[{"data":1,"prerenderedAt":5619},["ShallowReactive",2],{"site-info":3,"post-hello-jecision":99,"related-zh-TW-hello-jecision":530},{"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",{"zh":100,"en":339},{"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":320,"coverAlt":520,"date":322,"description":521,"draft":324,"extension":325,"featured":326,"lang":522,"meta":523,"navigation":326,"path":524,"seo":525,"slug":331,"stem":526,"tags":527,"translationKey":331,"updated":337,"__hash__":529},"blog\u002Fblog\u002Fen\u002Fhello-jecision.md","Moving the blog off a domain nobody could find",{"type":105,"value":343,"toc":512},[344,348,351,366,372,376,379,391,399,403,420,435,441,445,451,460,471,475,498,502,509],[108,345,347],{"id":346},"it-started-with-one-line-of-source-code","It started with one line of source code",[112,349,350],{},"I was sending an old article to a colleague, so I grabbed the page source, and stopped.",[112,352,353,354,356,357,359,360,362,363,365],{},"No ",[118,355,124],{}," on ",[118,358,120],{},". No ",[118,361,128],{},". No description, no Open Graph, no canonical, not a single line of JSON-LD. The math was worse: the source contained the literal string ",[118,364,132],{},", and only became typeset once JavaScript finished running.",[112,367,368,369],{},"Which means: ",[138,370,371],{},"to a search engine, that article never existed.",[108,373,375],{"id":374},"how-it-got-that-way","How it got that way",[112,377,378],{},"Not carelessness — we used the tool harder than it was designed to be used.",[112,380,381,382,384,385,387,388,390],{},"To make formulas render, we wrapped LaTeX in ",[118,383,159],{}," and replaced its ",[118,386,167],{}," with KaTeX output inside ",[118,389,163],{},". It looked perfect in the browser, so nobody questioned it.",[112,392,393,394,398],{},"The problem is that this couples two things: the content only exists after JavaScript runs, while SEO needs content that exists ",[395,396,397],"em",{},"before"," it does.",[108,400,402],{"id":401},"three-decisions","Three decisions",[112,404,405,408,409,411,412,414,415,411,417,419],{},[138,406,407],{},"Move the content, not the layout."," All 12 articles were converted to ",[118,410,183],{}," and ",[118,413,187],{},", handled by ",[118,416,191],{},[118,418,194],{}," at build time. Math is now part of the HTML.",[112,421,422,425,426,428,429,431,432,434],{},[138,423,424],{},"Let data decide the URL, not the filename."," The old filenames looked like ",[118,427,204],{},", and the resulting URL was ",[118,430,208],{}," — the numeric prefix silently vanished. Now a ",[118,433,212],{}," field in frontmatter owns the URL.",[112,436,437,440],{},[138,438,439],{},"Not a single old URL may break."," 28 redirects, including trailing-slash variants.",[108,442,444],{"id":443},"three-bugs-that-almost-broke-the-launch","Three bugs that almost broke the launch",[112,446,447,450],{},[138,448,449],{},"The eaten slug."," Discovered by fetching the live page, not by reading the code. Had we skipped that, we would have assumed a move is just copying files.",[112,452,453,456,457,459],{},[138,454,455],{},"hreflang pointing at 404s."," The i18n module generates alternates for every locale, but a Chinese-only article has no English version. ",[118,458,243],{}," failed with \"Article not found, linked from \u002Fblog\u002Fxxx\". The fix was not to disable the check, but to let each page declare which locales actually exist.",[112,461,462,467,468,470],{},[138,463,464,466],{},[118,465,255],{}," was HTML."," The build succeeded, the file existed, it had content — but it was a ",[118,469,260],{}," shim, not XML. Google would reject it outright. A post-build step now writes the sitemap index into that path.",[108,472,474],{"id":473},"where-we-are","Where we are",[272,476,477,480,486,492],{},[275,478,479],{},"Math is typeset server-side, with MathML in the markup",[275,481,482,483,485],{},"Every page has title, description, canonical, hreflang, OG; articles carry ",[118,484,283],{}," structured data",[275,487,488,489,491],{},"RSS, ",[118,490,290],{},", sitemap and redirects all in place",[275,493,494,495,497],{},"An ",[118,496,20],{}," script checks 25 pages for SEO regressions and broken links on every build: currently 0 errors, 0 warnings",[108,499,501],{"id":500},"the-one-line-takeaway","The one-line takeaway",[112,503,504,505,508],{},"Most of the work in a migration is not moving things — it is ",[138,506,507],{},"proving the things arrived",".",[112,510,511],{},"So we turned \"should be there\" into a check that runs on every build. People forget; scripts do not.",{"title":311,"searchDepth":312,"depth":312,"links":513},[514,515,516,517,518,519],{"id":346,"depth":312,"text":347},{"id":374,"depth":312,"text":375},{"id":401,"depth":312,"text":402},{"id":443,"depth":312,"text":444},{"id":473,"depth":312,"text":474},{"id":500,"depth":312,"text":501},"Diagram of moving from a temporary domain to the brand domain","The old site shipped every article with no title tag, description or canonical URL — and LaTeX that only appeared after JavaScript ran. Here is the move, and the three bugs that almost broke it.","en",{},"\u002Fblog\u002Fen\u002Fhello-jecision",{"title":341,"description":521},"blog\u002Fen\u002Fhello-jecision",[334,335,528],"refactor","B3ZG8h7XSdAA7KdGCvK_B8LStrBHlGoRZor2i2V_o7k",[531,5228],{"id":532,"title":533,"author":103,"body":534,"cover":5213,"coverAlt":5214,"date":5215,"description":5216,"draft":324,"extension":325,"featured":324,"lang":327,"meta":5217,"navigation":326,"path":5218,"seo":5219,"slug":5220,"stem":5221,"tags":5222,"translationKey":5220,"updated":5226,"__hash__":5227},"blog\u002Fblog\u002Fzh-TW\u002Fattention-as-kernel.md","Attention 其實是一個核平滑器",{"type":105,"value":535,"toc":5205},[536,539,542,545,548,944,1300,1334,2275,2494,2710,3254,3257,3260,3263,3269,3454,4094,4215,4594,4597,4600,4606,5032,5035,5038,5153,5185,5188,5191,5198,5201],[108,537,538],{"id":538},"一句話讓我想了很久",[112,540,541],{},"「Attention 就是一個核平滑器。」",[112,543,544],{},"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。",[108,546,547],{"id":547},"把公式拆開",[112,549,550,551,690,691,791,792,943],{},"給定查詢矩陣 ",[552,553,556,601],"span",{"className":554},[555],"katex",[552,557,560],{"className":558},[559],"katex-mathml",[561,562,564],"math",{"xmlns":563},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[565,566,567,596],"semantics",{},[568,569,570,574,578],"mrow",{},[571,572,573],"mi",{},"Q",[575,576,577],"mo",{},"∈",[579,580,581,585],"msup",{},[571,582,584],{"mathvariant":583},"double-struck","R",[568,586,587,590,593],{},[571,588,589],{},"n",[575,591,592],{},"×",[571,594,595],{},"d",[597,598,600],"annotation",{"encoding":599},"application\u002Fx-tex","Q \\in \\mathbb{R}^{n \\times d}",[552,602,606,632],{"className":603,"ariaHidden":605},[604],"katex-html","true",[552,607,610,615,620,625,629],{"className":608},[609],"base",[552,611],{"className":612,"style":614},[613],"strut","height:0.8778em;vertical-align:-0.1944em;",[552,616,573],{"className":617},[618,619],"mord","mathnormal",[552,621],{"className":622,"style":624},[623],"mspace","margin-right:0.2778em;",[552,626,577],{"className":627},[628],"mrel",[552,630],{"className":631,"style":624},[623],[552,633,635,639],{"className":634},[609],[552,636],{"className":637,"style":638},[613],"height:0.8491em;",[552,640,642,646],{"className":641},[618],[552,643,584],{"className":644},[618,645],"mathbb",[552,647,650],{"className":648},[649],"msupsub",[552,651,654],{"className":652},[653],"vlist-t",[552,655,658],{"className":656},[657],"vlist-r",[552,659,662],{"className":660,"style":638},[661],"vlist",[552,663,665,670],{"style":664},"top:-3.063em;margin-right:0.05em;",[552,666],{"className":667,"style":669},[668],"pstrut","height:2.7em;",[552,671,677],{"className":672},[673,674,675,676],"sizing","reset-size6","size3","mtight",[552,678,680,683,687],{"className":679},[618,676],[552,681,589],{"className":682},[618,619,676],[552,684,592],{"className":685},[686,676],"mbin",[552,688,595],{"className":689},[618,619,676],"、鍵矩陣 