[{"data":1,"prerenderedAt":5544},["ShallowReactive",2],{"site-info":3,"post-strata-125b-on-12gb":99,"related-zh-TW-strata-125b-on-12gb":626},{"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":624},{"id":101,"title":102,"author":103,"body":104,"cover":606,"coverAlt":607,"date":608,"description":609,"draft":610,"extension":611,"featured":610,"lang":612,"meta":613,"navigation":614,"ogImage":615,"path":616,"seo":617,"slug":618,"stem":619,"tags":620,"translationKey":618,"updated":624,"__hash__":625},"blog\u002Fblog\u002Fzh-TW\u002Fstrata-125b-on-12gb.md","125B 模型，12GB 顯卡，94 tok\u002Fs","Jecision",{"type":105,"value":106,"toc":592},"minimark",[107,128,132,135,138,141,149,156,160,163,179,182,186,193,196,199,203,209,212,268,271,274,280,287,291,294,297,300,303,364,371,376,383,386,392,398,404,407,410,416,423,430,434,437,444,447,495,498,505,513,516,537,540,543,546,553,566,569,589],[108,109,110],"blockquote",{},[111,112,113,114,121,122,127],"p",{},"這篇沒有我們的實測。數據來自 Qwen 官方發布、",[115,116,120],"a",{"href":117,"rel":118},"https:\u002F\u002Fgithub.com\u002FBlockYugaLabs\u002FStrata",[119],"nofollow","Strata 開源專案"," 的說明，以及一位對岸開發者的",[115,123,126],{"href":124,"rel":125},"https:\u002F\u002Fwww.bilibili.com\u002Fvideo\u002FBV1WwJy6EEuN\u002F",[119],"公開實測影片","。文中會標明哪個數字是誰給的。",[129,130,131],"h2",{"id":131},"三個數字",[111,133,134],{},"125B 的模型。12GB 的顯示卡。94 tok\u002Fs。",[111,136,137],{},"半年前，這三個數字放在一起是笑話。當時想在家用電腦跑 27B 的模型，量化後 20GB，加上 KV cache，你得有 24GB 的卡才勉強開到 128K 上下文。",[111,139,140],{},"現在 125B 跑在 12GB 上，而且不是「能跑」，是「比你讀字還快」。",[111,142,143,144,148],{},"真正變的不是模型變聰明，是",[145,146,147],"strong",{},"模型有多大，和硬體要多好，已經開始脫鉤","。",[111,150,151,152,155],{},"先講清楚前提：94 tok\u002Fs 是 RTX 5070（12GB）搭配 ",[145,153,154],{},"64GB 系統記憶體","的成績。只有 32GB 記憶體的話，只能跑砍半的 Coder 版本。這不是「插一張卡就好」的故事。",[129,157,159],{"id":158},"兩個限制被分別繞過","兩個限制，被分別繞過",[111,161,162],{},"大模型跑不動，卡在兩件事：",[164,165,166,173],"ol",{},[167,168,169,172],"li",{},[145,170,171],{},"裝不下","：權重是硬的。這顆模型加上 51B 的 n-gram 查表，量化後檔案還是 70–80GB。",[167,174,175,178],{},[145,176,177],{},"算不動","：每個 token 都要讓所有參數參與計算。",[111,180,181],{},"Qwen3.8-Flash-Next 處理了第二個，Strata 處理了第一個。",[129,183,185],{"id":184},"第一個限制不是每個專家都要上班","第一個限制：不是每個專家都要上班",[111,187,188,189,192],{},"Flash-Next 是 MoE（混合專家）架構。官方數字是 125B 總參數、",[145,190,191],{},"6B 激活","：每個 token 的計算量，大約等於一個 6B 模型。",[111,194,195],{},"打個比方：傳統稠密模型是「公司裡每個部門都得參與每一個案子」，MoE 是「只叫相關的幾個部門進來開會」。",[111,197,198],{},"所以「算不動」這件事，先解決了一半。",[129,200,202],{"id":201},"第二個限制把整台電腦當成一台機器","第二個限制：把整台電腦當成一台機器",[111,204,205,206,148],{},"但 6B 的計算量，不代表 6B 的記憶體——",[145,207,208],{},"125B 的權重還是得存在某個地方",[111,210,211],{},"Strata 的做法是不要全部塞進顯存，按「多常用」分三層放（見圖 1）：",[213,214,215,231],"table",{},[216,217,218],"thead",{},[219,220,221,225,228],"tr",{},[222,223,224],"th",{},"層",[222,226,227],{},"放什麼",[222,229,230],{},"佔用",[232,233,234,246,257],"tbody",{},[219,235,236,240,243],{},[237,238,239],"td",{},"顯存 VRAM",[237,241,242],{},"最常命中的專家",[237,244,245],{},"12–24 GB",[219,247,248,251,254],{},[237,249,250],{},"系統記憶體",[237,252,253],{},"全部專家（開機時載入）",[237,255,256],{},"約 47 GB",[219,258,259,262,265],{},[237,260,261],{},"SSD",[237,263,264],{},"其餘權重與 n-gram 查表",[237,266,267],{},"檔案總量 70–80 GB",[111,269,270],{},"（47GB 是那位開發者在 IQ3_S 量化下的實測；官方建議 64GB 記憶體就是為了裝下這一層。）",[111,272,273],{},"跑起來的時候，GPU 算它手上的專家；沒命中的，要嘛走 PCIe 搬進顯存，要嘛直接讓 CPU 用記憶體裡的權重算。",[111,275,276,277,148],{},"這就是為什麼 12GB 的卡能跑——",[145,278,279],{},"顯存不再是「裝不裝得下」的容器，而是「快取多大」的問題",[111,281,282],{},[283,284],"img",{"alt":285,"src":286},"如圖 1：三層記憶體階層——顯存放熱門專家、記憶體放全部專家、SSD 放冷門權重與 n-gram 查表","\u002Fimages\u002Fblog\u002Fstrata-125b-on-12gb\u002F01-memory-tiers.svg",[129,288,290],{"id":289},"還有一個加速器猜然後驗證","還有一個加速器：猜，然後驗證",[111,292,293],{},"除了省記憶體，Strata 還用了投機解碼：小模型先猜接下來幾個 token，大模型一次驗證。",[111,295,296],{},"猜對就省下好幾步，官方數字大約快 1.6–1.8 倍。",[129,298,299],{"id":299},"實際多快",[111,301,302],{},"官方在 RTX 5070（12GB 顯存、64GB RAM）上測到的：",[213,304,305,318],{},[216,306,307],{},[219,308,309,312,315],{},[222,310,311],{},"量化",[222,313,314],{},"寫答案",[222,316,317],{},"讀 prompt（32K）",[232,319,320,331,342,353],{},[219,321,322,325,328],{},[237,323,324],{},"Q2_0",[237,326,327],{},"94 tok\u002Fs",[237,329,330],{},"2,650 tok\u002Fs",[219,332,333,336,339],{},[237,334,335],{},"IQ2_XS",[237,337,338],{},"79 tok\u002Fs",[237,340,341],{},"2,090 tok\u002Fs",[219,343,344,347,350],{},[237,345,346],{},"IQ3_XXS",[237,348,349],{},"62 tok\u002Fs",[237,351,352],{},"1,750 tok\u002Fs",[219,354,355,358,361],{},[237,356,357],{},"IQ3_S",[237,359,360],{},"53 tok\u002Fs",[237,362,363],{},"1,620 tok\u002Fs",[111,365,366,367,370],{},"24GB 的卡更快。那位開發者在他的 4090 + 64GB RAM 上實測：",[145,368,369],{},"解碼約 140 tok\u002Fs、prefill 約 4,400 tok\u002Fs","，256K 上下文開滿也還跑得動。",[108,372,373],{},[111,374,375],{},"他第一次測到 203 tok\u002Fs，後來發現是 benchmark 沒把 reasoning 時間與快取命中算進去。修正後才是 140。這種「先懷疑自己的量測」比數字本身更有價值。",[111,377,378,379,382],{},"要注意上面那張表的「讀 prompt 2,650 tok\u002Fs」是快取熱起來之後的數字。",[145,380,381],{},"冷啟動第一次讀長文會慢很多","——官方文件寫的是每 30,000 tokens 約一分鐘，大約 500 tok\u002Fs。",[129,384,385],{"id":385},"代價",[111,387,388,391],{},[145,389,390],{},"一、量化到 2-bit。"," 94 tok\u002Fs 是 Q2_0 的數字，那是很兇的壓縮。要品質就得犧牲速度——IQ3_S 剩 53 tok\u002Fs。這是取捨，不是免費的。",[111,393,394,397],{},[145,395,396],{},"二、記憶體與啟動。"," 開機要把 35–55GB 載進記憶體，電腦會卡 1–3 分鐘；整個程序佔用超過 50GB。32GB 的機器只能跑 Coder 變體（砍掉一半專家），官方也提醒它在中文本表現較弱。",[111,399,400,403],{},[145,401,402],{},"三、一次一個請求。"," 預設不支援併發，第二個請求要排隊。開併發得調參數，12GB 的卡上會讓每個回答變慢。",[129,405,406],{"id":406},"調參的迷思",[111,408,409],{},"Strata 有不少參數可調：PCIE_FRAC（GPU 與 CPU 怎麼分攤計算）、CPU Workers、MTP 深度、KVGROW。",[111,411,412,413,148],{},"那位開發者花了大量時間做矩陣測試，他的結論是：",[145,414,415],{},"官方自動校準就夠了",[111,417,418,419,422],{},"他自己測出的「快 5%、快 9%」，多跑幾輪就消失，有時候還反過來。