477},[476],[360,17597,17599],{"className":17598},[481,482,483,484],[360,17600,634],{"className":17601,"style":736},[426,427,484],[360,17603,741],{"className":17604},[740],[360,17606,17608],{"className":17607},[465],[360,17609,17611],{"className":17610,"style":1339},[469],[360,17612],{},"，限制位移以避免穿透，替代IPC的CCD-aware線搜索。",[360,17615,17617],{"className":17616},[755],[360,17618,17620,17707],{"className":17619},[363],[360,17621,17623],{"className":17622},[367],[369,17624,17625],{"xmlns":371,"display":764},[373,17626,17627,17704],{},[376,17628,17629,17635,17637,17645,17648,17650,17652,17666,17668,17684,17686,17702],{},[628,17630,17631,17633],{},[379,17632,17044],{},[379,17634,634],{},[383,17636,796],{},[628,17638,17639,17642],{},[379,17640,17641],{},"γ",[379,17643,17644],{"mathvariant":6570},"P",[379,17646,17647],{},"min",[383,17649,775],{},[383,17651,779],{"stretchy":778},[628,17653,17654,17656],{},[379,17655,403],{},[376,17657,17658,17660,17662,17664],{},[379,17659,17647],{},[383,17661,775],{},[383,17663,784],{"separator":413},[379,17665,634],{},[383,17667,784],{"separator":413},[1206,17669,17670,17672,17682],{},[379,17671,403],{},[376,17673,17674,17676,17678,17680],{},[379,17675,17647],{},[383,17677,775],{},[383,17679,784],{"separator":413},[379,17681,634],{},[379,17683,10685],{},[383,17685,784],{"separator":413},[1206,17687,17688,17690,17700],{},[379,17689,403],{},[376,17691,17692,17694,17696,17698],{},[379,17693,17647],{},[383,17695,775],{},[383,17697,784],{"separator":413},[379,17699,634],{},[379,17701,6791],{},[383,17703,793],{"stretchy":778},[405,17705,17706],{"encoding":407},"b_v = \\gamma_{\\mathcal{P}} \\min(d_{\\min,v}, d_{\\min,v}^E, d_{\\min,v}^T)",[360,17708,17710,17765],{"className":17709,"ariaHidden":413},[412],[360,17711,17713,17716,17756,17759,17762],{"className":17712},[417],[360,17714],{"className":17715,"style":13668},[421],[360,17717,17719,17722],{"className":17718},[426],[360,17720,17044],{"className":17721},[426,427],[360,17723,17725],{"className":17724},[457],[360,17726,17728,17748],{"className":17727},[461,713],[360,17729,17731,17745],{"className":17730},[465],[360,17732,17734],{"className":17733,"style":7469},[469],[360,17735,17736,17739],{"style":1320},[360,17737],{"className":17738,"style":477},[476],[360,17740,17742],{"className":17741},[481,482,483,484],[360,17743,634],{"className":17744,"style":736},[426,427,484],[360,17746,741],{"className":17747},[740],[360,17749,17751],{"className":17750},[465],[360,17752,17754],{"className":17753,"style":1339},[469],[360,17755],{},[360,17757],{"className":17758,"style":432},[431],[360,17760,796],{"className":17761},[436],[360,17763],{"className":17764,"style":432},[431],[360,17766,17768,17772,17817,17820,17823,17826,17885,17888,17891,17963,17966,17969,18038],{"className":17767},[417],[360,17769],{"className":17770,"style":17771},[421],"height:1.2744em;vertical-align:-0.3831em;",[360,17773,17775,17779],{"className":17774},[426],[360,17776,17641],{"className":17777,"style":17778},[426,427],"margin-right:0.0556em;",[360,17780,17782],{"className":17781},[457],[360,17783,17785,17809],{"className":17784},[461,713],[360,17786,17788,17806],{"className":17787},[465],[360,17789,17791],{"className":17790,"style":8873},[469],[360,17792,17794,17797],{"style":17793},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[360,17795],{"className":17796,"style":477},[476],[360,17798,17800],{"className":17799},[481,482,483,484],[360,17801,17803],{"className":17802},[426,484],[360,17804,17644],{"className":17805,"style":8318},[426,6672,484],[360,17807,741],{"className":17808},[740],[360,17810,17812],{"className":17811},[465],[360,17813,17815],{"className":17814,"style":1339},[469],[360,17816],{},[360,17818],{"className":17819,"style":871},[431],[360,17821,17647],{"className":17822},[852],[360,17824,779],{"className":17825},[860],[360,17827,17829,17832],{"className":17828},[426],[360,17830,403],{"className":17831},[426,427],[360,17833,17835],{"className":17834},[457],[360,17836,17838,17877],{"className":17837},[461,713],[360,17839,17841,17874],{"className":17840},[465],[360,17842,17845],{"className":17843,"style":17844},[469],"height:0.3175em;",[360,17846,17847,17850],{"style":1320},[360,17848],{"className":17849,"style":477},[476],[360,17851,17853],{"className":17852},[481,482,483,484],[360,17854,17856,17868,17871],{"className":17855},[426,484],[360,17857,17859,17862,17865],{"className":17858},[852,484],[360,17860,524],{"className":17861},[484],[360,17863,1126],{"className":17864},[484],[360,17866,397],{"className":17867},[484],[360,17869,784],{"className":17870},[867,484],[360,17872,634],{"className":17873,"style":736},[426,427,484],[360,17875,741],{"className":17876},[740],[360,17878,17880],{"className":17879},[465],[360,17881,17883],{"className":17882,"style":2063},[469],[360,17884],{},[360,17886,784],{"className":17887},[867],[360,17889],{"className":17890,"style":871},[431],[360,17892,17894,17897],{"className":17893},[426],[360,17895,403],{"className":17896},[426,427],[360,17898,17900],{"className":17899},[457],[360,17901,17903,17954],{"className":17902},[461,713],[360,17904,17906,17951],{"className":17905},[465],[360,17907,17910,17940],{"className":17908,"style":17909},[469],"height:0.8913em;",[360,17911,17913,17916],{"style":17912},