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",[552,1304,1306,1320],{"className":1305},[555],[552,1307,1309],{"className":1308},[559],[561,1310,1311],{"xmlns":563},[565,1312,1313,1318],{},[568,1314,1315],{},[571,1316,1317],{},"i",[597,1319,1317],{"encoding":599},[552,1321,1323],{"className":1322,"ariaHidden":605},[604],[552,1324,1326,1330],{"className":1325},[609],[552,1327],{"className":1328,"style":1329},[613],"height:0.6595em;",[552,1331,1317],{"className":1332},[618,619]," 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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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",[552,2279,2281,2299],{"className":2280},[555],[552,2282,2284],{"className":2283},[559],[561,2285,2286],{"xmlns":563},[565,2287,2288,2296],{},[568,2289,2290],{},[820,2291,2292,2294],{},[571,2293,826],{},[571,2295,1372],{},[597,2297,2298],{"encoding":599},"v_j",[552,2300,2302],{"className":2301,"ariaHidden":605},[604],[552,2303,2305,2309],{"className":2304},[609],[552,2306],{"className":2307,"style":2308},[613],"height:0.7167em;vertical-align:-0.2861em;",[552,2310,2312,2315],{"className":2311},[618],[552,2313,826],{"className":2314,"style":928},[618,619],[552,2316,2318],{"className":2317},[649],[552,2319,2321,2341],{"className":2320},[653,905],[552,2322,2324,2338],{"className":2323},[657],[552,2325,2327],{"className":2326,"style":1508},[661],[552,2328,2329,2332],{"style":2235},[552,2330],{"className":2331,"style":669},[668],[552,2333,2335],{"className":2334},[673,674,675,676],[552,2336,1372],{"className":2337,"style":1615},[618,619,676],[552,2339,933],{"className":2340},[932],[552,2342,2344],{"className":2343},[657],[552,2345,2347],{"className":2346,"style":2254},[661],[552,2348],{}," 的加權和，權重是 ",[552,2351,2353,2371],{"className":2352},[555],[552,2354,2356],{"className":2355},[559],[561,2357,2358],{"xmlns":563},[565,2359,2360,2368],{},[568,2361,2362],{},[820,2363,2364,2366],{},[571,2365,1401],{},[571,2367,1317],{},[597,2369,2370],{"encoding":599},"q_i",[552,2372,2374],{"className":2373,"ariaHidden":605},[604],[552,2375,2377,2381],{"className":2376},[609],[552,2378],{"className":2379,"style":2380},[613],"height:0.625em;vertical-align:-0.1944em;",[552,2382,2384,2387],{"className":2383},[618],[552,2385,1401],{"className":2386,"style":928},[618,619],[552,2388,2390],{"className":2389},[649],[552,2391,2393,2413],{"className":2392},[653,905],[552,2394,2396,2410],{"className":2395},[657],[552,2397,2399],{"className":2398,"style":1508},[661],[552,2400,2401,2404],{"style":2235},[552,2402],{"className":2403,"style":669},[668],[552,2405,2407],{"className":2406},[673,674,675,676],[552,2408,1317],{"className":2409},[618,619,676],[552,2411,933],{"className":2412},[932],[552,2414,2416],{"className":2415},[657],[552,2417,2419],{"className":2418,"style":1530},[661],[552,2420],{},[552,2422,2424,2442],{"className":2423},[555],[552,2425,2427],{"className":2426},[559],[561,2428,2429],{"xmlns":563},[565,2430,2431,2439],{},[568,2432,2433],{},[820,2434,2435,2437],{},[571,2436,1410],{},[571,2438,1372],{},[597,2440,2441],{"encoding":599},"k_j",[552,2443,2445],{"className":2444,"ariaHidden":605},[604],[552,2446,2448,2452],{"className":2447},[609],[552,2449],{"className":2450,"style":2451},[613],"height:0.9805em;vertical-align:-0.2861em;",[552,2453,2455,2458],{"className":2454},[618],[552,2456,1410],{"className":2457,"style":1836},[618,619],[552,2459,2461],{"className":2460},[649],[552,2462,2464,2485],{"className":2463},[653,905],[552,2465,2467,2482],{"className":2466},[657],[552,2468,2470],{"className":2469,"style":1508},[661],[552,2471,2473,2476],{"style":2472},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[552,2474],{"className":2475,"style":669},[668],[552,2477,2479],{"className":2478},[673,674,675,676],[552,2480,1372],{"className":2481,"style":1615},[618,619,676],[552,2483,933],{"className":2484},[932],[552,2486,2488],{"className":2487},[657],[552,2489,2491],{"className":2490,"style":2254},[661],[552,2492],{}," 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。",[112,2495,2496,2497,2709],{},"如果把 ",[552,2498,2500,2554],{"className":2499},[555],[552,2501,2503],{"className":2502},[559],[561,2504,2505],{"xmlns":563},[565,2506,2507,2551],{},[568,2508,2509,2512,2514,2516,2518,2520,2522,2524,2526,2528,2530],{},[571,2510,2511],{},"κ",[575,2513,971],{"stretchy":970},[571,2515,1401],{},[575,2517,976],{"separator":605},[571,2519,1410],{},[575,2521,985],{"stretchy":970},[575,2523,988],{},[571,2525,1383],{},[575,2527,967],{},[995,2529,997],{},[568,2531,2532,2534,2540,2542,2545,2549],{},[575,2533,971],{"fence":605},[579,2535,2536,2538],{},[571,2537,1401],{},[571,2539,1015],{"mathvariant":963},[571,2541,1410],{},[571,2543,2544],{"mathvariant":963},"\u002F",[1017,2546,2547],{},[571,2548,595],{},[575,2550,985],{"fence":605},[597,2552,2553],{"encoding":599},"\\kappa(q, k) = \\exp\\!\\left(q^{\\top}k \u002F \\sqrt{d}\\right)",[552,2555,2557,2593],{"className":2556,"ariaHidden":605},[604],[552,2558,2560,2563,2566,2569,2572,2575,2578,2581,2584,2587,2590],{"className":2559},[609],[552,2561],{"className":2562,"style":1039},[613],[552,2564,2511],{"className":2565},[618,619],[552,2567,971],{"className":2568},[1051],[552,2570,1401],{"className":2571,"style":928},[618,619],[552,2573,976],{"className":2574},[1058],[552,2576],{"className":2577,"style":1062},[623],[552,2579,1410],{"className":2580,"style":1836},[618,619],[552,2582,985],{"className":2583},[1078],[552,2585],{"className":2586,"style":624},[623],[552,2588,988],{"className":2589},[628],[552,2591],{"className":2592,"style":624},[623],[552,2594,2596,2600,2603,2606,2609],{"className":2595},[609],[552,2597],{"className":2598,"style":2599},[613],"height:1.8em;vertical-align:-0.65em;",[552,2601,1383],{"className":2602},[1043],[552,2604],{"className":2605,"style":1104},[623],[552,2607],{"className":2608,"style":1062},[623],[552,2610,2612,2618,2650,2653,2656,2703],{"className":2611},[1111],[552,2613,2615],{"className":2614,"style":1116},[1051,1115],[552,2616,971],{"className":2617},[1120,1670],[552,2619,2621,2624],{"className":2620},[618],[552,2622,1401],{"className":2623,"style":928},[618,619],[552,2625,2627],{"className":2626},[649],[552,2628,2630],{"className":2629},[653],[552,2631,2633],{"className":2632},[657],[552,2634,2636],{"className":2635,"style":638},[661],[552,2637,2638,2641],{"style":664},[552,2639],{"className":2640,"style":669},[668],[552,2642,2644],{"className":2643},[673,674,675,676],[552,2645,2647],{"className":2646},[618,676],[552,2648,1015],{"className":2649},[618,676],[552,2651,1410],{"className":2652,"style":1836},[618,619],[552,2654,2544],{"className":2655},[618],[552,2657,2659],{"className":2658},[618,1154],[552,2660,2662,2695],{"className":2661},[653,905],[552,2663,2665,2692],{"className":2664},[657],[552,2666,2668,2680],{"className":2667,"style":1164},[661],[552,2669,2671,2674],{"className":2670,"style":1169},[1168],[552,2672],{"className":2673,"style":1147},[668],[552,2675,2677],{"className":2676,"style":1176},[618],[552,2678,595],{"className":2679},[618,619],[552,2681,2682,2685],{"style":1182},[552,2683],{"className":2684,"style":1147},[668],[552,2686,2688],{"className":2687,"style":1190},[1189],[1192,2689,2690],{"xmlns":1194,"width":1195,"height":1196,"viewBox":1197,"preserveAspectRatio":1198},[1200,2691],{"d":1202},[552,2693,933],{"className":2694},[932],[552,2696,2698],{"className":2697},[657],[552,2699,2701],{"className":2700,"style":1212},[661],[552,2702],{},[552,2704,2706],{"className":2705,"style":1116},[1078,1115],[552,2707,985],{"className":2708},[1120,1670]," 