原因是速度受當下生成的內容、專家路由狀態、MTP 命中率、GPU 快取影響——這些變異比參數本身還大。這是",[145,420,421],{},"在他那台機器上","的結論，換硬體要重新驗證。",[111,424,425,426,429],{},"有一個參數值得單獨提：",[145,427,428],{},"KVGROW","。預設開 256K 上下文時，引擎會先把整段 KV cache 的顯存預留起來，結果短對話也沒位置放熱專家，速度掉到最慢檔。KVGROW 讓 cache 跟著實際用量長大：短對話快，長對話才讓位。",[129,431,433],{"id":432},"所以該不該自架","所以，該不該自架？",[111,435,436],{},"先把興奮放一邊，算一算。匯率抓 1 美元 ≈ NT$31.5。",[111,438,439,440,443],{},"Qwen3.8-Flash 的 API 價格是 ",[145,441,442],{},"$0.16 \u002F 百萬 input token、$0.47 \u002F 百萬 output","——output 是 input 的三倍，這段先只算 input，方便比較。",[111,445,446],{},"一台跑得舒服的機器（4090 或同級 + 64GB RAM）大約台幣 8–10 萬。",[213,448,449,462],{},[216,450,451],{},[219,452,453,456,459],{},[222,454,455],{},"每月 input 用量",[222,457,458],{},"API 成本",[222,460,461],{},"自架電費",[232,463,464,475,485],{},[219,465,466,469,472],{},[237,467,468],{},"1,000 萬 token",[237,470,471],{},"約 NT$50",[237,473,474],{},"NT$250–370",[219,476,477,480,483],{},[237,478,479],{},"1 億 token",[237,481,482],{},"約 NT$500",[237,484,474],{},[219,486,487,490,493],{},[237,488,489],{},"10 億 token",[237,491,492],{},"約 NT$5,000",[237,494,474],{},[111,496,497],{},"電費怎麼算的：推論時約 350W，每天用兩小時、其餘時間閒置 80W，一個月約 74 度；台電累進電價抓每度 NT$3–5，就是每個月 250–370 元。24 小時滿載的話是 250 度左右，大約 NT$750–1,250。",[111,499,500,501,504],{},"看表就知道結論：",[145,502,503],{},"每月十億 token 以下，API 連電費都比較便宜","，還沒算機器折舊。",[111,506,507,508,512],{},"粗算回本點：",[509,510,511],"code",{},"10 萬 ÷ (5,000 − 370) ≈ 22 個月","——那是每月十億 token 的量才成立。對絕大多數中小企業，這個量級不會出現。",[111,514,515],{},"那自架還有什麼意義？三個：",[517,518,519,525,531],"ul",{},[167,520,521,524],{},[145,522,523],{},"隱私","：資料不出門。這是最難用錢換的",[167,526,527,530],{},[145,528,529],{},"離線","：工廠、船上、沒有網路的現場",[167,532,533,536],{},[145,534,535],{},"不受限","：不用擔心費率調整、政策變更、API 下架",[111,538,539],{},"如果你不屬於這三種，用 API 就好，而且會用得比自架開心。",[129,541,542],{"id":542},"真正的重點",[111,544,545],{},"這篇想說的不是「125B 好強」。",[111,547,548,549,552],{},"是",[145,550,551],{},"模型大小已經不能拿來判斷硬體需求了","。同樣 125B，你可以用 360GB 的伺服器跑，也可以用 12GB 的遊戲卡跑——差別只在於有沒有人幫你把記憶體階層調度好。",[111,554,555,556,560,561,565],{},"這也是我們做",[115,557,559],{"href":558},"\u002Fservices","地端部署","時真正在處理的問題：不是「這顆模型跑不跑得動」，而是「你的瓶頸在哪一層」。我們在",[115,562,564],{"href":563},"\u002Fwork\u002Fembedded-vision-background","資源受限裝置","上累積的經驗，換到 LLM 上其實是同一件事。",[111,567,568],{},"如果你正在評估要不要自架，先問三個問題：",[164,570,571,577,583],{},[167,572,573,576],{},[145,574,575],{},"每月 token 量有沒有到億級？"," 沒有，API 幾乎一定更划算。",[167,578,579,582],{},[145,580,581],{},"資料能不能出門？"," 不能，才輪到自架。",[167,584,585,588],{},[145,586,587],{},"有沒有 64GB 記憶體與一張 12GB 以上的卡？"," 沒有，連門檻都不到。",[111,590,591],{},"三題有兩題是「否」，答案就是 API。",{"title":593,"searchDepth":594,"depth":594,"links":595},"",2,[596,597,598,599,600,601,602,603,604,605],{"id":131,"depth":594,"text":131},{"id":158,"depth":594,"text":159},{"id":184,"depth":594,"text":185},{"id":201,"depth":594,"text":202},{"id":289,"depth":594,"text":290},{"id":299,"depth":594,"text":299},{"id":385,"depth":594,"text":385},{"id":406,"depth":594,"text":406},{"id":432,"depth":594,"text":433},{"id":542,"depth":594,"text":542},"\u002Fimages\u002Fblog\u002Fstrata-125b-on-12gb.svg","125B 模型跑在 12GB 顯卡上的示意圖","2026-10-24","Strata 讓 125B 的 Qwen3.8-Flash-Next 跑在 12GB 顯卡的遊戲機上，解碼 94 tok\u002Fs。拆解它靠什麼做到、代價在哪裡，以及中小企業該不該自架。",false,"md","zh-TW",{},true,"\u002Fimages\u002Fblog\u002Fstrata-125b-on-12gb.jpg","\u002Fblog\u002Fzh-tw\u002Fstrata-125b-on-12gb",{"title":102,"description":609},"strata-125b-on-12gb","blog\u002Fzh-TW\u002Fstrata-125b-on-12gb",[621,622,623],"local-llm","qwen","inference",null,"AzAFYoV9JKM6X1-qHOAhv64-PaZVkfvITS_J8_39_Oc",[627,851],{"id":628,"title":629,"author":103,"body":630,"cover":836,"coverAlt":837,"date":838,"description":839,"draft":610,"extension":611,"featured":614,"lang":612,"meta":840,"navigation":614,"ogImage":841,"path":842,"seo":843,"slug":844,"stem":845,"tags":846,"translationKey":844,"updated":624,"__hash__":850},"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",{"type":105,"value":631,"toc":828},[632,635,638,656,662,665,668,674,689,692,695,716,734,740,743,755,765,768,782,788,791,816,819,825],[129,633,634],{"id":634},"起因是一行原始碼",[111,636,637],{},"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。",[111,639,640,643,644,647,648,651,652,655],{},[509,641,642],{},"\u003Chtml>"," 沒有 ",[509,645,646],{},"lang","。沒有 ",[509,649,650],{},"\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 ",[509,653,654],{},"\\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。",[111,657,658,659],{},"也就是說：",[145,660,661],{},"對搜尋引擎來說，那篇文章從來沒有存在過。",[111,663,664],{},"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。",[129,666,667],{"id":667},"當初為什麼會變這樣",[111,669,670,671,148],{},"不是不用心，是",[145,672,673],{},"工具用得比它設計的方式更用力",[111,675,676,677,680,681,684,685,688],{},"當時為了讓公式能顯示，我們在 Markdown 裡塞了 ",[509,678,679],{},"\u003Cdiv class=\"latex-block\">","，然後在 ",[509,682,683],{},"onMounted"," 裡用 KaTeX 把它的 ",[509,686,687],{},"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。",[111,690,691],{},"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」。",[129,693,694],{"id":694},"搬家的三個決定",[111,696,697,700,701,704,705,708,709,704,712,715],{},[145,698,699],{},"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 ",[509,702,703],{},"$...$"," 與 ",[509,706,707],{},"$$...$$","，交給 ",[509,710,711],{},"remark-math",[509,713,714],{},"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。",[111,717,718,721,722,725,726,729,730,733],{},[145,719,720],{},"二、網址由資料決定，不由檔名決定。"," 舊站的檔名是 ",[509,723,724],{},"2604.04921v1.md","，結果網址變成 ",[509,727,728],{},"\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 ",[509,731,732],{},"slug","，檔名再亂都不影響網址。",[111,735,736,739],{},[145,737,738],{},"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。",[129,741,742],{"id":742},"三個差點讓上線失敗的意外",[111,744,745,748,749,751,752,754],{},[145,746,747],{},"第一個：slug 被吃掉。"