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[360,17914],{"className":17915,"style":477},[476],[360,17917,17919],{"className":17918},[481,482,483,484],[360,17920,17922,17934,17937],{"className":17921},[426,484],[360,17923,17925,17928,17931],{"className":17924},[852,484],[360,17926,524],{"className":17927},[484],[360,17929,1126],{"className":17930},[484],[360,17932,397],{"className":17933},[484],[360,17935,784],{"className":17936},[867,484],[360,17938,634],{"className":17939,"style":736},[426,427,484],[360,17941,17942,17945],{"style":4335},[360,17943],{"className":17944,"style":477},[476],[360,17946,17948],{"className":17947},[481,482,483,484],[360,17949,10685],{"className":17950,"style":12729},[426,427,484],[360,17952,741],{"className":17953},[740],[360,17955,17957],{"className":17956},[465],[360,17958,17961],{"className":17959,"style":17960},[469],"height:0.3831em;",[360,17962],{},[360,17964,784],{"className":17965},[867],[360,17967],{"className":17968,"style":871},[431],[360,17970,17972,17975],{"className":17971},[426],[360,17973,403],{"className":17974},[426,427],[360,17976,17978],{"className":17977},[457],[360,17979,17981,18030],{"className":17980},[461,713],[360,17982,17984,18027],{"className":17983},[465],[360,17985,17987,18016],{"className":17986,"style":17909},[469],[360,17988,17989,17992],{"style":17912},[360,17990],{"className":17991,"style":477},[476],[360,17993,17995],{"className":17994},[481,482,483,484],[360,17996,17998,18010,18013],{"className":17997},[426,484],[360,17999,18001,18004,18007],{"className":18000},[852,484],[360,18002,524],{"className":18003},[484],[360,18005,1126],{"className":18006},[484],[360,18008,397],{"className":18009},[484],[360,18011,784],{"className":18012},[867,484],[360,18014,634],{"className":18015,"style":736},[426,427,484],[360,18017,18018,18021],{"style":4335},[360,18019],{"className":18020,"style":477},[476],[360,18022,18024],{"className":18023},[481,482,483,484],[360,18025,6791],{"className":18026,"style":6960},[426,427,484],[360,18028,741],{"className":18029},[740],[360,18031,18033],{"className":18032},[465],[360,18034,18036],{"className":18035,"style":17960},[469],[360,18037],{},[360,18039,793],{"className":18040},[887],[5119,18042,18044],{"id":18043},"_5-關鍵演算法","5. 關鍵演算法",[272,18046,18047,18060],{},[275,18048,18049,16152,18052],{},[138,18050,18051],{},"接觸檢測演算法",[272,18053,18054,18057],{},[275,18055,18056],{},"頂點-面接觸檢測：使用BVH進行球查詢，並行計算接觸面和最小距離。",[275,18058,18059],{},"邊-邊接觸檢測：類似方法，處理邊之間的接觸。",[275,18061,18062,16152,18065],{},[138,18063,18064],{},"模擬管道",[272,18066,18067,18070],{},[275,18068,18069],{},"整合後向歐拉時間積分，結合VBD（Vertex Block Descent）求解器。",[275,18071,18072],{},"步驟包括接觸檢測、模擬求解和位移截斷，僅在頂點超出邊界時更新接觸檢測。",[5119,18074,18076],{"id":18075},"_6-實驗結果","6. 實驗結果",[272,18078,18079,18092,18117,18127],{},[275,18080,18081,16152,18084],{},[138,18082,18083],{},"效能提升",[272,18085,18086,18089],{},[275,18087,18088],{},"在GPU上實現即時大規模模擬，比IPC快100倍以上（最高達343倍）。",[275,18090,18091],{},"計算時間穩定，接近恆定預算。",[275,18093,18094,16152,18097],{},[138,18095,18096],{},"應用場景",[272,18098,18099,18105,18111],{},[275,18100,18101,18104],{},[138,18102,18103],{},"布料模擬","：處理50層布料碰撞、複雜打結、扭曲等極端情況，無穿透。",[275,18106,18107,18110],{},[138,18108,18109],{},"紗線級模擬","：防止紗線穿透和糾纏，保持結構完整性。",[275,18112,18113,18116],{},[138,18114,18115],{},"耦合模擬","：布料與人體或機器人互動，即時效能。",[275,18118,18119,16152,18122],{},[138,18120,18121],{},"收斂性",[272,18123,18124],{},[275,18125,18126],{},"OGC與VBD或牛頓法結合均快速收斂，VBD在GPU上尤其高效。",[275,18128,18129,16152,18132],{},[138,18130,18131],{},"與IPC比較",[272,18133,18134,18137],{},[275,18135,18136],{},"OGC避免IPC的數值阻尼問題，在低迭代數下保持動態性。",[275,18138,18139],{},"使用大接觸半徑時，OGC無偽影，而IPC會產生振盪和拉伸。",[5119,18141,18143],{"id":18142},"_7-侷限性與未來工作","7. 侷限性與未來工作",[272,18145,18146,18159],{},[275,18147,18148,16152,18151],{},[138,18149,18150],{},"侷限性",[272,18152,18153,18156],{},[275,18154,18155],{},"接觸力在凹區域邊界可能不連續，但實驗中未觀測到不穩定。",[275,18157,18158],{},"在高速、少碰撞場景中，位移邊界可能限制收斂速度，不如IPC的CCD高效。",[275,18160,18161,16152,18164],{},[138,18162,18163],{},"未來工作",[272,18165,18166,18169],{},[275,18167,18168],{},"智慧切換不同無穿透技術，或結合位移方向以 tighter 邊界。",[275,18170,18171],{},"與機器人、動畫和醫學模擬社群合作，改進實用性。",[5119,18173,18175],{"id":18174},"_8-結論","8. 結論",[272,18177,18178,18181],{},[275,18179,18180],{},"OGC提供了一種高效、穩定的無穿透接觸模型，透過幾何偏移和局部並行操作，顯著降低計算成本，適用於即時和大規模模擬。",[275,18182,18183],{},"程式碼將開源，促進後續研究和應用。",[112,18185,18186],{},"此彙整覆蓋了論文的核心內容，突出方法創新、技術細節和實驗結果。",{"title":311,"searchDepth":312,"depth":312,"links":18188},[18189,18190],{"id":16623,"depth":312,"text":16626},{"id":16630,"depth":312,"text":16613,"children":18191},[18192,18193,18194,18195,18196,18197,18198,18199],{"id":16636,"depth":4984,"text":16637},{"id":16664,"depth":4984,"text":16665},{"id":16692,"depth":4984,"text":16693},{"id":16730,"depth":4984,"text":16731},{"id":18043,"depth":4984,"text":18044},{"id":18075,"depth":4984,"text":18076},{"id":18142,"depth":4984,"text":18143},{"id":18174,"depth":4984,"text":18175},"\u002Fimages\u002Fblog\u002F2025-10-29.webp","2025-10-29","關於《Offset Geometric