當成一個核函數，上面這一坨就是",[552,2711,2713],{"className":2712},[947],[552,2714,2716,2806],{"className":2715},[555],[552,2717,2719],{"className":2718},[559],[561,2720,2721],{"xmlns":563,"display":956},[565,2722,2723,2803],{},[568,2724,2725,2731,2733,2801],{},[820,2726,2727,2729],{},[571,2728,1353],{},[571,2730,1317],{},[575,2732,988],{},[1003,2734,2735,2773],{},[568,2736,2737,2744,2746,2748,2754,2756,2762,2764,2767],{},[2738,2739,2740,2742],"munder",{},[575,2741,1367],{},[571,2743,1372],{},[571,2745,2511],{},[575,2747,971],{"stretchy":970},[820,2749,2750,2752],{},[571,2751,1401],{},[571,2753,1317],{},[575,2755,976],{"separator":605},[820,2757,2758,2760],{},[571,2759,1410],{},[571,2761,1372],{},[575,2763,985],{"stretchy":970},[995,2765,2766],{}," ",[820,2768,2769,2771],{},[571,2770,826],{},[571,2772,1372],{},[568,2774,2775,2781,2783,2785,2791,2793,2799],{},[2738,2776,2777,2779],{},[575,2778,1367],{},[571,2780,1372],{},[571,2782,2511],{},[575,2784,971],{"stretchy":970},[820,2786,2787,2789],{},[571,2788,1401],{},[571,2790,1317],{},[575,2792,976],{"separator":605},[820,2794,2795,2797],{},[571,2796,1410],{},[571,2798,1372],{},[575,2800,985],{"stretchy":970},[575,2802,976],{"separator":605},[597,2804,2805],{"encoding":599},"o_i = \\frac{\\sum_j \\kappa(q_i, k_j)\\, v_j}{\\sum_j \\kappa(q_i, k_j)} ,",[552,2807,2809,2864],{"className":2808,"ariaHidden":605},[604],[552,2810,2812,2815,2855,2858,2861],{"className":2811},[609],[552,2813],{"className":2814,"style":1489},[613],[552,2816,2818,2821],{"className":2817},[618],[552,2819,1353],{"className":2820},[618,619],[552,2822,2824],{"className":2823},[649],[552,2825,2827,2847],{"className":2826},[653,905],[552,2828,2830,2844],{"className":2829},[657],[552,2831,2833],{"className":2832,"style":1508},[661],[552,2834,2835,2838],{"style":1511},[552,2836],{"className":2837,"style":669},[668],[552,2839,2841],{"className":2840},[673,674,675,676],[552,2842,1317],{"className":2843},[618,619,676],[552,2845,933],{"className":2846},[932],[552,2848,2850],{"className":2849},[657],[552,2851,2853],{"className":2852,"style":1530},[661],[552,2854],{},[552,2856],{"className":2857,"style":624},[623],[552,2859,988],{"className":2860},[628],[552,2862],{"className":2863,"style":624},[623],[552,2865,2867,2871,3251],{"className":2866},[609],[552,2868],{"className":2869,"style":2870},[613],"height:2.6976em;vertical-align:-1.1218em;",[552,2872,2874,2877,3248],{"className":2873},[618],[552,2875],{"className":2876},[1051,1127],[552,2878,2880],{"className":2879},[1003],[552,2881,2883,3239],{"className":2882},[653,905],[552,2884,2886,3236],{"className":2885},[657],[552,2887,2890,3038,3046],{"className":2888,"style":2889},[661],"height:1.5758em;",[552,2891,2893,2896],{"style":2892},"top:-2.314em;",[552,2894],{"className":2895,"style":1147},[668],[552,2897,2899,2940,2943,2946,2949,2989,2992,2995,3035],{"className":2898},[618],[552,2900,2902,2905],{"className":2901},[1043],[552,2903,1367],{"className":2904,"style":1586},[1043,1584,1585],[552,2906,2908],{"className":2907},[649],[552,2909,2911,2932],{"className":2910},[653,905],[552,2912,2914,2929],{"className":2913},[657],[552,2915,2918],{"className":2916,"style":2917},[661],"height:0.162em;",[552,2919,2920,2923],{"style":1602},[552,2921],{"className":2922,"style":669},[668],[552,2924,2926],{"className":2925},[673,674,675,676],[552,2927,1372],{"className":2928,"style":1615},[618,619,676],[552,2930,933],{"className":2931},[932],[552,2933,2935],{"className":2934},[657],[552,2936,2938],{"className":2937,"style":1646},[661],[552,2939],{},[552,2941],{"className":2942,"style":1062},[623],[552,2944,2511],{"className":2945},[618,619],[552,2947,971],{"className":2948},[1051],[552,2950,2952,2955],{"className":2951},[618],[552,2953,1401],{"className":2954,"style":928},[618,619],[552,2956,2958],{"className":2957},[649],[552,2959,2961,2981],{"className":2960},[653,905],[552,2962,2964,2978],{"className":2963},[657],[552,2965,2967],{"className":2966,"style":1508},[661],[552,2968,2969,2972],{"style":2235},[552,2970],{"className":2971,"style":669},[668],[552,2973,2975],{"className":2974},[673,674,675,676],[552,2976,1317],{"className":2977},[618,619,676],[552,2979,933],{"className":2980},[932],[552,2982,2984],{"className":2983},[657],[552,2985,2987],{"className":2986,"style":1530},[661],[552,2988],{},[552,2990,976],{"className":2991},[1058],[552,2993],{"className":2994,"style":1062},[623],[552,2996,2998,3001],{"className":2997},[618],[552,2999,1410],{"className":3000,"style":1836},[618,619],[552,3002,3004],{"className":3003},[649],[552,3005,3007,3027],{"className":3006},[653,905],[552,3008,3010,3024],{"className":3009},[657],[552,3011,3013],{"className":3012,"style":1508},[661],[552,3014,3015,3018],{"style":2472},[552,3016],{"className":3017,"style":669},[668],[552,3019,3021],{"className":3020},[673,674,675,676],[552,3022,1372],{"className":3023,"style":1615},[618,619,676],[552,3025,933],{"className":3026},[932],[552,3028,3030],{"className":3029},[657],[552,3031,3033],{"className":3032,"style":2254},[661],[552,3034],{},[552,3036,985],{"className":3037},[1078],[552,3039,3040,3043],{"style":1217},[552,3041],{"className":3042,"style":1147},[668],[552,3044],{"className":3045,"style":1225},[1224],[552,3047,3049,3052],{"style":3048},"top:-3.8258em;",[552,3050],{"className":3051,"style":1147},[668],[552,3053,3055,3095,3098,3101,3104,3144,3147,3150,3190,3193,3196],{"className":3054},[618],[552,3056,3058,3061],{"className":3057},[1043],[552,3059,1367],{"className":3060,"style":1586},[1043,1584,1585],[552,3062,3064],{"className":3063},[649],[552,3065,3067,3087],{"className":3066},[653,905],[552,3068,3070,3084],{"className":3069},[657],[552,3071,3073],{"className":3072,"style":2917},[661],[552,3074,3075,3078],{"style":1602},[552,3076],{"className":3077,"style":669},[668],[552,3079,3081],{"className":3080},[673,674,675,676],[552,3082,1372],{"className":3083,"style":1615},[618,619,676],[552,3085,933],{"className":3086},[932],[552,3088,3090],{"className":3089},[657],[552,3091,3093],{"className":3092,"style":1646},[661],[552,3094],{},[552,3096],{"className":3097,"style":1062},[623],[552,3099,2511],{"className":3100},[618,619],[552,3102,971],{"className":3103},[1051],[552,3105,3107,3110],{"className":3106},[618],[552,3108,1401],{"className":3109,"style":928},[618,619],[552,3111,3113],{"className":3112},[649],[552,3114,3116,3136],{"className":3115},[653,905],[552,3117,3119,3133],{"className":3118},[657],[552,3120,3122],{"className":3121,"style":1508},[661],[552,3123,3124,3127],{"style":2235},[552,3125],{"className":3126,"style":669},[668],[552,3128,3130],{"className":3129},[673,674,675,676],[552,3131,1317],{"className":3132},[618,619,676],[552,3134,933],{"className":3135},[932],[552,3137,3139],{"className":3138},[657],[552,3140,3142],{"className":3141,"style":1530},[661],[552,3143],{},[552,3145,976],{"className":3146},[1058],[552,3148],{"className":3149,"style":1062},[623],[552,3151,3153,3156],{"className":3152},[618],[552,3154,1410],{"className":3155,"style":1836},[618,619],[552,3157,3159],{"className":3158},[649],[552,3160,3162,3182],{"className":3161},[653,905],[552,3163,3165,3179],{"className":3164},[657],[552,3166,3168],{"className":3167,"style":1508},[661],[552,3169,3170,3173],{"style":2472},[552,3171],{"className":3172,"style":669},[668],[552,3174,3176],{"className":3175},[673,674,675,676],[552,3177,1372],{"className":3178,"style":1615},[618,619,676],[552,3180,933],{"className":3181},[932],[552,3183,3185],{"className":3184},[657],[552,3186,3188],{"className":3187,"style":2254},[661],[552,3189],{},[552,3191,985],{"className":3192},[1078],[552,3194],{"className":3195,"style":1062},[623],[552,3197,3199,3202],{"className":3198},[618],[552,3200,826],{"className":3201,"style":928},[618,619],[552,3203,3205],{"className":3204},[649],[552,3206,3208,3228],{"className":3207},[653,905],[552,3209,3211,3225],{"className":3210},[657],[552,3212,3214],{"className":3213,"style":1508},[661],[552,3215,3216,3219],{"style":2235},[552,3217],{"className":3218,"style":669},[668],[552,3220,3222],{"className":3221},[673,674,675,676],[552,3223,1372],{"className":3224,"style":1615},[618,619,676],[552,3226,933],{"className":3227},[932],[552,3229,3231],{"className":3230},[657],[552,3232,3234],{"className":3233,"style":2254},[661],[552,3235],{},[552,3237,933],{"className":3238},[932],[552,3240,3242],{"className":3241},[657],[552,3243,3246],{"className":3244,"style":3245},[661],"height:1.1218em;",[552,3247],{},[552,3249],{"className":3250},[1078,1127],[552,3252,976],{"className":3253},[1058],[112,3255,3256],{},"——非參數回歸裡的 Nadaraya–Watson 估計。