," 這件事是在舊站發現的——",[509,750,724],{}," 變成 ",[509,753,728],{},"。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。",[111,756,757,760,761,764],{},[145,758,759],{},"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 ",[509,762,763],{},"nuxt generate"," 的時候 prerender 直接掛掉，錯誤訊息長得像「Article not found，linked from \u002Fblog\u002Fxxx」。",[111,766,767],{},"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。",[111,769,770,777,778,781],{},[145,771,772,773,776],{},"第三個：",[509,774,775],{},"\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 ",[509,779,780],{},"\u003Cmeta http-equiv=\"refresh\">"," 的轉址頁，不是 XML。Google 會直接判定 sitemap 無效。",[111,783,784,785,787],{},"後來在 build 後加了一步，把 sitemap index 的內容直接寫成 ",[509,786,775],{},"，才真的能提交。",[129,789,790],{"id":790},"現在的樣子",[517,792,793,796,803,810],{},[167,794,795],{},"公式在伺服器端就排版完成，原始碼裡有 MathML",[167,797,798,799,802],{},"每一頁都有 title、description、canonical、hreflang、og，文章另有 ",[509,800,801],{},"BlogPosting"," 結構化資料",[167,804,805,806,809],{},"RSS、",[509,807,808],{},"llms.txt","、sitemap、301 對照都有",[167,811,812,813,815],{},"還寫了一支 ",[509,814,20],{}," 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",[129,817,818],{"id":818},"一句話總結",[111,820,821,822,148],{},"搬家最花時間的不是搬，是",[145,823,824],{},"確認搬過去的東西真的還在",[111,826,827],{},"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":593,"searchDepth":594,"depth":594,"links":829},[830,831,832,833,834,835],{"id":634,"depth":594,"text":634},{"id":667,"depth":594,"text":667},{"id":694,"depth":594,"text":694},{"id":742,"depth":594,"text":742},{"id":790,"depth":594,"text":790},{"id":818,"depth":594,"text":818},"\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。",{},"\u002Fimages\u002Fblog\u002Fhello-jecision.jpg","\u002Fblog\u002Fzh-tw\u002Fhello-jecision",{"title":629,"description":839},"hello-jecision","blog\u002Fzh-TW\u002Fhello-jecision",[847,848,849],"nuxt","seo","重構","vTfvwFIhG6P612a8EVZKVEoCJfLUHgQBkyLmUnoWKe0",{"id":852,"title":853,"author":103,"body":854,"cover":5528,"coverAlt":5529,"date":5530,"description":5531,"draft":610,"extension":611,"featured":610,"lang":612,"meta":5532,"navigation":614,"ogImage":5533,"path":5534,"seo":5535,"slug":5536,"stem":5537,"tags":5538,"translationKey":5536,"updated":5542,"__hash__":5543},"blog\u002Fblog\u002Fzh-TW\u002Fattention-as-kernel.md","Attention 其實是一個核平滑器",{"type":105,"value":855,"toc":5520},[856,859,862,865,868,1264,1621,1655,2596,2815,3031,3575,3578,3581,3584,3590,3775,4415,4536,4915,4918,4921,4927,5353,5356,5359,5468,5500,5503,5506,5513,5516],[129,857,858],{"id":858},"一句話讓我想了很久",[111,860,861],{},"「Attention 就是一個核平滑器。」",[111,863,864],{},"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。",[129,866,867],{"id":867},"把公式拆開",[111,869,870,871,1010,1011,1111,1112,1263],{},"給定查詢矩陣 ",[872,873,876,921],"span",{"className":874},[875],"katex",[872,877,880],{"className":878},[879],"katex-mathml",[881,882,884],"math",{"xmlns":883},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[885,886,887,916],"semantics",{},[888,889,890,894,898],"mrow",{},[891,892,893],"mi",{},"Q",[895,896,897],"mo",{},"∈",[899,900,901,905],"msup",{},[891,902,904],{"mathvariant":903},"double-struck","R",[888,906,907,910,913],{},[891,908,909],{},"n",[895,911,912],{},"×",[891,914,915],{},"d",[917,918,920],"annotation",{"encoding":919},"application\u002Fx-tex","Q \\in \\mathbb{R}^{n \\times d}",[872,922,926,952],{"className":923,"ariaHidden":925},[924],"katex-html","true",[872,927,930,935,940,945,949],{"className":928},[929],"base",[872,931],{"className":932,"style":934},[933],"strut","height:0.8778em;vertical-align:-0.1944em;",[872,936,893],{"className":937},[938,939],"mord","mathnormal",[872,941],{"className":942,"style":944},[943],"mspace","margin-right:0.2778em;",[872,946,897],{"className":947},[948],"mrel",[872,950],{"className":951,"style":944},[943],[872,953,955,959],{"className":954},[929],[872,956],{"className":957,"style":958},[933],"height:0.8491em;",[872,960,962,966],{"className":961},[938],[872,963,904],{"className":964},[938,965],"mathbb",[872,967,970],{"className":968},[969],"msupsub",[872,971,974],{"className":972},[973],"vlist-t",[872,975,978],{"className":976},[977],"vlist-r",[872,979,982],{"className":980,"style":958},[981],"vlist",[872,983,985,990],{"style":984},"top:-3.063em;margin-right:0.05em;",[872,986],{"className":987,"style":989},[988],"pstrut","height:2.7em;",[872,991,997],{"className":992},[993,994,995,996],"sizing","reset-size6","size3","mtight",[872,998,1000,1003,1007],{"className":999},[938,996],[872,1001,909],{"className":1002},[938,939,996],[872,1004,912],{"className":1005},[1006,996],"mbin",[872,1008,915],{"className":1009},[938,939,996],"、鍵矩陣 ",[872,1012,1014,1044],{"className":1013},[875],[872,1015,1017],{"className":1016},[879],[881,1018,1019],{"xmlns":883},[885,1020,1021,1041],{},[888,1022,1023,1026,1028],{},[891,1024,1025],{},"K",[895,1027,897],{},[899,1029,1030,1032],{},[891,1031,904],{"mathvariant":903},[888,1033,1034,1037,1039],{},[891,1035,1036],{},"m",[895,1038,912],{},[891,1040,915],{},[917,1042,1043],{"encoding":919},"K \\in \\mathbb{R}^{m \\times