Contact》論文的詳細條列式彙整，涵蓋方法、演算法和實驗結果。",{},"\u002Fblog\u002Fzh-tw\u002Foffset-geometric-contact",{"title":16613,"description":18202},"offset-geometric-contact","blog\u002Fzh-TW\u002Foffset-geometric-contact",[5424,18209,18210],"physics","collision","B4BJiTsJ1K8WoHkfI7I8185qrUCE5ZkbEjjchoD0tAo",{"id":18213,"title":18214,"author":103,"body":18215,"cover":18844,"coverAlt":18214,"date":18845,"description":18846,"draft":324,"extension":325,"featured":324,"lang":327,"meta":18847,"navigation":326,"path":18848,"seo":18849,"slug":18850,"stem":18851,"tags":18852,"translationKey":18850,"updated":337,"__hash__":18855},"blog\u002Fblog\u002Fzh-TW\u002Farxiv-2506-13777v1.md","複雜城市系統的實體資訊AI綜述",{"type":105,"value":18216,"toc":18821},[18217,18225,18229,18233,18266,18268,18272,18276,18395,18397,18401,18405,18513,18515,18519,18523,18527,18547,18551,18571,18575,18595,18599,18619,18623,18637,18641,18655,18659,18673,18677,18691,18693,18697,18701,18704,18724,18728,18765,18769,18785,18789,18800,18802,18806],[108,18218,18220,18221],{"id":18219},"論文位置httpsarxivorghtml250613777v1","論文位置：",[5055,18222,18223],{"href":18223,"rel":18224},"https:\u002F\u002Farxiv.org\u002Fhtml\u002F2506.13777v1",[5059],[5119,18226,18228],{"id":18227},"a-survey-of-physics-informed-ai-for-complex-urban-systems","A Survey of Physics-Informed AI for Complex Urban Systems",[108,18230,18232],{"id":18231},"_1-引言-introduction","1. 引言 (Introduction)",[272,18234,18235,18241,18247,18252,18257,18260,18263],{},[275,18236,18237,18240],{},[138,18238,18239],{},"背景","：物理與AI的融合是科學研究的新範式，2024年諾貝爾物理學獎和化學獎都授予AI相關成果，凸顯了這一趨勢。",[275,18242,18243,18246],{},[138,18244,18245],{},"城市系統的複雜性","：傳統實體或純資料驅動方法難以應對城市系統的多維動態互動挑戰。",[275,18248,18249,18251],{},[138,18250,15841],{},"：如何將物理規律（如流體力學、熱傳導、社會力模型）與AI的數據驅動能力結合，以提升城市系統的預測準確性、可解釋性和決策能力。",[275,18253,18254,4023],{},[138,18255,18256],{},"論文貢獻",[275,18258,18259],{},"提出新的PIAI方法分類法（三大典範、七種方法）；",[275,18261,18262],{},"系統性回顧PIAI在八大城市子系統的應用；",[275,18264,18265],{},"指出目前限制並規劃未來研究方向。",[5479,18267],{},[108,18269,18271],{"id":18270},"_2-物理資訊ai方法-physics-informed-ai-methods","2. 物理資訊AI方法 (Physics-informed AI Methods)",[5119,18273,18275],{"id":18274},"三大典範與七種方法","三大典範與七種方法：",[4848,18277,18278,18293],{},[4851,18279,18280],{},[4854,18281,18282,18285,18288,18290],{},[4857,18283,18284],{},"範式",[4857,18286,18287],{},"方法",[4857,18289,15961],{},[4857,18291,18292],{},"典型技術",[4864,18294,18295,18311,18324,18337,18353,18366,18382],{},[4854,18296,18297,18302,18305,18308],{},[4869,18298,18299],{},[138,18300,18301],{},"實體整合AI",[4869,18303,18304],{},"1. PINN with Loss Function",[4869,18306,18307],{},"將物理約束作為損失項嵌入神經網路",[4869,18309,18310],{},"PINN, CPINN, PCNN",[4854,18312,18313,18315,18318,18321],{},[4869,18314],{},[4869,18316,18317],{},"2. PINN with Weight Initialization",[4869,18319,18320],{},"用物理知識初始化網路權重，提升訓練效率",[4869,18322,18323],{},"遷移學習、自我監督預訓練",[4854,18325,18326,18328,18331,18334],{},[4869,18327],{},[4869,18329,18330],{},"3. PINN with Architecture Design",[4869,18332,18333],{},"將物理結構嵌入網路架構，增強可解釋性",[4869,18335,18336],{},"對稱性編碼、哈密頓結構",[4854,18338,18339,18344,18347,18350],{},[4869,18340,18341],{},[138,18342,18343],{},"物理-AI混合整合",[4869,18345,18346],{},"4. Sequential Physics-AI Ensemble",[4869,18348,18349],{},"物理與AI模型順序協作，分階段處理",[4869,18351,18352],{},"殘差建模、管道架構",[4854,18354,18355,18357,18360,18363],{},[4869,18356],{},[4869,18358,18359],{},"5. Parallel Physics-AI Ensemble",[4869,18361,18362],{},"物理與AI模型並行處理，結果融合",[4869,18364,18365],{},"動態加權、多模型輸出組合",[4854,18367,18368,18373,18376,18379],{},[4869,18369,18370],{},[138,18371,18372],{},"AI整合物理",[4869,18374,18375],{},"6. AI-Discovered Physics Model",[4869,18377,18378],{},"用AI從資料自動發現物理規律",[4869,18380,18381],{},"SINDy框架、符號迴歸",[4854,18383,18384,18386,18389,18392],{},[4869,18385],{},[4869,18387,18388],{},"7. Neural Physics Model",[4869,18390,18391],{},"以AI模組取代物理模型中難以建模的部分",[4869,18393,18394],{},"神經網路取代湍流閉合項",[5479,18396],{},[108,18398,18400],{"id":18399},"_3-複雜城市系統及涉及的物理定律-complex-urban-systems-and-involved-physical-laws","3. 複雜城市系統及涉及的物理定律 (Complex Urban Systems and Involved Physical Laws)",[5119,18402,18404],{"id":18403},"七大子系統及其核心物理原理","七大子系統及其核心物理原理：",[4848,18406,18407,18420],{},[4851,18408,18409],{},[4854,18410,18411,18414,18417],{},[4857,18412,18413],{},"子系統",[4857,18415,18416],{},"物理定律\u002F模型",[4857,18418,18419],{},"應用範例",[4864,18421,18422,18435,18448,18461,18474,18487,18500],{},[4854,18423,18424,18429,18432],{},[4869,18425,18426],{},[138,18427,18428],{},"能源",[4869,18430,18431],{},"電力潮流方程式、壓力平衡方程式",[4869,18433,18434],{},"電網最佳化、再生能源整合",[4854,18436,18437,18442,18445],{},[4869,18438,18439],{},[138,18440,18441],{},"環境",[4869,18443,18444],{},"高斯煙羽模型、SWMM水文模型",[4869,18446,18447],{},"空氣品質預測、洪水模擬",[4854,18449,18450,18455,18458],{},[4869,18451,18452],{},[138,18453,18454],{},"經濟",[4869,18456,18457],{},"網路流模型、Huff消費選擇模型",[4869,18459,18460],{},"供應鏈最佳化、市場需求預測",[4854,18462,18463,18468,18471],{},[4869,18464,18465],{},[138,18466,18467],{},"交通",[4869,18469,18470],{},"LWR交通流模型、跟馳模型",[4869,18472,18473],{},"交通狀態估計、號誌控制",[4854,18475,18476,18481,18484],{},[4869,18477,18478],{},[138,18479,18480],{},"資訊",[4869,18482,18483],{},"香農容量公式、射線追蹤模型",[4869,18485,18486],{},"社群網路資訊傳播預測",[4854,18488,18489,18494,18497],{},[4869,18490,18491],{},[138,18492,18493],{},"公共服務",[4869,18495,18496],{},"SIR流行病模型、社會力模型",[4869,18498,18499],{},"疫情預測、人群疏散模擬",[4854,18501,18502,18507,18510],{},[4869,18503,18504],{},[138,18505,18506],{},"緊急管理",[4869,18508,18509],{},"社會力模型、時空衰減模型",[4869,18511,18512],{},"災害動力學建模、災後人口流動預測",[5479,18514],{},[108,18516,18518],{"id":18517},"_4-實體資訊ai在複雜城市系統中的應用-piai-in-complex-urban-systems","4. 