一模一樣。",[112,3258,3259],{},"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。",[108,3261,3262],{"id":3262},"知道這件事有什麼用",[112,3264,3265,3266,153],{},"不是為了數學上的優雅。是因為它會告訴你",[138,3267,3268],{},"哪裡可以砍",[112,3270,3271,3272,3302,3303,3453],{},"如果 ",[552,3273,3275,3289],{"className":3274},[555],[552,3276,3278],{"className":3277},[559],[561,3279,3280],{"xmlns":563},[565,3281,3282,3286],{},[568,3283,3284],{},[571,3285,2511],{},[597,3287,3288],{"encoding":599},"\\kappa",[552,3290,3292],{"className":3291,"ariaHidden":605},[604],[552,3293,3295,3299],{"className":3294},[609],[552,3296],{"className":3297,"style":3298},[613],"height:0.4306em;",[552,3300,2511],{"className":3301},[618,619]," 可以寫成有限維特徵映射的內積，也就是 ",[552,3304,3306,3354],{"className":3305},[555],[552,3307,3309],{"className":3308},[559],[561,3310,3311],{"xmlns":563},[565,3312,3313,3351],{},[568,3314,3315,3317,3319,3321,3323,3325,3327,3330,3333,3335,3337,3343,3345,3347,3349],{},[571,3316,2511],{},[575,3318,971],{"stretchy":970},[571,3320,1401],{},[575,3322,976],{"separator":605},[571,3324,1410],{},[575,3326,985],{"stretchy":970},[575,3328,3329],{},"≈",[571,3331,3332],{},"ϕ",[575,3334,971],{"stretchy":970},[571,3336,1401],{},[579,3338,3339,3341],{},[575,3340,985],{"stretchy":970},[571,3342,1015],{"mathvariant":963},[571,3344,3332],{},[575,3346,971],{"stretchy":970},[571,3348,1410],{},[575,3350,985],{"stretchy":970},[597,3352,3353],{"encoding":599},"\\kappa(q,k) \\approx \\phi(q)^{\\top}\\phi(k)",[552,3355,3357,3393],{"className":3356,"ariaHidden":605},[604],[552,3358,3360,3363,3366,3369,3372,3375,3378,3381,3384,3387,3390],{"className":3359},[609],[552,3361],{"className":3362,"style":1039},[613],[552,3364,2511],{"className":3365},[618,619],[552,3367,971],{"className":3368},[1051],[552,3370,1401],{"className":3371,"style":928},[618,619],[552,3373,976],{"className":3374},[1058],[552,3376],{"className":3377,"style":1062},[623],[552,3379,1410],{"className":3380,"style":1836},[618,619],[552,3382,985],{"className":3383},[1078],[552,3385],{"className":3386,"style":624},[623],[552,3388,3329],{"className":3389},[628],[552,3391],{"className":3392,"style":624},[623],[552,3394,3396,3400,3403,3406,3409,3441,3444,3447,3450],{"className":3395},[609],[552,3397],{"className":3398,"style":3399},[613],"height:1.0991em;vertical-align:-0.25em;",[552,3401,3332],{"className":3402},[618,619],[552,3404,971],{"className":3405},[1051],[552,3407,1401],{"className":3408,"style":928},[618,619],[552,3410,3412,3415],{"className":3411},[1078],[552,3413,985],{"className":3414},[1078],[552,3416,3418],{"className":3417},[649],[552,3419,3421],{"className":3420},[653],[552,3422,3424],{"className":3423},[657],[552,3425,3427],{"className":3426,"style":638},[661],[552,3428,3429,3432],{"style":664},[552,3430],{"className":3431,"style":669},[668],[552,3433,3435],{"className":3434},[673,674,675,676],[552,3436,3438],{"className":3437},[618,676],[552,3439,1015],{"className":3440},[618,676],[552,3442,3332],{"className":3443},[618,619],[552,3445,971],{"className":3446},[1051],[552,3448,1410],{"className":3449,"style":1836},[618,619],[552,3451,985],{"className":3452},[1078],"，那麼",[552,3455,3457],{"className":3456},[947],[552,3458,3460,3564],{"className":3459},[555],[552,3461,3463],{"className":3462},[559],[561,3464,3465],{"xmlns":563,"display":956},[565,3466,3467,3561],{},[568,3468,3469,3475,3477,3559],{},[820,3470,3471,3473],{},[571,3472,1353],{},[571,3474,1317],{},[575,3476,3329],{},[1003,3478,3479,3523],{},[568,3480,3481,3483,3485,3491,3497,3503,3505,3507,3513,3515],{},[571,3482,3332],{},[575,3484,971],{"stretchy":970},[820,3486,3487,3489],{},[571,3488,1401],{},[571,3490,1317],{},[579,3492,3493,3495],{},[575,3494,985],{"stretchy":970},[571,3496,1015],{"mathvariant":963},[2738,3498,3499,3501],{},[575,3500,1367],{},[571,3502,1372],{},[571,3504,3332],{},[575,3506,971],{"stretchy":970},[820,3508,3509,3511],{},[571,3510,1410],{},[571,3512,1372],{},[575,3514,985],{"stretchy":970},[1397,3516,3517,3519,3521],{},[571,3518,826],{},[571,3520,1372],{},[571,3522,1015],{"mathvariant":963},[568,3524,3525,3527,3529,3535,3541,3547,3549,3551,3557],{},[571,3526,3332],{},[575,3528,971],{"stretchy":970},[820,3530,3531,3533],{},[571,3532,1401],{},[571,3534,1317],{},[579,3536,3537,3539],{},[575,3538,985],{"stretchy":970},[571,3540,1015],{"mathvariant":963},[2738,3542,3543,3545],{},[575,3544,1367],{},[571,3546,1372],{},[571,3548,3332],{},[575,3550,971],{"stretchy":970},[820,3552,3553,3555],{},[571,3554,1410],{},[571,3556,1372],{},[575,3558,985],{"stretchy":970},[571,3560,508],{"mathvariant":963},[597,3562,3563],{"encoding":599},"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)} .",[552,3565,3567,3623],{"className":3566,"ariaHidden":605},[604],[552,3568,3570,3574,3614,3617,3620],{"className":3569},[609],[552,3571],{"className":3572,"style":3573},[613],"height:0.6331em;vertical-align:-0.15em;",[552,3575,3577,3580],{"className":3576},[618],[552,3578,1353],{"className":3579},[618,619],[552,3581,3583],{"className":3582},[649],[552,3584,3586,3606],{"className":3585},[653,905],[552,3587,3589,3603],{"className":3588},[657],[552,3590,3592],{"className":3591,"style":1508},[661],[552,3593,3594,3597],{"style":1511},[552,3595],{"className":3596,"style":669},[668],[552,3598,3600],{"className":3599},[673,674,675,676],[552,3601,1317],{"className":3602},[618,619,676],[552,3604,933],{"className":3605},[932],[552,3607,3609],{"className":3608},[657],[552,3610,3612],{"className":3611,"style":1530},[661],[552,3613],{},[552,3615],{"className":3616,"style":624},[623],[552,3618,3329],{"className":3619},[628],[552,3621],{"className":3622,"style":624},[623],[552,3624,3626,3630,4091],{"className":3625},[609],[552,3627],{"className":3628,"style":3629},[613],"height:2.7967em;vertical-align:-1.1218em;",[552,3631,3633,3636,4088],{"className":3632},[618],[552,3634],{"className":3635},[1051,1127],[552,3637,3639],{"className":3638},[1003],[552,3640,3642,4080],{"className":3641},[653,905],[552,3643,3645,4077],{"className":3644},[657],[552,3646,3649,3832,3840],{"className":3647,"style":3648},[661],"height:1.6749em;",[552,3650,3651,3654],{"style":2892},[552,3652],{"className":3653,"style":1147},[668],[552,3655,3657,3660,3663,3703,3737,3740,3780,3783,3786,3789,3829],{"className":3656},[618],[552,3658,3332],{"className":3659},[618,619],[552,3661,971],{"className":3662},[1051],[552,3664,3666,3669],{"className":3665},[618],[552,3667,1401],{"className":3668,"style":928},[618,619],[552,3670,3672],{"className":3671},[649],[552,3673,3675,3695],{"className":3674},[653,905],[552,3676,3678,3692],{"className":3677},[657],[552,3679,3681],{"className":3680,"style":1508},[661],[552,3682,3683,3686],{"style":2235},[552,3684],{"className":3685,"style":