d}",[872,1045,1047,1067],{"className":1046,"ariaHidden":925},[924],[872,1048,1050,1054,1058,1061,1064],{"className":1049},[929],[872,1051],{"className":1052,"style":1053},[933],"height:0.7224em;vertical-align:-0.0391em;",[872,1055,1025],{"className":1056,"style":1057},[938,939],"margin-right:0.0715em;",[872,1059],{"className":1060,"style":944},[943],[872,1062,897],{"className":1063},[948],[872,1065],{"className":1066,"style":944},[943],[872,1068,1070,1073],{"className":1069},[929],[872,1071],{"className":1072,"style":958},[933],[872,1074,1076,1079],{"className":1075},[938],[872,1077,904],{"className":1078},[938,965],[872,1080,1082],{"className":1081},[969],[872,1083,1085],{"className":1084},[973],[872,1086,1088],{"className":1087},[977],[872,1089,1091],{"className":1090,"style":958},[981],[872,1092,1093,1096],{"style":984},[872,1094],{"className":1095,"style":989},[988],[872,1097,1099],{"className":1098},[993,994,995,996],[872,1100,1102,1105,1108],{"className":1101},[938,996],[872,1103,1036],{"className":1104},[938,939,996],[872,1106,912],{"className":1107},[1006,996],[872,1109,915],{"className":1110},[938,939,996]," 與值矩陣 ",[872,1113,1115,1150],{"className":1114},[875],[872,1116,1118],{"className":1117},[879],[881,1119,1120],{"xmlns":883},[885,1121,1122,1147],{},[888,1123,1124,1127,1129],{},[891,1125,1126],{},"V",[895,1128,897],{},[899,1130,1131,1133],{},[891,1132,904],{"mathvariant":903},[888,1134,1135,1137,1139],{},[891,1136,1036],{},[895,1138,912],{},[1140,1141,1142,1144],"msub",{},[891,1143,915],{},[891,1145,1146],{},"v",[917,1148,1149],{"encoding":919},"V \\in \\mathbb{R}^{m \\times d_v}",[872,1151,1153,1172],{"className":1152,"ariaHidden":925},[924],[872,1154,1156,1159,1163,1166,1169],{"className":1155},[929],[872,1157],{"className":1158,"style":1053},[933],[872,1160,1126],{"className":1161,"style":1162},[938,939],"margin-right:0.2222em;",[872,1164],{"className":1165,"style":944},[943],[872,1167,897],{"className":1168},[948],[872,1170],{"className":1171,"style":944},[943],[872,1173,1175,1178],{"className":1174},[929],[872,1176],{"className":1177,"style":958},[933],[872,1179,1181,1184],{"className":1180},[938],[872,1182,904],{"className":1183},[938,965],[872,1185,1187],{"className":1186},[969],[872,1188,1190],{"className":1189},[973],[872,1191,1193],{"className":1192},[977],[872,1194,1196],{"className":1195,"style":958},[981],[872,1197,1198,1201],{"style":984},[872,1199],{"className":1200,"style":989},[988],[872,1202,1204],{"className":1203},[993,994,995,996],[872,1205,1207,1210,1213],{"className":1206},[938,996],[872,1208,1036],{"className":1209},[938,939,996],[872,1211,912],{"className":1212},[1006,996],[872,1214,1216,1219],{"className":1215},[938,996],[872,1217,915],{"className":1218},[938,939,996],[872,1220,1222],{"className":1221},[969],[872,1223,1226,1254],{"className":1224},[973,1225],"vlist-t2",[872,1227,1229,1249],{"className":1228},[977],[872,1230,1233],{"className":1231,"style":1232},[981],"height:0.1645em;",[872,1234,1236,1240],{"style":1235},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[872,1237],{"className":1238,"style":1239},[988],"height:2.5em;",[872,1241,1245],{"className":1242},[993,1243,1244,996],"reset-size3","size1",[872,1246,1146],{"className":1247,"style":1248},[938,939,996],"margin-right:0.0359em;",[872,1250,1253],{"className":1251},[1252],"vlist-s","​",[872,1255,1257],{"className":1256},[977],[872,1258,1261],{"className":1259,"style":1260},[981],"height:0.143em;",[872,1262],{},"，scaled dot-product attention 是",[872,1265,1268],{"className":1266},[1267],"katex-display",[872,1269,1271,1351],{"className":1270},[875],[872,1272,1274],{"className":1273},[879],[881,1275,1277],{"xmlns":883,"display":1276},"block",[885,1278,1279,1348],{},[888,1280,1281,1285,1288,1292,1294,1297,1299,1301,1303,1306,1309,1312,1314,1318,1343,1345],{},[891,1282,1284],{"mathvariant":1283},"normal","Attn",[895,1286,1287],{},"⁡",[895,1289,1291],{"stretchy":1290},"false","(",[891,1293,893],{},[895,1295,1296],{"separator":925},",",[891,1298,1025],{},[895,1300,1296],{"separator":925},[891,1302,1126],{},[895,1304,1305],{"stretchy":1290},")",[895,1307,1308],{},"=",[891,1310,1311],{"mathvariant":1283},"softmax",[895,1313,1287],{},[1315,1316,1317],"mtext",{}," ⁣",[888,1319,1320,1322,1341],{},[895,1321,1291],{"fence":925},[1323,1324,1325,1336],"mfrac",{},[888,1326,1327,1329],{},[891,1328,893],{},[899,1330,1331,1333],{},[891,1332,1025],{},[891,1334,1335],{"mathvariant":1283},"⊤",[1337,1338,1339],"msqrt",{},[891,1340,915],{},[895,1342,1305],{"fence":925},[891,1344,1126],{},[891,1346,1347],{"mathvariant":1283},".",[917,1349,1350],{"encoding":919},"\\operatorname{Attn}(Q, K, V) = \\operatorname{softmax}\\!\\left(\\frac{QK^{\\top}}{\\sqrt{d}}\\right) V .",[872,1352,1354,1409],{"className":1353,"ariaHidden":925},[924],[872,1355,1357,1361,1369,1373,1376,1380,1384,1387,1390,1393,1396,1400,1403,1406],{"className":1356},[929],[872,1358],{"className":1359,"style":1360},[933],"height:1em;vertical-align:-0.25em;",[872,1362,1365],{"className":1363},[1364],"mop",[872,1366,1284],{"className":1367},[938,1368],"mathrm",[872,1370,1291],{"className":1371},[1372],"mopen",[872,1374,893],{"className":1375},[938,939],[872,1377,1296],{"className":1378},[1379],"mpunct",[872,1381],{"className":1382,"style":1383},[943],"margin-right:0.1667em;",[872,1385,1025],{"className":1386,"style":1057},[938,939],[872,1388,1296],{"className":1389},[1379],[872,1391],{"className":1392,"style":1383},[943],[872,1394,1126],{"className":1395,"style":1162},[938,939],[872,1397,1305],{"className":1398},[1399],"mclose",[872,1401],{"className":1402,"style":944},[943],[872,1404,1308],{"className":1405},[948],[872,1407],{"className":1408,"style":944},[943],[872,1410,1412,1416,1422,1426,1429,1612,1615,1618],{"className":1411},[929],[872,1413],{"className":1414,"style":1415},[933],"height:2.4761em;vertical-align:-0.95em;",[872,1417,1419],{"className":1418},[1364],[872,1420,1311],{"className":1421},[938,1368],[872,1423],{"className":1424,"style":1425},[943],"margin-right:-0.1667em;",[872,1427],{"className":1428,"style":1383},[943],[872,1430,1433,1442,1606],{"className":1431},[1432],"minner",[872,1434,1438],{"className":1435,"style":1437},[1372,1436],"delimcenter","top:0em;",[872,1439,1291],{"className":1440},[1441,995],"delimsizing",[872,1443,1445,1449,1603],{"className":1444},[938],[872,1446],{"className":1447},[1372,1448],"nulldelimiter",[872,1450,1452],{"className":1451},[1323],[872,1453,1455,1594],{"className":1454},[973,1225],[872,1456,1458,1591],{"className":1457},[977],[872,1459,1462,1536,1547],{"className":1460,"style":1461},[981],"height:1.5261em;",[872,1463,1465,1469],{"style":1464},"top:-2.1778em;",[872,1466],{"className":1467,"style":1468},[988],"height:3em;",[872,1470,1472],{"className":1471},[938],[872,1473,1476],{"className":1474},[938,1475],"sqrt",[872,1477,1479,1527],{"className":1478},[973,1225],[872,1480,1482,1524],{"className":1481},[977],[872,1483,1486,1501],{"className":1484,"style":1485},[981],"height:0.9322em;",[872,1487,1491,1494],{"className":1488,"style":1490},[1489],"svg-align","top:-3em;",[872,1492],{"className":1493,"style":1468},[988],[872,1495,1498],{"className":1496,"style":1497},[938],"padding-left:0.833em;",[872,1499,915],{"className":1500},[938,939],[872,1502,1504,1507],{"style":1503},"top:-2.8922em;",[872,1505],{"className":1506,"style":1468},[988],[872,1508,1512],{"className":1509,"style":1511},[1510],"hide-tail","min-width:0.853em;height:1.08em;",[1513,1514,1520],"svg",{"xmlns":1515,"width":1516,"height":1517,"viewBox":1518,"preserveAspectRatio":1519},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[1521,1522],"path",{"d":1523},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[872,1525,1253],{"className":1526},[1252],[872,1528,1530],{"className":1529},[977],[872,1531,1534],{"className":1532,"style":1533},[981],"height:0.1078em;",[872,1535],{},[872,1537,1539,1542],{"style":1538},"top:-3.23em;",[872,1540],{"className":1541,"style":1468},[988],[872,1543],{"className":1544,"style":1546},[1545],"frac-line","border-bottom-width:0.04em;",[872,1548,1550,1553],{"style":1549},"top:-3.677em;",[872,1551],{"className":1552,"style":1468},[988],[872,1554,1556,1559],{"className":1555},[938],[872,1557,893],{"className":1558},[938,939],[872,1560,1562,1565],{"className":1561},[938],[872,1563,1025],{"className":1564,"style":1057},[938,939],[872,1566,1568],{"className":1567},[969],[872,1569,1571],{"className":1570},[973],[872,1572,1574],{"className":1573},[977],[872,1575,1577],{"className":1576,"style":958},[981],[872,1578,1579,1582],{"style":984},[872,1580],{"className":1581,"style":989},[988],[872,1583,1585],{"className":1584},[993,994,995,996],[872,1586,1588],{"className":1587},[938,996],[872,1589,1335],{"className":1590},[938,996],[872,1592,1253],{"className":1593},[1252],[872,1595,1597],{"className":1596},[977],[872,1598,1601],{"className":1599,"style":1600},[981],"height:0.93em;",[872,1602],{},[872,1604],{"className":1605},[1399,1448],[872,1607,1609],{"className":1608,"style":1437},[1399,1436],[872,1610,1305],{"className":1611},[1441,995],[872,1613],{"className":1614,"style":1383},[943],[872,1616,1126],{"className":1617,"style":1162},[938,939],[872,1619,1347],{"className":1620},[938],[111,1622,1623,1624,1654],{},"看整體沒感覺，把它逐項展開。第 ",[872,1625,1627,1641],{"className":1626},[875],[872,1628,1630],{"className":1629},[879],[881,1631,1632],{"xmlns":883},[885,1633,1634,1639],{},[888,1635,1636],{},[891,1637,1638],{},"i",[917,1640,1638],{"encoding":919},[872,1642,1644],{"className":1643,"ariaHidden":925},[924],[872,1645,1647,1651],{"className":1646},[929],[872,1648],{"className":1649,"style":1650},[933],"height:0.6595em;",[872,1652,1638],{"className":1653},[938,939]," 個輸出向量是",[872,1656,1658],{"className":1657},[1267],[872,1659,1661,1801],{"className":1660},[875],[872,1662,1664],{"className":1663},[879],[881,1665,1666],{"xmlns":883,"display":1276},[885,1667,1668,1798],{},[888,1669,1670,1677,1679,1796],{},[1140,1671,1672,1675],{},[891,1673,1674],{},"o",[891,1676,1638],{},[895,1678,1308],{},[1323,1680,1681,1746],{},[888,1682,1683,1702,1705,1707,1709,1740],{},[1684,1685,1686,1689,1700],"munderover",{},[895,1687,1688],{},"∑",[888,1690,1691,1694,1696],{},[891,1692,1693],{},"j",[895,1695,1308],{},[1697,1698,1699],"mn",{},"1",[891,1701,1036],{},[891,1703,1704],{},"exp",[895,1706,1287],{},[1315,1708,1317],{},[888,1710,1711,1713,1738],{},[895,1712,1291],{"fence":925},[1323,1714,1715,1734],{},[888,1716,1717,1727],{},[1718,1719,1720,1723,1725],"msubsup",{},[891,1721,1722],{},"q",[891,1724,1638],{},[891,1726,1335],{"mathvariant":1283},[1140,1728,1729,1732],{},[891,1730,1731],{},"k",[891,1733,1693],{},[1337,1735,1736],{},[891,1737,915],{},[895,1739,1305],{"fence":925},[1140,1741,1742,1744],{},[891,1743,1146],{},[891,1745,1693],{},[888,1747,1748,1762,1764,1766,1768],{},[1684,1749,1750,1752,1760],{},[895,1751,1688],{},[888,1753,1754,1756,1758],{},[891,1755,1693],{},[895,1757,1308],{},[1697,1759,1699],{},[891,1761,1036],{},[891,1763,1704],{},[895,1765,1287],{},[1315,1767,1317],{},[888,1769,1770,1772,1794],{},[895,1771,1291],{"fence":925},[1323,1773,1774,1790],{},[888,1775,1776,1784],{},[1718,1777,1778,1780,1782],{},[891,1779,1722],{},[891,1781,1638],{},[891,1783,1335],{"mathvariant":1283},[1140,1785,1786,1788],{},[891,1787,1731],{},[891,1789,1693],{},[1337,1791,1792],{},[891,1793,915],{},[895,1795,1305],{"fence":925},[891,1797,1347],{"mathvariant":1283},[917,1799,1800],{"encoding":919},"o_i = \\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)} .",[872,1802,1804,1863],{"className":1803,"ariaHidden":925},[924],[872,1805,1807,1811,1854,1857,1860],{"className":1806},[929],[872,1808],{"className":1809,"style":1810},[933],"height:0.5806em;vertical-align:-0.15em;",[872,1812,1814,1817],{"className":1813},[938],[872,1815,1674],{"className":1816},[938,939],[872,1818,1820],{"className":1819},[969],[872,1821,1823,1845],{"className":1822},[973,1225],[872,1824,1826,1842],{"className":1825},[977],[872,1827,1830],{"className":1828,"style":1829},[981],"height:0.3117em;",[872,1831,1833,1836],{"style":1832},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[872,1834],{"className":1835,"style":989},[988],[872,1837,1839],{"className":1838},[993,994,995,996],[872,1840,1638],{"className":1841},[938,939,996],[872,1843,1253],{"className":1844},[1252],[872,1846,1848],{"className":1847},[977],[872,1849,1852],{"className":1850,"style":1851},[981],"height:0.15em;",[872,1853],{},[872,1855],{"className":1856,"style":944},[943],[872,1858,1308],{"className":1859},[948],[872,1861],{"className":1862,"style":944},[943],[872,1864,1866,1870,2593],{"className":1865},[929],[872,1867],{"className":1868,"style":1869},[933],"height:3.8892em;vertical-align:-1.69em;",[872,1871,1873,1876,2590],{"className":1872},[938],[872,1874],{"className":1875},[1372,1448],[872,1877,1879],{"className":1878},[1323],[872,1880,1882,2581],{"className":1881},[973,1225],[872,1883,1885,2578],{"className":1884},[977],[872,1886,1889,2216,2225],{"className":1887,"style":1888},[981],"height:2.1991em;",[872,1890,1892,1896],{"style":1891},"top:-2.1191em;",[872,1893],{"className":1894,"style":1895},[988],"height:3.1591em;",[872,1897,1899,1970,1973,1976,1979,1982],{"className":1898},[938],[872,1900,1902,1908],{"className":1901},[1364],[872,1903,1688],{"className":1904,"style":1907},[1364,1905,1906],"op-symbol","small-op","position:relative;top:0em;",[872,1909,1911],{"className":1910},[969],[872,1912,1914,1961],{"className":1913},[973,1225],[872,1915,1917,1958],{"className":1916},[977],[872,1918,1921,1943],{"className":1919,"style":1920},[981],"height:0.8043em;",[872,1922,1924,1927],{"style":1923},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[872,1925],{"className":1926,"style":989},[988],[872,1928,1930],{"className":1929},[993,994,995,996],[872,1931,1933,1937,1940],{"className":1932},[938,996],[872,1934,1693],{"className":1935,"style":1936},[938,939,996],"margin-right:0.0572em;",[872,1938,1308],{"className":1939},[948,996],[872,1941,1699],{"className":1942},[938,996],[872,1944,1946,1949],{"style":1945},"top:-3.2029em;margin-right:0.05em;",[872,1947],{"className":1948,"style":989},[988],[872,1950,1952],{"className":1951},[993,994,995,996],[872,1953,1955],{"className":1954},[938,996],[872,1956,1036],{"className":1957},[938,939,996],[872,1959,1253],{"className":1960},[1252],[872,1962,1964],{"className":1963},[977],[872,1965,1968],{"className":1966,"style":1967},[981],"height:0.4358em;",[872,1969],{},[872,1971],{"className":1972,"style":1383},[943],[872,1974,1704],{"className":1975},[1364],[872,1977],{"className":1978,"style":1425},[943],[872,1980],{"className":1981,"style":1383},[943],[872,1983,1985,1992,2210],{"className":1984},[1432],[872,1986,1988],{"className":1987,"style":1437},[1372,1436],[872,1989,1291],{"className":1990},[1441,1991],"size2",[872,1993,1995,1998,2207],{"className":1994},[938],[872,1996],{"className":1997},[1372,1448],[872,1999,2001],{"className":2000},[1323],[872,2002,2004,2198],{"className":2003},[973,1225],[872,2005,2007,2195],{"className":2006},[977],[872,2008,2011,2073,2081],{"className":2009,"style":2010},[981],"height:1.1242em;",[872,2012,2014,2017],{"style":2013},"top:-2.5335em;",[872,2015],{"className":2016,"style":1468},[988],[872,2018,2020],{"className":2019},[993,994,995,996],[872,2021,2023],{"className":2022},[938,996],[872,2024,2026],{"className":2025},[938,1475,996],[872,2027,2029,2064],{"className":2028},[973,1225],[872,2030,2032,2061],{"className":2031},[977],[872,2033,2036,2048],{"className":2034,"style":2035},[981],"height:0.9378em;",[872,2037,2039,2042],{"className":2038,"style":1490},[1489],[872,2040],{"className":2041,"style":1468},[988],[872,2043,2045],{"className":2044,"style":1497},[938,996],[872,2046,915],{"className":2047},[938,939,996],[872,2049,2051,2054],{"style":2050},"top:-2.8978em;",[872,2052],{"className":2053,"style":1468},[988],[872,2055,2057],{"className":2056,"style":1511},[1510,996],[1513,2058,2059],{"xmlns":1515,"width":1516,"height":1517,"viewBox":1518,"preserveAspectRatio":1519},[1521,2060],{"d":1523},[872,2062,1253],{"className":2063},[1252],[872,2065,2067],{"className":2066},[977],[872,2068,2071],{"className":2069,"style":2070},[981],"height:0.1022em;",[872,2072],{},[872,2074,2075,2078],{"style":1538},[872,2076],{"className":2077,"style":1468},[988],[872,2079],{"className":2080,"style":1546},[1545],[872,2082,2084,2087],{"style":2083},"top:-3.5356em;",[872,2085],{"className":2086,"style":1468},[988],[872,2088,2090],{"className":2089},[993,994,995,996],[872,2091,2093,2151],{"className":2092},[938,996],[872,2094,2096,2099],{"className":2095},[938,996],[872,2097,1722],{"className":2098,"style":1248},[938,939,996],[872,2100,2102],{"className":2101},[969],[872,2103,2105,2142],{"className":2104},[973,1225],[872,2106,2108,2139],{"className":2107},[977],[872,2109,2112,2124],{"className":2110,"style":2111},[981],"height:0.8408em;",[872,2113,2115,2118],{"style":2114},"top:-2.1777em;margin-left:-0.0359em;margin-right:0.0714em;",[872,2116],{"className":2117,"style":1239},[988],[872,2119,2121],{"className":2120},[993,1243,1244,996],[872,2122,1638],{"className":2123},[938,939,996],[872,2125,2127,2130],{"style":2126},"top:-2.8448