實體資訊AI在複雜城市系統中的應用 (PIAI in Complex Urban Systems)",[5119,18520,18522],{"id":18521},"分領域應用摘要","分領域應用摘要：",[5454,18524,18526],{"id":18525},"_41-能源系統","4.1 能源系統",[272,18528,18529,18535,18541],{},[275,18530,18531,18534],{},[138,18532,18533],{},"發電","：PINN用於風力發電場重建、太陽能輸出預測；",[275,18536,18537,18540],{},[138,18538,18539],{},"輸電","：結合基爾霍夫定律的神經網路加速最優潮流計算；",[275,18542,18543,18546],{},[138,18544,18545],{},"用電","：整合價格-需求關係、熱力學模型預測EV充電需求。",[5454,18548,18550],{"id":18549},"_42-環境系統","4.2 環境系統",[272,18552,18553,18559,18565],{},[275,18554,18555,18558],{},[138,18556,18557],{},"空氣","：擴散-平流方程式與神經網路結合提升污染物預測精度；",[275,18560,18561,18564],{},[138,18562,18563],{},"水","：PINN求解Richards方程式模擬土壤水流動；",[275,18566,18567,18570],{},[138,18568,18569],{},"碳","：實體資訊網路整合遙感資料估算碳儲量。",[5454,18572,18574],{"id":18573},"_43-交通系統","4.3 交通系統",[272,18576,18577,18583,18589],{},[275,18578,18579,18582],{},[138,18580,18581],{},"狀態估計","：LWR\u002FARZ模型與深度學習融合，從稀疏感測器資料推斷交通狀態；",[275,18584,18585,18588],{},[138,18586,18587],{},"流量預測","：時空微分方程網路（STDEN）建模潛在能量場驅動流量變化；",[275,18590,18591,18594],{},[138,18592,18593],{},"控制","：物理約束強化學習用於信號燈和匝道控制最佳化。",[5454,18596,18598],{"id":18597},"_44-經濟系統","4.4 經濟系統",[272,18600,18601,18607,18613],{},[275,18602,18603,18606],{},[138,18604,18605],{},"製造","：PINN解決熱傳導問題，優化工業流程；",[275,18608,18609,18612],{},[138,18610,18611],{},"金融","：Black-Scholes方程式與神經網路結合用於選擇權定價；",[275,18614,18615,18618],{},[138,18616,18617],{},"商品與POI","：結合規模定律與軌跡流程圖預測公司成長與使用者移動模式。",[5454,18620,18622],{"id":18621},"_45-資訊系統","4.5 資訊系統",[272,18624,18625,18631],{},[275,18626,18627,18630],{},[138,18628,18629],{},"資訊傳播","：基於熱傳導方程式建模社交網路資訊密度流；",[275,18632,18633,18636],{},[138,18634,18635],{},"觀點演化","：神經ODE模擬社群網路中使用者觀點的動態變化。",[5454,18638,18640],{"id":18639},"_46-公共服務","4.6 公共服務",[272,18642,18643,18649],{},[275,18644,18645,18648],{},[138,18646,18647],{},"公共衛生","：EINN（流行病學資訊神經網路）融合多源資料提升疫情預測；",[275,18650,18651,18654],{},[138,18652,18653],{},"公共安全","：社會力模型與AI結合優化人群模擬和車輛軌跡預測。",[5454,18656,18658],{"id":18657},"_47-緊急管理","4.7 緊急管理",[272,18660,18661,18667],{},[275,18662,18663,18666],{},[138,18664,18665],{},"災害預測","：PINN用於洪水峰值深度預測、地震前兆偵測；",[275,18668,18669,18672],{},[138,18670,18671],{},"人群模擬","：物理知識增強的生成模型模擬災時人群移動。",[5454,18674,18676],{"id":18675},"_48-城市系統整體","4.8 城市系統整體",[272,18678,18679,18685],{},[275,18680,18681,18684],{},[138,18682,18683],{},"跨領域耦合","：圖神經網路耦合洪水動力學與交通模型，模擬極端天氣下的城市運作；",[275,18686,18687,18690],{},[138,18688,18689],{},"多尺度整合","：從建築級熱模型到區域氣候模式的統一PIAI框架。",[5479,18692],{},[108,18694,18696],{"id":18695},"_5-討論與開放問題-discussions-and-open-problems","5. 討論與開放問題 (Discussions and Open Problems)",[5119,18698,18700],{"id":18699},"_51-方法選擇指南","5.1 方法選擇指南",[112,18702,18703],{},"選擇PIAI方法需考慮：",[272,18705,18706,18712,18718],{},[275,18707,18708,18711],{},[138,18709,18710],{},"物理知識可用性","（強約束 vs 弱約束）；",[275,18713,18714,18717],{},[138,18715,18716],{},"資料可用性","（豐富 vs 稀疏）；",[275,18719,18720,18723],{},[138,18721,18722],{},"任務目標","（即時性、準確性、可解釋性）。",[5119,18725,18727],{"id":18726},"_52-當前限制與未來方向","5.2 當前限制與未來方向",[272,18729,18730,18736,18742,18748,18753,18756,18759,18762],{},[275,18731,18732,18735],{},[138,18733,18734],{},"理論保障不足","：PINN的收斂性和泛化性缺乏嚴格證明；",[275,18737,18738,18741],{},[138,18739,18740],{},"運算效率","：訓練成本高，難以即時部署；",[275,18743,18744,18747],{},[138,18745,18746],{},"知識表示","：缺乏對因果、規則等非數值知識的整合；",[275,18749,18750,4023],{},[138,18751,18752],{},"未來方向",[275,18754,18755],{},"發展多解析度網路架構；",[275,18757,18758],{},"結合主動學習與遷移學習；",[275,18760,18761],{},"探索PIAI與LLM的融合；",[275,18763,18764],{},"建構城市數位孿生和通用世界模型。",[5119,18766,18768],{"id":18767},"_53-關鍵發現","5.3 關鍵發現",[272,18770,18771,18782],{},[275,18772,18773,18774,18777,18778,18781],{},"城市系統是PIAI的天然試驗場，因其兼具",[138,18775,18776],{},"物理規律性","與",[138,18779,18780],{},"人類活動不確定性","；",[275,18783,18784],{},"資料豐富的任務偏向AI主導方法，而決策支援任務更需要物理可解釋性。",[5119,18786,18788],{"id":18787},"_54-開放問題","5.4 開放問題",[272,18790,18791,18794,18797],{},[275,18792,18793],{},"如何為高維複雜環境中的PIAI建立理論根基？",[275,18795,18796],{},"如何建構統一框架整合確定性、隨機性和基於Agent的模型？",[275,18798,18799],{},"如何實現跨城市子系統的可遷移PIAI模型？",[5479,18801],{},[108,18803,18805],{"id":18804},"_6-結論-conclusion","6. 結論 (Conclusion)",[272,18807,18808,18815,18818],{},[275,18809,18810,18811,18814],{},"PIAI透過融合物理與AI，為城市系統建模提供了",[138,18812,18813],{},"更可靠、可解釋、自適應","的解決方案。",[275,18816,18817],{},"論文提出的分類法與應用綜述為後續研究提供了清晰路線圖。",[275,18819,18820],{},"未來需跨領域合作，推動PIAI在真實城市環境中的部署與驗證。",{"title":311,"searchDepth":312,"depth":312,"links":18822},[18823,18827,18828,18831,18834,18837,18843],{"id":18219,"depth":312,"text":18824,"children":18825},"論文位置：https:\u002F\u002Farxiv.org\u002Fhtml\u002F2506.13777v1",[18826],{"id":18227,"depth":4984,"text":18228},{"id":18231,"depth":312,"text":18232},{"id":18270,"depth":312,"text":18271,"children":18829},[18830],{"id":18274,"depth":4984,"text":18275},{"id":18399,"depth":312,"text":18400,"children":18832},[18833],{"id":18403,"depth":4984,"text":18404},{"id":18517,"depth":312,"text":18518,"children":18835},[18836],{"id":18521,"depth":4984,"text":18522},{"id":18695,"depth":312,"text":18696,"children":18838},[18839,18840,18841,18842],{"id":18699,"depth":4984,"text":18700},{"id":18726,"depth":4984,"text":18727},{"id":18767,"depth":4984,"text":18768},{"id":18787,"depth":4984,"text":18788},{"id":18804,"depth":312,"text":18805},"\u002Fimages\u002Fblog\u002F2506.13777.1.webp","2025-09-16","城市系統的物理資訊 AI 綜述：整理這篇 survey 