669},[668],[552,3687,3689],{"className":3688},[673,674,675,676],[552,3690,1317],{"className":3691},[618,619,676],[552,3693,933],{"className":3694},[932],[552,3696,3698],{"className":3697},[657],[552,3699,3701],{"className":3700,"style":1530},[661],[552,3702],{},[552,3704,3706,3709],{"className":3705},[1078],[552,3707,985],{"className":3708},[1078],[552,3710,3712],{"className":3711},[649],[552,3713,3715],{"className":3714},[653],[552,3716,3718],{"className":3717},[657],[552,3719,3722],{"className":3720,"style":3721},[661],"height:0.7751em;",[552,3723,3725,3728],{"style":3724},"top:-2.989em;margin-right:0.05em;",[552,3726],{"className":3727,"style":669},[668],[552,3729,3731],{"className":3730},[673,674,675,676],[552,3732,3734],{"className":3733},[618,676],[552,3735,1015],{"className":3736},[618,676],[552,3738],{"className":3739,"style":1062},[623],[552,3741,3743,3746],{"className":3742},[1043],[552,3744,1367],{"className":3745,"style":1586},[1043,1584,1585],[552,3747,3749],{"className":3748},[649],[552,3750,3752,3772],{"className":3751},[653,905],[552,3753,3755,3769],{"className":3754},[657],[552,3756,3758],{"className":3757,"style":2917},[661],[552,3759,3760,3763],{"style":1602},[552,3761],{"className":3762,"style":669},[668],[552,3764,3766],{"className":3765},[673,674,675,676],[552,3767,1372],{"className":3768,"style":1615},[618,619,676],[552,3770,933],{"className":3771},[932],[552,3773,3775],{"className":3774},[657],[552,3776,3778],{"className":3777,"style":1646},[661],[552,3779],{},[552,3781],{"className":3782,"style":1062},[623],[552,3784,3332],{"className":3785},[618,619],[552,3787,971],{"className":3788},[1051],[552,3790,3792,3795],{"className":3791},[618],[552,3793,1410],{"className":3794,"style":1836},[618,619],[552,3796,3798],{"className":3797},[649],[552,3799,3801,3821],{"className":3800},[653,905],[552,3802,3804,3818],{"className":3803},[657],[552,3805,3807],{"className":3806,"style":1508},[661],[552,3808,3809,3812],{"style":2472},[552,3810],{"className":3811,"style":669},[668],[552,3813,3815],{"className":3814},[673,674,675,676],[552,3816,1372],{"className":3817,"style":1615},[618,619,676],[552,3819,933],{"className":3820},[932],[552,3822,3824],{"className":3823},[657],[552,3825,3827],{"className":3826,"style":2254},[661],[552,3828],{},[552,3830,985],{"className":3831},[1078],[552,3833,3834,3837],{"style":1217},[552,3835],{"className":3836,"style":1147},[668],[552,3838],{"className":3839,"style":1225},[1224],[552,3841,3842,3845],{"style":3048},[552,3843],{"className":3844,"style":1147},[668],[552,3846,3848,3851,3854,3894,3926,3929,3969,3972,3975,3978,4018,4021],{"className":3847},[618],[552,3849,3332],{"className":3850},[618,619],[552,3852,971],{"className":3853},[1051],[552,3855,3857,3860],{"className":3856},[618],[552,3858,1401],{"className":3859,"style":928},[618,619],[552,3861,3863],{"className":3862},[649],[552,3864,3866,3886],{"className":3865},[653,905],[552,3867,3869,3883],{"className":3868},[657],[552,3870,3872],{"className":3871,"style":1508},[661],[552,3873,3874,3877],{"style":2235},[552,3875],{"className":3876,"style":669},[668],[552,3878,3880],{"className":3879},[673,674,675,676],[552,3881,1317],{"className":3882},[618,619,676],[552,3884,933],{"className":3885},[932],[552,3887,3889],{"className":3888},[657],[552,3890,3892],{"className":3891,"style":1530},[661],[552,3893],{},[552,3895,3897,3900],{"className":3896},[1078],[552,3898,985],{"className":3899},[1078],[552,3901,3903],{"className":3902},[649],[552,3904,3906],{"className":3905},[653],[552,3907,3909],{"className":3908},[657],[552,3910,3912],{"className":3911,"style":638},[661],[552,3913,3914,3917],{"style":664},[552,3915],{"className":3916,"style":669},[668],[552,3918,3920],{"className":3919},[673,674,675,676],[552,3921,3923],{"className":3922},[618,676],[552,3924,1015],{"className":3925},[618,676],[552,3927],{"className":3928,"style":1062},[623],[552,3930,3932,3935],{"className":3931},[1043],[552,3933,1367],{"className":3934,"style":1586},[1043,1584,1585],[552,3936,3938],{"className":3937},[649],[552,3939,3941,3961],{"className":3940},[653,905],[552,3942,3944,3958],{"className":3943},[657],[552,3945,3947],{"className":3946,"style":2917},[661],[552,3948,3949,3952],{"style":1602},[552,3950],{"className":3951,"style":669},[668],[552,3953,3955],{"className":3954},[673,674,675,676],[552,3956,1372],{"className":3957,"style":1615},[618,619,676],[552,3959,933],{"className":3960},[932],[552,3962,3964],{"className":3963},[657],[552,3965,3967],{"className":3966,"style":1646},[661],[552,3968],{},[552,3970],{"className":3971,"style":1062},[623],[552,3973,3332],{"className":3974},[618,619],[552,3976,971],{"className":3977},[1051],[552,3979,3981,3984],{"className":3980},[618],[552,3982,1410],{"className":3983,"style":1836},[618,619],[552,3985,3987],{"className":3986},[649],[552,3988,3990,4010],{"className":3989},[653,905],[552,3991,3993,4007],{"className":3992},[657],[552,3994,3996],{"className":3995,"style":1508},[661],[552,3997,3998,4001],{"style":2472},[552,3999],{"className":4000,"style":669},[668],[552,4002,4004],{"className":4003},[673,674,675,676],[552,4005,1372],{"className":4006,"style":1615},[618,619,676],[552,4008,933],{"className":4009},[932],[552,4011,4013],{"className":4012},[657],[552,4014,4016],{"className":4015,"style":2254},[661],[552,4017],{},[552,4019,985],{"className":4020},[1078],[552,4022,4024,4027],{"className":4023},[618],[552,4025,826],{"className":4026,"style":928},[618,619],[552,4028,4030],{"className":4029},[649],[552,4031,4033,4068],{"className":4032},[653,905],[552,4034,4036,4065],{"className":4035},[657],[552,4037,4039,4051],{"className":4038,"style":638},[661],[552,4040,4042,4045],{"style":4041},"top:-2.4413em;margin-left:-0.0359em;margin-right:0.05em;",[552,4043],{"className":4044,"style":669},[668],[552,4046,4048],{"className":4047},[673,674,675,676],[552,4049,1372],{"className":4050,"style":1615},[618,619,676],[552,4052,4053,4056],{"style":664},[552,4054],{"className":4055,"style":669},[668],[552,4057,4059],{"className":4058},[673,674,675,676],[552,4060,4062],{"className":4061},[618,676],[552,4063,1015],{"className":4064},[618,676],[552,4066,933],{"className":4067},[932],[552,4069,4071],{"className":4070},[657],[552,4072,4075],{"className":4073,"style":4074},[661],"height:0.3948em;",[552,4076],{},[552,4078,933],{"className":4079},[932],[552,4081,4083],{"className":4082},[657],[552,4084,4086],{"className":4085,"style":3245},[661],[552,4087],{},[552,4089],{"className":4090},[1078,1127],[552,4092,508],{"className":4093},[618],[112,4095,4096,4097,4146,4147,4214],{},"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 ",[552,4098,4100,4123],{"className":4099},[555],[552,4101,4103],{"className":4102},[559],[561,4104,4105],{"xmlns":563},[565,4106,4107,4120],{},[568,4108,4109,4112,4114,4116,4118],{},[571,4110,4111],{},"O",[575,4113,971],{"stretchy":970},[571,4115,589],{},[571,4117,716],{},[575,4119,985],{"stretchy":970},[597,4121,4122],{"encoding":599},"O(nm)",[552,4124,4126],{"className":4125,"ariaHidden":605},[604],[552,4127,4129,4132,4136,4139,4143],{"className":4128},[609],[552,4130],{"className":4131,"style":1039},[613],[552,4133,4111],{"className":4134,"style":4135},[618,619],"margin-right:0.0278em;",[552,4137,971],{"className":4138},[1051],[552,4140,4142],{"className":4141},[618,619],"nm",[552,4144,985],{"className":4145},[1078]," 掉到 ",[552,4148,4150,4175],{"className":4149},[555],[552,4151,4153],{"className":4152},[559],[561,4154,4155],{"xmlns":563},[565,4156,4157,4172],{},[568,4158,4159,4161,4163,4165,4168,4170],{},[571,4160,4111],{},[575,4162,971],{"stretchy":970},[571,4164,589],{},[575,4166,4167],{},"+",[571,4169,716],{},[575,4171,985],{"stretchy":970},[597,4173,4174],{"encoding":599},"O(n + m)",[552,4176,4178,4202],{"className":4177,"ariaHidden":605},[604],[552,4179,4181,4184,4187,4190,4193,4196,4199],{"className":4180},[609],[552,4182],{"className":4183,"style":1039},[613],[552,4185,4111],{"className":4186,"style":4135},[618,619],[552,4188,971],{"className":4189},[1051],[552,4191,589],{"className":4192},[618,619],[552,4194],{"className":4195,"style":842},[623],[552,4197,4167],{"className":4198},[686],[552,4200],{"className":4201,"style":842},[623],[552,4203,4205,4208,4211],{"className":4204},[609],[552,4206],{"className":4207,"style":1039},[613],[552,4209,716],{"className":4210},[618,619],[552,4212,985],{"className":4213},[1078],"：",[552,4216,4218],{"className":4217},[947],[552,4219,4221,4326],{"className":4220},[555],[552,4222,4224],{"className":4223},[559],[561,4225,4226],{"xmlns":563,"display":956},[565,4227,4228,4323],{},[4229,4230,4234,4273],"mtable",{"rowspacing":4231,"columnalign":4232,"columnspacing":4233},"0.25em","right left","0em",[4235,4236,4237,4247],"mtr",{},[4238,4239,4240],"mtd",{},[4241,4242,4244],"mstyle",{"scriptlevel":4243,"displaystyle":605},"0",[995,4245,4246],{},"標準 attention",[4238,4248,4249],{},[4241,4250,4251],{"scriptlevel":4243,"displaystyle":605},[568,4252,4253,4255,4258,4261,4263,4265,4267,4269,4271],{},[568,4254],{},[575,4256,4257],{},":",[623,4259],{"width":4260},"1em",[571,4262,4111],{},[575,4264,971],{"stretchy":970},[571,4266,589],{},[571,4268,716],{},[571,4270,595],{},[575,4272,985],{"stretchy":970},[4235,4274,4275,4282],{},[4238,4276,4277],{},[4241,4278,4279],{"scriptlevel":4243,"displaystyle":605},[995,4280,4281],{},"線性 attention",[4238,4283,4284],{},[4241,4285,4286],{"scriptlevel":4243,"displaystyle":605},[568,4287,4288,4290,4292,4294,4296,4298],{},[568,4289],{},[575,4291,4257],{},[623,4293],{"width":4260},[571,4295,4111],{},[995,4297,997],{},[568,4299,4300,4302,4304,4311,4313,4315,4321],{},[575,4301,971],{"fence":605},[571,4303,589],{},[579,4305,4306,4308],{},[571,4307,595],{},[1376,4309,4310],{},"2",[575,4312,4167],{},[571,4314,716],{},[579,4316,4317,4319],{},[571,4318,595],{},[1376,4320,4310],{},[575,4322,985],{"fence":605},[597,4324,4325],{"encoding":599},"\\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}",[552,4327,4329],{"className":4328,"ariaHidden":605},[604],[552,4330,4332,4336],{"className":4331},[609],[552,4333],{"className":4334,"style":4335},[613],"height:2.7241em;vertical-align:-1.1121em;",[552,4337,4339],{"className":4338},[618],[552,4340,4342,4409],{"className":4341},[4229],[552,4343,4346],{"className":4344},[4345],"col-align-r",[552,4347,4349,4400],{"className":4348},[653,905],[552,4350,4352,4397],{"className":4351},[657],[552,4353,4356,4378],{"className":4354,"style":4355},[661],"height:1.6121em;",[552,4357,4359,4362],{"style":4358},"top:-3.7721em;",[552,4360],{"className":4361,"style":1147},[668],[552,4363,4365],{"className":4364},[618],[552,4366,4369,4374],{"className":4367},[618,4368],"text",[552,4370,4373],{"className":4371},[618,4372],"cjk_fallback","標準",[552,4375,4377],{"className":4376},[618]," attention",[552,4379,4381,4384],{"style":4380},"top:-2.2479em;",[552,4382],{"className":4383,"style":1147},[668],[552,4385,4387],{"className":4386},[618],[552,4388,4390,4394],{"className":4389},[618,4368],[552,4391,4393],{"className":4392},[618,4372],"線性",[552,4395,4377],{"className":4396},[618],[552,4398,933],{"className":4399},[932],[552,4401,4403],{"className":4402},[657],[552,4404,4407],{"className":4405,"style":4406},[661],"height:1.1121em;",[552,4408],{},[552,4410,4413],{"className":4411},[4412],"col-align-l",[552,4414,4416,4586],{"className":4415},[653,905],[552,4417,4419,4583],{"className":4418},[657],[552,4420,4422,4461],{"className":4421,"style":4355},[661],[552,4423,4424,4427],{"style":4358},[552,4425],{"className":4426,"style":1147},[668],[552,4428,4430,4433,4436,4439,4443,4446,4449,4452,4455,4458],{"className":4429},[618],[552,4431],{"className":4432},[618],[552,4434],{"className":4435,"style":624},[623],[552,4437,4257],{"className":4438},[628],[552,4440],{"className":4441,"style":4442},[623],"margin-right:1em;",[552,4444],{"className":4445,"style":624},[623],[552,4447,4111],{"className":4448,"style":4135},[618,619],[552,4450,971],{"className":4451},[1051],[552,4453,4142],{"className":4454},[618,619],[552,4456,595],{"className":4457},[618,619],[552,4459,985],{"className":4460},[1078],[552,4462,4463,4466],{"style":4380},[552,4464],{"className":4465,"style":1147},[668],[552,4467,4469,4472,4475,4478,4481,4484,4487,4490,4493],{"className":4468},[618],[552,4470],{"className":4471},[618],[552,4473],{"className":4474,"style":624},[623],[552,4476,4257],{"className":4477},[628],[552,4479],{"className":4480,"style":4442},[623],[552,4482],{"className":4483,"style":624},[623],[552,4485,4111],{"className":4486,"style":4135},[618,619],[552,4488],{"className":4489,"style":1104},[623],[552,4491],{"className":4492,"style":1062},[623],[552,4494,4496,4502,4505,4536,4539,4542,4545,4548,4577],{"className":4495},[1111],[552,4497,4499],{"className":4498,"style":1116},[1051,1115],[552,4500,971],{"className":4501},[1120,924],[552,4503,589],{"className":4504},[618,619],[552,4506,4508,4511],{"className":4507},[618],[552,4509,595],{"className":4510},[618,619],[552,4512,4514],{"className":4513},[649],[552,4515,4517],{"className":4516},[653],[552,4518,4520],{"className":4519},[657],[552,4521,4524],{"className":4522,"style":4523},[661],"height:0.8641em;",[552,4525,4527,4530],{"style":4526},"top:-3.113em;margin-right:0.05em;",[552,4528],{"className":4529,"style":669},[668],[552,4531,4533],{"className":4532},[673,674,675,676],[552,4534,4310],{"className":4535},[618,676],[552,4537],{"className":4538,"style":842},[623],[552,4540,4167],{"className":4541},[686],[552,4543],{"className":4544,"style":842},[623],[552,4546,716],{"className":4547},[618,619],[552,4549,4551,4554],{"className":4550},[618],[552,4552,595],{"className":4553},[618,619],[552,4555,4557],{"className":4556},[649],[552,4558,4560],{"className":4559},[653],[552,4561,4563],{"className":4562},[657],[552,4564,4566],{"className":4565,"style":4523},[661],[552,4567,4568,4571],{"style":4526},[552,4569],{"className":4570,"style":669},[668],[552,4572,4574],{"className":4573},[673,674,675,676],[552,4575,4310],{"className":4576},[618,676],[552,4578,4580],{"className":4579,"style":1116},[1078,1115],[552,4581,985],{"className":4582},[1120,924],[552,4584,933],{"className":4585},[932],[552,4587,4589],{"className":4588},[657],[552,4590,4592],{"className":4591,"style":4406},[661],[552,4593],{},[112,4595,4596],{},"這就是 linear attention 那一整家族在做的事。",[108,4598,4599],{"id":4599},"代價在分母",[112,4601,4602,4603],{},"這裡有個很容易被忽略的地方：",[138,4604,4605],{},"誤差最大的地方不是分子，是分母。",[112,4607,4608,4609,4841,4842,5031],{},"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，",[552,4610,4612,4658],{"className":4611},[555],[552,4613,4615],{"className":4614},[559],[561,4616,4617],{"xmlns":563},[565,4618,4619,4655],{},[568,4620,4621,4623,4625,4631,4637,4643,4645,4647,4653],{},[571,4622,3332],{},[575,4624,971],{"stretchy":970},[820,4626,4627,4629],{},[571,4628,1401],{},[571,4630,1317],{},[579,4632,4633,4635],{},[575,4634,985],{"stretchy":970},[571,4636,1015],{"mathvariant":963},[820,4638,4639,4641],{},[575,4640,1367],{},[571,4642,1372],{},[571,4644,3332],{},[575,4646,971],{"stretchy":970},[820,4648,4649,4651],{},[571,4650,1410],{},[571,4652,1372],{},[575,4654,985],{"stretchy":970},[597,4656,4657],{"encoding":599},"\\phi(q_i)^{\\top}\\sum_j \\phi(k_j)",[552,4659,4661],{"className":4660,"ariaHidden":605},[604],[552,4662,4664,4668,4671,4674,4714,4746,4749,4789,4792,4795,4798,4838],{"className":4663},[609],[552,4665],{"className":4666,"style":4667},[613],"height:1.2849em;vertical-align:-0.4358em;",[552,4669,3332],{"className":4670},[618,619],[552,4672,971],{"className":4673},[1051],[552,4675,4677,4680],{"className":4676},[618],[552,4678,1401],{"className":4679,"style":928},[618,619],[552,4681,4683],{"className":4682},[649],[552,4684,4686,4706],{"className":4685},[653,905],[552,4687,4689,4703],{"className":4688},[657],[552,4690,4692],{"className":4691,"style":1508},[661],[552,4693,4694,4697],{"style":2235},[552,4695],{"className":4696,"style":669},[668],[552,4698,4700],{"className":4699},[673,674,675,676],[552,4701,1317],{"className":4702},[618,619,676],[552,4704,933],{"className":4705},[932],[552,4707,4709],{"className":4708},[657],[552,4710,4712],{"className":4711,"style":1530},[661],[552,4713],{},[552,4715,4717,4720],{"className":4716},[1078],[552,4718,985],{"className":4719},[1078],[552,4721,4723],{"className":4722},[649],[552,4724,4726],{"className":4725},[653],[552,4727,4729],{"className":4728},[657],[552,4730,4732],{"className":4731,"style":638},[661],[552,4733,4734,4737],{"style":664},[552,4735],{"className":4736,"style":669},[668],[552,4738,4740],{"className":4739},[673,674,675,676],[552,4741,4743],{"className":4742},[618,676],[552,4744,1015],{"className":4745},[618,676],[552,4747],{"className":4748,"style":1062},[623],[552,4750,4752,4755],{"className":4751},[1043],[552,4753,1367],{"className":4754,"style":1586},[1043,1584,1585],[552,4756,4758],{"className":4757},[649],[552,4759,4761,4781],{"className":4760},[653,905],[552,4762,4764,4778],{"className":4763},[657],[552,4765,4767],{"className":4766,"style":2917},[661],[552,4768,4769,4772],{"style":1602},[552,4770],{"className":4771,"style":669},[668],[552,4773,4775],{"className":4774},[673,674,675,676],[552,4776,1372],{"className":4777,"style":1615},[618,619,676],[552,4779,933],{"className":4780},[932],[552,4782,4784],{"className":4783},[657],[552,4785,4787],{"className":4786,"style":1646},[661],[552,4788],{},[552,4790],{"className":4791,"style":1062},[623],[552,4793,3332],{"className":4794},[618,619],[552,4796,971],{"className":4797},[1051],[552,4799,4801,4804],{"className":4800},[618],[552,4802,1410],{"className":4803,"style":1836},[618,619],[552,4805,4807],{"className":4806},[649],[552,4808,4810,4830],{"className":4809},[653,905],[552,4811,4813,4827],{"className":4812},[657],[552,4814,4816],{"className":4815,"style":1508},[661],[552,4817,4818,4821],{"style":2472},[552,4819],{"className":4820,"style":669},[668],[552,4822,4824],{"className":4823},[673,674,675,676],[552,4825,1372],{"className":4826,"style":1615},[618,619,676],[552,4828,933],{"className":4829},[932],[552,4831,4833],{"className":4832},[657],[552,4834,4836],{"className":4835,"style":2254},[661],[552,4837],{},[552,4839,985],{"className":4840},[1078]," 和真正的 ",[552,4843,4845,4883],{"className":4844},[555],[552,4846,4848],{"className":4847},[559],[561,4849,4850],{"xmlns":563},[565,4851,4852,4880],{},[568,4853,4854,4860,4862,4864,4870,4872,4878],{},[820,4855,4856,4858],{},[575,4857,1367],{},[571,4859,1372],{},[571,4861,2511],{},[575,4863,971],{"stretchy":970},[820,4865,4866,4868],{},[571,4867,1401],{},[571,4869,1317],{},[575,4871,976],{"separator":605},[820,4873,4874,4876],{},[571,4875,1410],{},[571,4877,1372],{},[575,4879,985],{"stretchy":970},[597,4881,4882],{"encoding":599},"\\sum_j \\kappa(q_i,k_j)",[552,4884,4886],{"className":4885,"ariaHidden":605},[604],[552,4887,4889,4893,4933,4936,4939,4942,4982,4985,4988,5028],{"className":4888},[609],[552,4890],{"className":4891,"style":4892},[613],"height:1.1858em;vertical-align:-0.4358em;",[552,4894,4896,4899],{"className":4895},[1043],[552,4897,1367],{"className":4898,"style":1586},[1043,1584,1585],[552,4900,4902],{"className":4901},[649],[552,4903,4905,4925],{"className":4904},[653,905],[552,4906,4908,4922],{"className":4907},[657],[552,4909,4911],{"className":4910,"style":2917},[661],[552,4912,4913,4916],{"style":1602},[552,4914],{"className":4915,"style":669},[668],[552,4917,4919],{"className":4918},[673,674,675,676],[552,4920,1372],{"className":4921,"style":1615},[618,619,676],[552,4923,933],{"className":4924},[932],[552,4926,4928],{"className":4927},[657],[552,4929,4931],{"className":4930,"style":1646},[661],[552,4932],{},[552,4934],{"className":4935,"style":1062},[623],[552,4937,2511],{"className":4938},[618,619],[552,4940,971],{"className":4941},[1051],[552,4943,4945,4948],{"className":4944},[618],[552,4946,1401],{"className":4947,"style":928},[618,619],[552,4949,4951],{"className":4950},[649],[552,4952,4954,4974],{"className":4953},[653,905],[552,4955,4957,4971],{"className":4956},[657],[552,4958,4960],{"className":4959,"style":1508},[661],[552,4961,4962,4965],{"style":2235},[552,4963],{"className":4964,"style":669},[668],[552,4966,4968],{"className":4967},[673,674,675,676],[552,4969,1317],{"className":4970},[618,619,676],[552,4972,933],{"className":4973},[932],[552,4975,4977],{"className":4976},[657],[552,4978,4980],{"className":4979,"style":1530},[661],[552,4981],{},[552,4983,976],{"className":4984},[1058],[552,4986],{"className":4987,"style":1062},[623],[552,4989,4991,4994],{"className":4990},[618],[552,4992,1410],{"className":4993,"style":1836},[618,619],[552,4995,4997],{"className":4996},[649],[552,4998,5000,5020],{"className":4999},[653,905],[552,5001,5003,5017],{"className":5002},[657],[552,5004,5006],{"className":5005,"style":1508},[661],[552,5007,5008,5011],{"style":2472},[552,5009],{"className":5010,"style":669},[668],[552,5012,5014],{"className":5013},[673,674,675,676],[552,5015,1372],{"className":5016,"style":1615},[618,619,676],[552,5018,933],{"className":5019},[932],[552,5021,5023],{"className":5022},[657],[552,5024,5026],{"className":5025,"style":2254},[661],[552,5027],{},[552,5029,985],{"className":5030},[1078]," 差得越多，輸出就越不可信。",[112,5033,5034],{},"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。",[108,5036,5037],{"id":5037},"實務上的三個檢查點",[5039,5040,5041,5054],"table",{},[5042,5043,5044],"thead",{},[5045,5046,5047,5051],"tr",{},[5048,5049,5050],"th",{},"檢查",[5048,5052,5053],{},"為什麼",[5055,5056,5057,5066,5145],"tbody",{},[5045,5058,5059,5063],{},[5060,5061,5062],"td",{},"分母的相對誤差",[5060,5064,5065],{},"比分子更致命，因為它會整體縮放輸出",[5045,5067,5068,5142],{},[5060,5069,5070,5071],{},"特徵維度 ",[552,5072,5074,5092],{"className":5073},[555],[552,5075,5077],{"className":5076},[559],[561,5078,5079],{"xmlns":563},[565,5080,5081,5089],{},[568,5082,5083],{},[820,5084,5085,5087],{},[571,5086,595],{},[571,5088,3332],{},[597,5090,5091],{"encoding":599},"d_\\phi",[552,5093,5095],{"className":5094,"ariaHidden":605},[604],[552,5096,5098,5101],{"className":5097},[609],[552,5099],{"className":5100,"style":2451},[613],[552,5102,5104,5107],{"className":5103},[618],[552,5105,595],{"className":5106},[618,619],[552,5108,5110],{"className":5109},[649],[552,5111,5113,5134],{"className":5112},[653,905],[552,5114,5116,5131],{"className":5115},[657],[552,5117,5120],{"className":5118,"style":5119},[661],"height:0.3361em;",[552,5121,5122,5125],{"style":1511},[552,5123],{"className":5124,"style":669},[668],[552,5126,5128],{"className":5127},[673,674,675,676],[552,5129,3332],{"className":5130},[618,619,676],[552,5132,933],{"className":5133},[932],[552,5135,5137],{"className":5136},[657],[552,5138,5140],{"className":5139,"style":2254},[661],[552,5141],{},[5060,5143,5144],{},"太少則近似粗糙，太多則省不下計算",[5045,5146,5147,5150],{},[5060,5148,5149],{},"序列長度分佈",[5060,5151,5152],{},"短序列幾乎沒有收益，長序列才值得換",[5154,5155,5159],"pre",{"className":5156,"code":5157,"language":5158,"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",[118,5160,5161,5168,5173,5179],{"__ignoreMap":311},[552,5162,5165],{"class":5163,"line":5164},"line",1,[552,5166,5167],{},"# 概念示意：先把 K、V 的累加量算好\n",[552,5169,5170],{"class":5163,"line":312},[552,5171,5172],{},"kv = einsum(\"m d, m e -> d