em;margin-right:0.0714em;",[872,2128],{"className":2129,"style":1239},[988],[872,2131,2133],{"className":2132},[993,1243,1244,996],[872,2134,2136],{"className":2135},[938,996],[872,2137,1335],{"className":2138},[938,996],[872,2140,1253],{"className":2141},[1252],[872,2143,2145],{"className":2144},[977],[872,2146,2149],{"className":2147,"style":2148},[981],"height:0.3223em;",[872,2150],{},[872,2152,2154,2158],{"className":2153},[938,996],[872,2155,1731],{"className":2156,"style":2157},[938,939,996],"margin-right:0.0315em;",[872,2159,2161],{"className":2160},[969],[872,2162,2164,2186],{"className":2163},[973,1225],[872,2165,2167,2183],{"className":2166},[977],[872,2168,2171],{"className":2169,"style":2170},[981],"height:0.3281em;",[872,2172,2174,2177],{"style":2173},"top:-2.357em;margin-left:-0.0315em;margin-right:0.0714em;",[872,2175],{"className":2176,"style":1239},[988],[872,2178,2180],{"className":2179},[993,1243,1244,996],[872,2181,1693],{"className":2182,"style":1936},[938,939,996],[872,2184,1253],{"className":2185},[1252],[872,2187,2189],{"className":2188},[977],[872,2190,2193],{"className":2191,"style":2192},[981],"height:0.2819em;",[872,2194],{},[872,2196,1253],{"className":2197},[1252],[872,2199,2201],{"className":2200},[977],[872,2202,2205],{"className":2203,"style":2204},[981],"height:0.538em;",[872,2206],{},[872,2208],{"className":2209},[1399,1448],[872,2211,2213],{"className":2212,"style":1437},[1399,1436],[872,2214,1305],{"className":2215},[1441,1991],[872,2217,2219,2222],{"style":2218},"top:-3.3891em;",[872,2220],{"className":2221,"style":1895},[988],[872,2223],{"className":2224,"style":1546},[1545],[872,2226,2228,2231],{"style":2227},"top:-4.1991em;",[872,2229],{"className":2230,"style":1895},[988],[872,2232,2234,2297,2300,2303,2306,2309,2533,2536],{"className":2233},[938],[872,2235,2237,2240],{"className":2236},[1364],[872,2238,1688],{"className":2239,"style":1907},[1364,1905,1906],[872,2241,2243],{"className":2242},[969],[872,2244,2246,2289],{"className":2245},[973,1225],[872,2247,2249,2286],{"className":2248},[977],[872,2250,2252,2272],{"className":2251,"style":1920},[981],[872,2253,2254,2257],{"style":1923},[872,2255],{"className":2256,"style":989},[988],[872,2258,2260],{"className":2259},[993,994,995,996],[872,2261,2263,2266,2269],{"className":2262},[938,996],[872,2264,1693],{"className":2265,"style":1936},[938,939,996],[872,2267,1308],{"className":2268},[948,996],[872,2270,1699],{"className":2271},[938,996],[872,2273,2274,2277],{"style":1945},[872,2275],{"className":2276,"style":989},[988],[872,2278,2280],{"className":2279},[993,994,995,996],[872,2281,2283],{"className":2282},[938,996],[872,2284,1036],{"className":2285},[938,939,996],[872,2287,1253],{"className":2288},[1252],[872,2290,2292],{"className":2291},[977],[872,2293,2295],{"className":2294,"style":1967},[981],[872,2296],{},[872,2298],{"className":2299,"style":1383},[943],[872,2301,1704],{"className":2302},[1364],[872,2304],{"className":2305,"style":1425},[943],[872,2307],{"className":2308,"style":1383},[943],[872,2310,2312,2318,2527],{"className":2311},[1432],[872,2313,2315],{"className":2314,"style":1437},[1372,1436],[872,2316,1291],{"className":2317},[1441,1991],[872,2319,2321,2324,2524],{"className":2320},[938],[872,2322],{"className":2323},[1372,1448],[872,2325,2327],{"className":2326},[1323],[872,2328,2330,2516],{"className":2329},[973,1225],[872,2331,2333,2513],{"className":2332},[977],[872,2334,2337,2395,2403],{"className":2335,"style":2336},[981],"height:1.1591em;",[872,2338,2339,2342],{"style":2013},[872,2340],{"className":2341,"style":1468},[988],[872,2343,2345],{"className":2344},[993,994,995,996],[872,2346,2348],{"className":2347},[938,996],[872,2349,2351],{"className":2350},[938,1475,996],[872,2352,2354,2387],{"className":2353},[973,1225],[872,2355,2357,2384],{"className":2356},[977],[872,2358,2360,2372],{"className":2359,"style":2035},[981],[872,2361,2363,2366],{"className":2362,"style":1490},[1489],[872,2364],{"className":2365,"style":1468},[988],[872,2367,2369],{"className":2368,"style":1497},[938,996],[872,2370,915],{"className":2371},[938,939,996],[872,2373,2374,2377],{"style":2050},[872,2375],{"className":2376,"style":1468},[988],[872,2378,2380],{"className":2379,"style":1511},[1510,996],[1513,2381,2382],{"xmlns":1515,"width":1516,"height":1517,"viewBox":1518,"preserveAspectRatio":1519},[1521,2383],{"d":1523},[872,2385,1253],{"className":2386},[1252],[872,2388,2390],{"className":2389},[977],[872,2391,2393],{"className":2392,"style":2070},[981],[872,2394],{},[872,2396,2397,2400],{"style":1538},[872,2398],{"className":2399,"style":1468},[988],[872,2401],{"className":2402,"style":1546},[1545],[872,2404,2406,2409],{"style":2405},"top:-3.5102em;",[872,2407],{"className":2408,"style":1468},[988],[872,2410,2412],{"className":2411},[993,994,995,996],[872,2413,2415,2473],{"className":2414},[938,996],[872,2416,2418,2421],{"className":2417},[938,996],[872,2419,1722],{"className":2420,"style":1248},[938,939,996],[872,2422,2424],{"className":2423},[969],[872,2425,2427,2464],{"className":2426},[973,1225],[872,2428,2430,2461],{"className":2429},[977],[872,2431,2434,2446],{"className":2432,"style":2433},[981],"height:0.927em;",[872,2435,2437,2440],{"style":2436},"top:-2.214em;margin-left:-0.0359em;margin-right:0.0714em;",[872,2438],{"className":2439,"style":1239},[988],[872,2441,2443],{"className":2442},[993,1243,1244,996],[872,2444,1638],{"className":2445},[938,939,996],[872,2447,2449,2452],{"style":2448},"top:-2.931em;margin-right:0.0714em;",[872,2450],{"className":2451,"style":1239},[988],[872,2453,