如何把物理約束接進都市尺度的預測模型，以及它點出的資料、尺度與可解釋性缺口。",{},"\u002Fblog\u002Fzh-tw\u002Farxiv-2506-13777v1",{"title":18214,"description":18846},"arxiv-2506-13777v1","blog\u002Fzh-TW\u002Farxiv-2506-13777v1",[5424,18853,18854],"urban","survey","MX39x2Lyu08AZhatQPB9H_RGdiocApt5nsn4PPt8w_c",{"id":18857,"title":18858,"author":103,"body":18859,"cover":18954,"coverAlt":18858,"date":20126,"description":20127,"draft":324,"extension":325,"featured":324,"lang":327,"meta":20128,"navigation":326,"path":20129,"seo":20130,"slug":20131,"stem":20132,"tags":20133,"translationKey":20131,"updated":337,"__hash__":20136},"blog\u002Fblog\u002Fzh-TW\u002Farxiv-2005-03448v3.md","基於稀缺資料的物理資訊學習：從資料中發現控制方程式",{"type":105,"value":18860,"toc":20111},[18861,18868,18872,18875,18877,18907,18923,18925,18928,18939,18942,18947,18949,18955,18959,18967,18971,18976,19630,19635,19646,19651,19665,19670,19678,19682,19755,19760,19768,19774,20093,20097],[108,18862,18220,18864],{"id":18863},"論文位置httpsarxivorgabs200503448",[5055,18865,18866],{"href":18866,"rel":18867},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2005.03448",[5059],[5119,18869,18871],{"id":18870},"physics-informed-learning-of-governing-equations-from-scarce-data","Physics-informed learning of governing equations from scarce data",[5119,18873,18874],{"id":18874},"內容簡易翻譯",[108,18876,15764],{"id":15764},[112,18878,18879,18882,18885,18886,18889,18891,18892,18895,18898,18900,18901,18903,18904,18906],{},[138,18880,18881],{},"陳鑷（Zhao Chen）",[18883,18884,1187],"sup",{},"，*",[138,18887,18888],{},"劉陽（Yang Liu）",[18883,18890,4119],{},"，",[138,18893,18894],{},"孫浩（Hao Sun）",[18883,18896,18897],{},"1,3",[18883,18899,1187],{}," 美國東北大學土木與環境工程系（波士頓）\n",[18883,18902,4119],{}," 美國東北大學機械與工業工程系（波士頓）\n",[18883,18905,6508],{}," 麻省理工學院土木與環境工程系（劍橋）",[5180,18908,18909],{},[112,18910,18911],{},[5450,18912,18913,18914,18918,18919],{},"通訊作者信箱: ",[5055,18915,18917],{"href":18916},"mailto:yang1.liu@northeastern.edu","yang1.liu@northeastern.edu","; ",[5055,18920,18922],{"href":18921},"mailto:h.sun@northeastern.edu","h.sun@northeastern.edu",[108,18924,5464],{"id":5464},[112,18926,18927],{},"利用數據揭示描述複雜物理系統行為的基本控制定律或方程，能顯著推動多學科領域的建模、模擬與理解。本文提出一種新型物理資訊深度學習架構（PiDL），用於從稀缺且含雜訊的資料中發現非線性時空系統的控制偏微分方程（PDE）。此方法深度融合了深度神經網路（DNN）的表示學習能力、物理嵌入、自動微分與稀疏迴歸技術，實現三大目標：",[5193,18929,18930,18933,18936],{},[275,18931,18932],{},"逼近系統變數解",[275,18934,18935],{},"計算關鍵導數項",[275,18937,18938],{},"辨識構成PDE結構與明確表達的關鍵導數項及參數",[112,18940,18941],{},"透過數值與實驗驗證，此方法在資料稀缺性和雜訊幹擾下對多種PDE系統（考慮不同初\u002F邊值條件）的發現具有高效性與穩健性。該框架為實際應用中難以獲取大規模精確資料集的問題提供了閉式模型發現的潛力。",[112,18943,18944,18946],{},[138,18945,5476],{},"：控制方程式發現、物理資訊深度學習、偏微分方程（PDE）",[108,18948,5483],{"id":5483},[112,18950,18951],{},[5790,18952],{"alt":18953,"src":18954},"image","\u002Fimages\u002Fblog\u002F2005.03448v3.webp",[5119,18956,18958],{"id":18957},"_1-問題背景","1. 問題背景",[272,18960,18961,18964],{},[275,18962,18963],{},"傳統PDE建模依賴第一原理（如守恆定律），但許多複雜系統的解析描述未知",[275,18965,18966],{},"資料驅動方法（如SINDy）受限於資料品質與數量，數值微分對雜訊敏感",[5119,18968,18970],{"id":18969},"_2-pidl框架創新","2. PiDL框架創新",[112,18972,18973,4023],{},[138,18974,18975],{},"損失函數",[360,18977,18979],{"className":18978},[755],[360,18980,18982,19071],{"className":18981},[363],[360,18983,18985],{"className":18984},[367],[369,18986,18987],{"xmlns":371,"display":764},[373,18988,18989,19068],{},[376,18990,18991,18993,18995,18997,18999,19002,19004,19006,19022,19024,19026,19041,19043,19046],{},[379,18992,8453],{"mathvariant":6570},[383,18994,779],{"stretchy":778},[379,18996,6596],{},[383,18998,784],{"separator":413},[379,19000,19001],{"mathvariant":771},"Λ",[383,19003,793],{"stretchy":778},[383,19005,796],{},[2547,19007,19008,19019],{},[2547,19009,19010,19016],{},[628,19011,19012,19014],{},[379,19013,8453],{"mathvariant":6570},[379,19015,403],{},[383,19017,19018],{"stretchy":413},"⏟",[803,19020,19021],{},"數據損失",[383,19023,3976],{},[379,19025,12963],{},[2547,19027,19028,19038],{},[2547,19029,19030,19036],{},[628,19031,19032,19034],{},[379,19033,8453],{"mathvariant":6570},[379,19035,112],{},[383,19037,19018],{"stretchy":413},[803,19039,19040],{},"物理損失",[383,19042,3976],{},[379,19044,19045],{},"β",[2547,19047,19048,19065],{},[2547,19049,19050,19063],{},[376,19051,19052,19055,19057],{},[379,19053,19054],{"mathvariant":771},"∥",[379,19056,19001],{"mathvariant":771},[628,19058,19059,19061],{},[379,19060,19054],{"mathvariant":771},[1185,19062,4052],{},[383,19064,19018],{"stretchy":413},[803,19066,19067],{},"稀疏正則",[405,19069,19070],{"encoding":407},"\\mathcal{L}(\\theta, \\Lambda) = \\underbrace{\\mathcal{L}_d}_{\\text{數據損失}} + \\alpha \\underbrace{\\mathcal{L}_p}_{\\text{物理損失}} + \\beta \\underbrace{\\|\\Lambda\\|_0}_{\\text{稀疏正則}}",[360,19072,19074,19110,19289,19461],{"className":19073,"ariaHidden":413},[412],[360,19075,19077,19080,19083,19086,19089,19092,19095,19098,19101,19104,19107],{"className":19076},[417],[360,19078],{"className":19079,"style":848},[421],[360,19081,8453],{"className":19082},[426,6672],[360,19084,779],{"className":19085},[860],[360,19087,6596],{"className":19088,"style":3944},[426,427],[360,19090,784],{"className":19091},[867],[360,19093],{"className":19094,"style":871},[431],[360,19096,19001],{"className":19097},[426],[360,19099,793],{"className":19100},[887],[360,19102],{"className":19103,"style":432},[431],[360,19105,796],{"className":19106},[436],[360,19108],{"className":19109,"style":432},[431],[360,19111,19113,19117,19280,19283,19286],{"className":19112},[417],[360,19114],{"className":19115,"style":19116},[421],"height:2.1597em;vertical-align:-1.4763em;",[360,19118,19120],{"className":19119},[920,2547],[360,19121,19123,19271],{"className":19122},[461,713],[360,19124,19126,19268],{"className":19125},[465],[360,19127,19129,19147],{"className":19128,"style":5226},[469],[360,19130,19132,19135],{"style":19131},"top:-1.5237em;",[360,19133],{"className":19134,"style":956},[476],[360,19136,19138],{"className":19137},[481,482,483,484],[360,19139,19141],{"className":19140},[426,484],[360,19142,19144],{"className":19143},[426,4177,484],[360,19145,19021],{"className":19146},[426,4181,484],[360,19148,19149,19152],{"style":978},[360,19150],{"className":19151,"style":956},[476],[360,19153,19155],{"className":19154},[920,2547],[360,19156,19158,19259],{"className":19157},[461,713],[360,19159,19161,19256],{"className":19160},[465],[360,19162,19164,19208],{"className":19163,"style":5226},[469],[360,19165,19168,19171],{"className":19166,"style":19167},[977],"top:-2.202em;",[360,19169],{"className":19170,"style":956},[476],[360,19172,19176,19188,19198],{"className":19173,"style":19175},[19174],"stretchy","height:0.548em;min-width:1.6em;",[360,19177,19181],{"className":19178,"style":19180},[19179],"brace-left","height:0.548em;",[1001,19182,19185],{"xmlns":1003,"width":1004,"height":19183,"viewBox":19184,"preserveAspectRatio":1007},"0.548em","0 0 400000 548",[1009,19186],{"d":19187},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[360,19189,19192],{"className":19190,"style":19180},[19191],"brace-center",[1001,19193,19195],{"xmlns":1003,"width":1004,"height":19183,"viewBox":19184,"preserveAspectRatio":19194},"xMidYMin slice",[1009,19196],{"d":19197},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[360,19199,19202],{"className":19200,"style":19180},[19201],"brace-right",[1001,19203,19205],{"xmlns":1003,"width":1004,"height":19183,"viewBox":19184,"preserveAspectRatio":19204},"xMaxYMin slice",[1009,19206],{"d":19207},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[360,19209,19210,19213],{"style":978},[360,19211],{"className":19212,"style":956},[476],[360,19214,19216],{"className":19215},[426],[360,19217,19219,19222],{"className":19218},[426],[360,19220,8453],{"className":19221},[426,6672],[360,19223,19225],{"className":19224},[457],[360,19226,19228,19248],{"className":19227},[461,713],[360,19229,19231,19245],{"className":19230},[465],[360,19232,19234],{"className":19233,"style":4928},[469],[360,19235,19236,19239],{"style":1320},[360,19237],{"className":19238,"style":477},[476],[360,19240,19242],{"className":19241},[481,482,483,484],[360,19243,403],{"className":19244},[426,427,484],[360,19246,741],{"className":19247},[740],[360,19249,19251],{"className":19250},[465],[360,19252,19254],{"className":19253,"style":1339},[469],[360,19255],{},[360,19257,741],{"className":19258},[740],[360,19260,19262],{"className":19261},[465],[360,19263,19266],{"className":19264,"style":19265},[469],"height:0.798em;",[360,19267],{},[360,19269,741],{"className":19270},[740],[360,19272,19274],{"className":19273},[465],[360,19275,19278],{"className":19276,"style":19277},[469],"height:1.4763em;",[360,19279],{},[360,19281],{"className":19282,"style":650},[431],[360,19284,3976],{"className":19285},[494],[360,19287],{"className":19288,"style":650},[431],[360,19290,19292,19296,19299,19302,19452,19455,19458],{"className":19291},[417],[360,19293],{"className":19294,"style":19295},[421],"height:2.2958em;vertical-align:-1.6124em;",[360,19297,12963],{"className":19298,"style":13056},[426,427],[360,19300],{"className":19301,"style":871},[431],[360,19303,19305],{"className":19304},[920,2547],[360,19306,19308,19443],{"className":19307},[461,713],[360,19309,19311,19440],{"className":19310},[465],[360,19312,19314,19332],{"className":19313,"style":5226},[469],[360,19315,19317,19320],{"style":19316},"top:-1.3876em;",[360,19318],{"className":19319,"style":956},[476],[360,19321,19323],{"className":19322},[481,482,483,484],[360,19324,19326],{"className":19325},[426,484],[360,19327,19329],{"className":19328},[426,4177,484],[360,19330,19040],{"className":19331},[426,4181,484],[360,19333,19334,19337],{"style":978},[360,19335],{"className":19336,"style":956},[476],[360,19338,19340],{"className":19339},[920,2547],[360,19341,19343,19431],{"className":19342},[461,713],[360,19344,19346,19428],{"className":19345},[465],[360,19347,19349,19380],{"className":19348,"style":5226},[469],[360,19350,19353,19356],{"className":19351,"style":19352},[977],"top:-2.0659em;",[360,19354],{"className":19355,"style":956},[476],[360,19357,19359,19366,19373],{"className":19358,"style":19175},[19174],[360,19360,19362],{"className":19361,"style":19180},[19179],[1001,19363,19364],{"xmlns":1003,"width":1004,"height":19183,"viewBox":19184,"preserveAspectRatio":1007},[1009,19365],{"d":19187},[360,19367,19369],{"className":19368,"style":19180},[19191],[1001,19370,19371],{"xmlns":1003,"width":1004,"height":19183,"viewBox":19184,"preserveAspectRatio":19194},[1009,19372],{"d":19197},[360,19374,19376],{"className":19375,"style":19180},[19201],[1001,19377,19378],{"xmlns":1003,"width":1004,"height":19183,"viewBox":19184,"preserveAspectRatio":19204},[1009,19379],{"d":19207},[360,19381,19382,19385],{"style":978},[360,19383],{"className":19384,"style":956},[476],[360,19386,19388],{"className":19387},[426],[360,19389,19391,19394],{"className":19390},[426],[360,19392,8453],{"className":19393},[426,6672],[360,19395,19397],{"className":19396},[457],[360,19398,19400,19420],{"className":19399},[461,713],[360,19401,19403,19417],{"className":19402},[465],[360,19404,19406],{"className":19405,"style":7469},[469],[360,19407,19408,19411],{"style":1320},[360,19409],{"className":19410,"style":477},[476],[360,19412,19414],{"className":19413},[481,482,483,484],[360,19415,112],{"className":19416},[426,427,484],[360,19418,741],{"className":19419},[740],[360,19421,19423],{"className":19422},[465],[360,19424,19426],{"className":19425,"style":2063},[469],[360,19427],{},[360,19429,741],{"className":19430},[740],[360,19432,19434],{"className":19433},[465],[360,19435,19438],{"className":19436,"style":19437},[469],"height:0.9341em;",[360,19439],{},[360,19441,741],{"className":19442},[740],[360,19444,19446],{"className":19445},[465],[360,19447,19450],{"className":19448,"style":19449},[469],"height:1.6124em;",[360,19451],{},[360,19453],{"className":19454,"style":650},[431],[360,19456,3976],{"className":19457},[494],[360,19459],{"className":19460,"style":650},[431],[360,19462,19464,19468,19472,19475],{"className":19463},[417],[360,19465],{"className":19466,"style":19467},[421],"height:2.3263em;vertical-align:-1.5763em;",[360,19469,19045],{"className":19470,"style":19471},[426,427],"margin-right:0.0528em;",[360,19473],{"className":19474,"style":871},[431],[360,19476,19478],{"className":19477},[920,2547],[360,19479,19481,19621],{"className":19480},[461,713],[360,19482,19484,19618],{"className":19483},[465],[360,19485,19488,19506],{"className":19486,"style":19487},[469],"heig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DNN直接建模解函數",[275,19660,19661,19664],{},[138,19662,19663],{},"多重資料集處理","：\n└─ \"根-分支\"DNN結構處理不同初邊值條件\n    ├─ 共享根網路擷取共同特徵\n    └─ 獨立分支網路適應特定條件",[112,19666,19667,4023],{},[138,19668,19669],{},"最佳化演算法（ADO）",[272,19671,19672,19675],{},[275,19673,19674],{},"交替方向優化逐步訓練DNN參數(θ)與PDE係數(Λ)",[275,19676,19677],{},"解決ℓ₀正規的NP難問題",[5119,19679,19681],{"id":19680},"_3-實驗驗證","3. 實驗驗證",[4848,19683,19684,19700],{},[4851,19685,19686],{},[4854,19687,19688,19691,19694,19697],{},[4857,19689,19690],{},"PDE類型",[4857,19692,19693],{},"雜訊等級",[4857,19695,19696],{},"辨識誤差",[4857,19698,19699],{},"資料使用量",[4864,19701,19702,19716,19729,19742],{},[4854,19703,19704,19707,19710,19713],{},[4869,19705,19706],{},"Burgers方程式",[4869,19708,19709],{},"10%",[4869,19711,19712],{},"1.15%",[4869,19714,19715],{},"1.95%",[4854,19717,19718,19721,19723,19726],{},[4869,19719,19720],{},"KS方程式（混沌）",[4869,19722,19709],{},[4869,19724,19725],{},"0.71%",[4869,19727,19728],{},"12.3%",[4854,19730,19731,19734,19736,19739],{},[4869,19732,19733],{},"Navier-Stokes方程式",[4869,19735,19709],{},[4869,19737,19738],{},"1.40%",[4869,19740,19741],{},"0.22%",[4854,19743,19744,19747,19749,19752],{},[4869,19745,19746],{},"Schrödinger方程式",[4869,19748,19709],{},[4869,19750,19751],{},"2.31%",[4869,19753,19754],{},"37.5%",[112,19756,19757,4023],{},[138,19758,19759],{},"多重資料集場景",[272,19761,19762,19765],{},[275,19763,19764],{},"Burgers方程式（小黏度衝擊波）",[275,19766,19767],{},"Fitzhugh-Nagumo神經元模型",[112,19769,19770,19773],{},[138,19771,19772],{},"真實實驗","：\n細胞遷移增殖實驗發現Fisher-Kolmogorov模型：",[360,19775,19777],{"className":19776},[755],[360,19778,19780,19837],{"className":19779},[363],[360,19781,19783],{"className":19782},[367],[369,19784,19785],{"xmlns":371,"display":764},[373,19786,19787,19834],{},[376,19788,19789,19796,19798,19800,19810,19812,19818,19820,19822,19828],{},[628,19790,19791,19794],{},[379,19792,19793],{},"ρ",[379,19795,6804],{},[383,19797,796],{},[379,19799,17641],{},[628,19801,19802,19804],{},[379,19803,19793],{},[376,19805,19806,19808],{},[379,19807,6582],{},[379,19809,6582],{},[383,19811,3976],{},[628,19813,19814,19816],{},[379,19815,9076],{},[1185,19817,1187],{},[379,19819,19793],{},[383,19821,3976],{},[628,19823,19824,19826],{},[379,19825,9076],{},[1185,19827,4119],{},[387,19829,19830,19832],{},[379,19831,19793],{},[1185,19833,4119],{},[405,19835,19836],{"encoding":407},"\\rho_t 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優勢與意義",[112,20098,5658,20099,20102,20103,20106,20107,20110],{},[138,20100,20101],{},"突破限制","：資料稀缺(\u003C1%原網格)+高雜訊(10%)下精確發現PDE\n✅ ",[138,20104,20105],{},"物理可解釋性","：DNN提供全域逼近，稀疏PDE約束網絡\n✅ ",[138,20108,20109],{},"應用潛力","：流體力學\u002F生物遷移\u002F氣候預測等高成本實測領域",{"title":311,"searchDepth":312,"depth":312,"links":20112},[20113,20118,20119,20120],{"id":18863,"depth":312,"text":20114,"children":20115},"論文位置：https:\u002F\u002Farxiv.org\u002Fabs\u002F2005.03448",[20116,20117],{"id":18870,"depth":4984,"text":18871},{"id":18874,"depth":4984,"text":18874},{"id":15764,"depth":312,"text":15764},{"id":5464,"depth":312,"text":5464},{"id":5483,"depth":312,"text":5483,"children":20121},[20122,20123,20124,20125],{"id":18957,"depth":4984,"text":18958},{"id":18969,"depth":4984,"text":18970},{"id":19680,"depth":4984,"text":19681},{"id":20095,"depth":4984,"text":20096},"2025-07-09","PiDL 用稀疏又帶噪音的觀測資料反推非線性時空系統的偏微分方程；本文整理它的網路架構、訓練策略，以及在 Burgers 與 Navier–Stokes 