e\", phi(K), V)   # O(m d^2)\n",[552,5174,5176],{"class":5163,"line":5175},3,[552,5177,5178],{},"z  = phi(K).sum(dim=0)                       # O(m d)\n",[552,5180,5182],{"class":5163,"line":5181},4,[552,5183,5184],{},"out = einsum(\"n d, d e -> n e\", phi(Q), kv) \u002F einsum(\"n d, d -> n\", phi(Q), z)\n",[108,5186,5187],{"id":5187},"給下一個要調這個模型的人",[112,5189,5190],{},"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。",[112,5192,5193,5194,5197],{},"因為換成 linear attention 不是「更快的 attention」，而是",[138,5195,5196],{},"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。",[112,5199,5200],{},"知道自己在付什麼，比知道自己在用什麼重要。",[5202,5203,5204],"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":5206},[5207,5208,5209,5210,5211,5212],{"id":538,"depth":312,"text":538},{"id":547,"depth":312,"text":547},{"id":3262,"depth":312,"text":3262},{"id":4599,"depth":312,"text":4599},{"id":5037,"depth":312,"text":5037},{"id":5187,"depth":312,"text":5187},"\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":533,"description":5216},"attention-as-kernel","blog\u002Fzh-TW\u002Fattention-as-kernel",[5223,5224,5225],"transformer","數學","推論","2026-10-22","ipCLaHTj0DVQiZoXlJ6DPzX7hl68udhNuMH4wa4gke8",{"id":5229,"title":5230,"author":103,"body":5231,"cover":5606,"coverAlt":5230,"date":5607,"description":5608,"draft":324,"extension":325,"featured":324,"lang":327,"meta":5609,"navigation":326,"path":5610,"seo":5611,"slug":5612,"stem":5613,"tags":5614,"translationKey":5612,"updated":337,"__hash__":5618},"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":5232,"toc":5596},[5233,5243,5251,5257,5263,5267,5270,5273,5287,5293,5297,5303,5309,5314,5370,5379,5383,5436,5456,5463,5477,5481,5484,5516,5520,5546,5550,5573,5588],[112,5234,5235,5238,5239,5242],{},[138,5236,5237],{},"研究單位","：騰訊微信 AI × 清華大學\n",[138,5240,5241],{},"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。",[112,5244,5245],{},[5246,5247,5248],"a",{"href":5248,"rel":5249},"https:\u002F\u002Farxiv.org\u002Fhtml\u002F2510.27688v1",[5250],"nofollow",[112,5252,5253],{},[5246,5254,5255],{"href":5255,"rel":5256},"https:\u002F\u002Fshaochenze.github.io\u002Fblog\u002F2025\u002FCALM\u002F",[5250],[112,5258,5259],{},[5246,5260,5261],{"href":5261,"rel":5262},"https:\u002F\u002Fgithub.com\u002Fmahmoodlab\u002Fclam",[5250],[108,5264,5266],{"id":5265},"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",[112,5268,5269],{},"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。",[112,5271,5272],{},"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：",[272,5274,5275,5281],{},[275,5276,5277,5280],{},[138,5278,5279],{},"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。",[275,5282,5283,5286],{},[138,5284,5285],{},"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。",[112,5288,5289,5292],{},[138,5290,5291],{},"結論","：必須轉向高維連續路線，從根本上改變生成機制。",[108,5294,5296],{"id":5295},"二calm-核心思路與原理","二、CALM 核心思路與原理",[112,5298,5299,5302],{},[138,5300,5301],{},"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。",[112,5304,5305,5308],{},[138,5306,5307],{},"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。",[5310,5311,5313],"h3",{"id":5312},"傳統範式-vs-calm-連續模型","傳統範式 vs. CALM 連續模型",[5039,5315,5316,5329],{},[5042,5317,5318],{},[5045,5319,5320,5323,5326],{},[5048,5321,5322],{},"維度",[5048,5324,5325],{},"傳統離散模型 (Discrete)",[5048,5327,5328],{},"CALM 連續模型 (Continuous)",[5055,5330,5331,5344,5357],{},[5045,5332,5333,5338,5341],{},[5060,5334,5335],{},[138,5336,5337],{},"擴容方式",[5060,5339,5340],{},"詞表指數級膨脹，算力爆炸",[5060,5342,5343],{},"增加向量維度，算力平滑增長",[5045,5345,5346,5351,5354],{},[5060,5347,5348],{},[138,5349,5350],{},"計算目標",[5060,5352,5353],{},"從數十萬詞中選一個機率",[5060,5355,5356],{},"在連續空間中直接預測精確座標",[5045,5358,5359,5364,5367],{},[5060,5360,5361],{},[138,5362,5363],{},"序列長度",[5060,5365,5366],{},"極長序列 T",[5060,5368,5369],{},"短序列 T\u002FK，步數銳減",[5371,5372,5373],"blockquote",{},[112,5374,5375,5378],{},[138,5376,5377],{},"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。",[108,5380,5382],{"id":5381},"三calm-的四大技術支柱","三、CALM 的四大技術支柱",[5384,5385,5386,5422,5431],"ol",{},[275,5387,5388,5391,5392,5421],{},[138,5389,5390],{},"高保真自編碼器 (Autoencoder)","\n將 ",[552,5393,5395,5408],{"className":5394},[555],[552,5396,5398],{"className":5397},[559],[561,5399,5400],{"xmlns":563},[565,5401,5402,5406],{},[568,5403,5404],{},[571,5405,705],{},[597,5407,705],{"encoding":599},[552,5409,5411],{"className":5410,"ariaHidden":605},[604],[552,5412,5414,5418],{"className":5413},[609],[552,5415],{"className":5416,"style":5417},[613],"height:0.6833em;",[552,5419,705],{"className":5420,"style":737},[618,619]," 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。",[275,5423,5424,5427,5428,153],{},[138,5425,5426],{},"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",[138,5429,5430],{},"單步生成",[275,5432,5433],{},[138,5434,5435],{},"連續空間魯棒性三大魔法",[272,5437,5438,5444,5450],{},[275,5439,5440,5443],{},[138,5441,5442],{},"變分正則化 (Variational Regularization)","：熨平潛空間（Latent space）中的褶皺與斷層。",[275,5445,5446,5449],{},[138,5447,5448],{},"KL 裁減 (KL Clipping)","：防止潛空間維度嚴重塌陷。",[275,5451,5452,5455],{},[138,5453,5454],{},"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。",[5384,5457,5458],{"start":5181},[275,5459,5460],{},[138,5461,5462],{},"無機率評估體系 BrierLM & 溫度控制",[272,5464,5465,5471],{},[275,5466,5467,5470],{},[138,5468,5469],{},"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。",[275,5472,5473,5476],{},[138,5474,5475],{},"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。",[108,5478,5480],{"id":5479},"四流程總覽","四、流程總覽",[112,5482,5483],{},"整個運作閉環如下，速度遠超傳統的逐字生成：",[5384,5485,5486,5492,5498,5504,5510],{},[275,5487,5488,5491],{},[138,5489,5490],{},"輸入","：離散 Token 序列。",[275,5493,5494,5497],{},[138,5495,5496],{},"壓縮","：編碼器將每 K 個 Token 壓縮為 1 個連續向量。",[275,5499,5500,5503],{},[138,5501,5502],{},"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。",[275,5505,5506,5509],{},[138,5507,5508],{},"預測","：生成頭直接單步預測出下一個目標連續向量。",[275,5511,5512,5515],{},[138,5513,5514],{},"還原","：解碼器將預測出的連續向量無損還原為 K 個離散 Token。",[108,5517,5519],{"id":5518},"五實驗結果與性能優勢","五、實驗結果與性能優勢",[272,5521,5522,5528,5534,5540],{},[275,5523,5524,5527],{},[138,5525,5526],{},"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。",[275,5529,5530,5533],{},[138,5531,5532],{},"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。",[275,5535,5536,5539],{},[138,5537,5538],{},"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。",[275,5541,5542,5545],{},[138,5543,5544],{},"「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。",[108,5547,5549],{"id":5548},"六總結與展望","六、總結與展望",[112,5551,5552,5553,5556,5557,5560,5561,5564,5565,5568,5569,5572],{},"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",[138,5554,5555],{},"魯棒自編碼器"," (高保真壓縮) → ",[138,5558,5559],{},"頻寬倍增"," (步數縮短至 1\u002FK) → ",[138,5562,5563],{},"單步生成引擎"," (極速推理) → ",[138,5566,5567],{},"拒絕採樣"," (多樣性可調) → ",[138,5570,5571],{},"BrierLM"," (公正尺標)。",[112,5574,5575,5576,5579,5580,5583,5584,5587],{},"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",[138,5577,5578],{},"多模態連續生成","、",[138,5581,5582],{},"強化學習整合","以及",[138,5585,5586],{},"終端設備部署","開啟了全新的紀元。",[5371,5589,5590],{},[112,5591,5592,5595],{},[138,5593,5594],{},"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":5597},[5598,5599,5602,5603,5604,5605],{"id":5265,"depth":312,"text":5266},{"id":5295,"depth":312,"text":5296,"children":5600},[5601],{"id":5312,"depth":5175,"text":5313},{"id":5381,"depth":312,"text":5382},{"id":5479,"depth":312,"text":5480},{"id":5518,"depth":312,"text":5519},{"id":5548,"depth":312,"text":5549},"\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":5230,"description":5608},"arxiv-2510-27688v1","blog\u002Fzh-TW\u002Farxiv-2510-27688v1",[5615,5616,5617],"paper","llm","efficiency","YXovptigvOJLPbjjtVgdCae3S6_ZcSZxJw3mH-BUBnc",1792775650058]