上的實測表現。",{},"\u002Fblog\u002Fzh-tw\u002Farxiv-2005-03448v3",{"title":18858,"description":20127},"arxiv-2005-03448v3","blog\u002Fzh-TW\u002Farxiv-2005-03448v3",[5424,20134,20135],"physics-informed","pde","_lKytP4yva134MGngrv1HaVh0_rypK39ea4Sq0NPTS4",{"id":20138,"title":20139,"author":103,"body":20140,"cover":20378,"coverAlt":20139,"date":20379,"description":20380,"draft":324,"extension":325,"featured":324,"lang":327,"meta":20381,"navigation":326,"path":20382,"seo":20383,"slug":20384,"stem":20385,"tags":20386,"translationKey":20384,"updated":337,"__hash__":20388},"blog\u002Fblog\u002Fzh-TW\u002Fgemini-for-workspace.md","Gemini for Workspace 提示指南精要",{"type":105,"value":20141,"toc":20366},[20142,20145,20148,20193,20196,20234,20237,20240,20254,20257,20271,20274,20288,20291,20305,20308,20322,20325,20359],[112,20143,20144],{},"本文總結 Google Workspace 中 Gemini AI 的提示工程核心技巧與應用場景，幫助用戶高效利用生成式 AI 提升生產力。",[108,20146,20147],{"id":20147},"有效提示四要素",[5193,20149,20150,20161,20171,20182],{},[275,20151,20152,20155,20156,20158],{},[138,20153,20154],{},"角色 (Persona)","：明確指定 AI 扮演的角色",[15767,20157],{},[118,20159,20160],{},"例：You are a program manager in tech industry...",[275,20162,20163,20165,20166,20168],{},[138,20164,14710],{},"：使用動詞明確指令",[15767,20167],{},[118,20169,20170],{},"例：Draft an executive summary email...",[275,20172,20173,20176,20177,20179],{},[138,20174,20175],{},"上下文 (Context)","：提供背景資訊",[15767,20178],{},[118,20180,20181],{},"例：based on project docs in Drive...",[275,20183,20184,20187,20188,20190],{},[138,20185,20186],{},"格式 (Format)","：指定輸出結構",[15767,20189],{},[118,20191,20192],{},"例：Limit to bullet points",[108,20194,20195],{"id":20195},"六大核心提示技巧",[272,20197,20198,20204,20210,20216,20222,20228],{},[275,20199,20200,20203],{},[138,20201,20202],{},"🗣️ 自然語言","：用完整句子表達，如日常對話",[275,20205,20206,20209],{},[138,20207,20208],{},"🎯 具體明確","：提供充足上下文（可通過@引用文件）",[275,20211,20212,20215],{},[138,20213,20214],{},"✂️ 簡潔清晰","：避免專業術語，平均有效提示約21詞",[275,20217,20218,20221],{},[138,20219,20220],{},"🔁 迭代優化","：通過對話逐步精煉結果",[275,20223,20224,20227],{},[138,20225,20226],{},"📂 活用文件","：整合 Google Drive\u002FDocs 內容",[275,20229,20230,20233],{},[138,20231,20232],{},"✍️ 提示編輯","：用「Make this a power prompt:」優化原始提示",[108,20235,20236],{"id":20236},"關鍵應用場景",[5119,20238,20239],{"id":20239},"行政支援",[272,20241,20242,20248],{},[275,20243,20244,20245],{},"會議議程：",[118,20246,20247],{},"Plan a 3-day offsite with team bonding activities",[275,20249,20250,20251],{},"郵件管理：",[118,20252,20253],{},"Summarize emails from [manager] last week",[5119,20255,20256],{"id":20256},"客戶服務",[272,20258,20259,20265],{},[275,20260,20261,20262],{},"客訴處理：",[118,20263,20264],{},"Draft empathetic response for damaged goods complaint",[275,20266,20267,20268],{},"自助資源：",[118,20269,20270],{},"Create \"Resolve Common Issues\" self-service guide",[5119,20272,20273],{"id":20273},"高管應用",[272,20275,20276,20282],{},[275,20277,20278,20279],{},"簡報製作：",[118,20280,20281],{},"Generate investor pitch outline for [product] launch",[275,20283,20284,20285],{},"趨勢分析：",[118,20286,20287],{},"Summarize top 5 emerging tech trends in [industry]",[5119,20289,20290],{"id":20290},"行銷創作",[272,20292,20293,20299],{},[275,20294,20295,20296],{},"內容生成：",[118,20297,20298],{},"Write 3 social posts about [product] for [audience]",[275,20300,20301,20302],{},"視覺設計：",[118,20303,20304],{},"Create retro-cool logo for coffee\u002Fgaming cafe",[5119,20306,20307],{"id":20307},"專案管理",[272,20309,20310,20316],{},[275,20311,20312,20313],{},"追蹤系統：",[118,20314,20315],{},"Create issue tracker with status\u002Fowner\u002Factions",[275,20317,20318,20319],{},"會議摘要：",[118,20320,20321],{},"Extract action items from meeting transcript",[108,20323,20324],{"id":20324},"進階提示技巧",[272,20326,20327,20335,20343,20351],{},[275,20328,20329,4023,20332],{},[138,20330,20331],{},"角色扮演",[118,20333,20334],{},"You are head of creative at ad agency...",[275,20336,20337,4023,20340],{},[138,20338,20339],{},"輸出限制",[118,20341,20342],{},"Keep responses under 100 words",[275,20344,20345,4023,20348],{},[138,20346,20347],{},"語調控制",[118,20349,20350],{},"Use formal tone for executive communication",[275,20352,20353,4023,20356],{},[138,20354,20355],{},"迭代優化",[118,20357,20358],{},"Refine previous response to be more data-driven",[112,20360,20361],{},[5055,20362,20365],{"href":20363,"rel":20364},"https:\u002F\u002Fservices.google.com\u002Ffh\u002Ffiles\u002Fmisc\u002Fgemini-for-google-workspace-prompting-guide-101.pdf",[5059],"獲取完整指南",{"title":311,"searchDepth":312,"depth":312,"links":20367},[20368,20369,20370,20377],{"id":20147,"depth":312,"text":20147},{"id":20195,"depth":312,"text":20195},{"id":20236,"depth":312,"text":20236,"children":20371},[20372,20373,20374,20375,20376],{"id":20239,"depth":4984,"text":20239},{"id":20256,"depth":4984,"text":20256},{"id":20273,"depth":4984,"text":20273},{"id":20290,"depth":4984,"text":20290},{"id":20307,"depth":4984,"text":20307},{"id":20324,"depth":312,"text":20324},"\u002Fimages\u002Fblog\u002Fgemini-workspace-guide.webp","2025-06-05","把 Google Workspace 裡 Gemini 的提示技巧整理成可直接照抄的模板：角色設定、上下文邊界、輸出格式，以及最常見的四種失敗模式。",{},"\u002Fblog\u002Fzh-tw\u002Fgemini-for-workspace",{"title":20139,"description":20380},"gemini-for-workspace","blog\u002Fzh-TW\u002Fgemini-for-workspace",[20387,44],"gemini","xKxfdHg5Nuv0-FjsDBBBi6RgOp7EQVs73rZI2Y_9aVQ",1792775650057]