2455],{"className":2454},[993,1243,1244,996],[872,2456,2458],{"className":2457},[938,996],[872,2459,1335],{"className":2460},[938,996],[872,2462,1253],{"className":2463},[1252],[872,2465,2467],{"className":2466},[977],[872,2468,2471],{"className":2469,"style":2470},[981],"height:0.286em;",[872,2472],{},[872,2474,2476,2479],{"className":2475},[938,996],[872,2477,1731],{"className":2478,"style":2157},[938,939,996],[872,2480,2482],{"className":2481},[969],[872,2483,2485,2505],{"className":2484},[973,1225],[872,2486,2488,2502],{"className":2487},[977],[872,2489,2491],{"className":2490,"style":2170},[981],[872,2492,2493,2496],{"style":2173},[872,2494],{"className":2495,"style":1239},[988],[872,2497,2499],{"className":2498},[993,1243,1244,996],[872,2500,1693],{"className":2501,"style":1936},[938,939,996],[872,2503,1253],{"className":2504},[1252],[872,2506,2508],{"className":2507},[977],[872,2509,2511],{"className":2510,"style":2192},[981],[872,2512],{},[872,2514,1253],{"className":2515},[1252],[872,2517,2519],{"className":2518},[977],[872,2520,2522],{"className":2521,"style":2204},[981],[872,2523],{},[872,2525],{"className":2526},[1399,1448],[872,2528,2530],{"className":2529,"style":1437},[1399,1436],[872,2531,1305],{"className":2532},[1441,1991],[872,2534],{"className":2535,"style":1383},[943],[872,2537,2539,2542],{"className":2538},[938],[872,2540,1146],{"className":2541,"style":1248},[938,939],[872,2543,2545],{"className":2544},[969],[872,2546,2548,2569],{"className":2547},[973,1225],[872,2549,2551,2566],{"className":2550},[977],[872,2552,2554],{"className":2553,"style":1829},[981],[872,2555,2557,2560],{"style":2556},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[872,2558],{"className":2559,"style":989},[988],[872,2561,2563],{"className":2562},[993,994,995,996],[872,2564,1693],{"className":2565,"style":1936},[938,939,996],[872,2567,1253],{"className":2568},[1252],[872,2570,2572],{"className":2571},[977],[872,2573,2576],{"className":2574,"style":2575},[981],"height:0.2861em;",[872,2577],{},[872,2579,1253],{"className":2580},[1252],[872,2582,2584],{"className":2583},[977],[872,2585,2588],{"className":2586,"style":2587},[981],"height:1.69em;",[872,2589],{},[872,2591],{"className":2592},[1399,1448],[872,2594,1347],{"className":2595},[938],[111,2597,2598,2599,2670,2671,704,2742,2814],{},"左邊的分子是一堆 ",[872,2600,2602,2620],{"className":2601},[875],[872,2603,2605],{"className":2604},[879],[881,2606,2607],{"xmlns":883},[885,2608,2609,2617],{},[888,2610,2611],{},[1140,2612,2613,2615],{},[891,2614,1146],{},[891,2616,1693],{},[917,2618,2619],{"encoding":919},"v_j",[872,2621,2623],{"className":2622,"ariaHidden":925},[924],[872,2624,2626,2630],{"className":2625},[929],[872,2627],{"className":2628,"style":2629},[933],"height:0.7167em;vertical-align:-0.2861em;",[872,2631,2633,2636],{"className":2632},[938],[872,2634,1146],{"className":2635,"style":1248},[938,939],[872,2637,2639],{"className":2638},[969],[872,2640,2642,2662],{"className":2641},[973,1225],[872,2643,2645,2659],{"className":2644},[977],[872,2646,2648],{"className":2647,"style":1829},[981],[872,2649,2650,2653],{"style":2556},[872,2651],{"className":2652,"style":989},[988],[872,2654,2656],{"className":2655},[993,994,995,996],[872,2657,1693],{"className":2658,"style":1936},[938,939,996],[872,2660,1253],{"className":2661},[1252],[872,2663,2665],{"className":2664},[977],[872,2666,2668],{"className":2667,"style":2575},[981],[872,2669],{}," 的加權和，權重是 ",[872,2672,2674,2692],{"className":2673},[875],[872,2675,2677],{"className":2676},[879],[881,2678,2679],{"xmlns":883},[885,2680,2681,2689],{},[888,2682,2683],{},[1140,2684,2685,2687],{},[891,2686,1722],{},[891,2688,1638],{},[917,2690,2691],{"encoding":919},"q_i",[872,2693,2695],{"className":2694,"ariaHidden":925},[924],[872,2696,2698,2702],{"className":2697},[929],[872,2699],{"className":2700,"style":2701},[933],"height:0.625em;vertical-align:-0.1944em;",[872,2703,2705,2708],{"className":2704},[938],[872,2706,1722],{"className":2707,"style":1248},[938,939],[872,2709,2711],{"className":2710},[969],[872,2712,2714,2734],{"className":2713},[973,1225],[872,2715,2717,2731],{"className":2716},[977],[872,2718,2720],{"className":2719,"style":1829},[981],[872,2721,2722,2725],{"style":2556},[872,2723],{"className":2724,"style":989},[988],[872,2726,2728],{"className":2727},[993,994,995,996],[872,2729,1638],{"className":2730},[938,939,996],[872,2732,1253],{"className":2733},[1252],[872,2735,2737],{"className":2736},[977],[872,2738,2740],{"className":2739,"style":1851},[981],[872,2741],{},[872,2743,2745,2763],{"className":2744},[875],[872,2746,2748],{"className":2747},[879],[881,2749,2750],{"xmlns":883},[885,2751,2752,2760],{},[888,2753,2754],{},[1140,2755,2756,2758],{},[891,2757,1731],{},[891,2759,1693],{},[917,2761,2762],{"encoding":919},"k_j",[872,2764,2766],{"className":2765,"ariaHidden":925},[924],[872,2767,2769,2773],{"className":2768},[929],[872,2770],{"className":2771,"style":2772},[933],"height:0.9805em;vertical-align:-0.2861em;",[872,2774,2776,2779],{"className":2775},[938],[872,2777,1731],{"className":2778,"style":2157},[938,939],[872,2780,2782],{"className":2781},[969],[872,2783,2785,2806],{"className":2784},[973,1225],[872,2786,2788,2803],{"className":2787},[977],[872,2789,2791],{"className":2790,"style":1829},[981],[872,2792,2794,2797],{"style":2793},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[872,2795],{"className":2796,"style":989},[988],[872,2798,2800],{"className":2799},[993,994,995,996],[872,2801,1693],{"className":2802,"style":1936},[938,939,996],[872,2804,1253],{"className":2805},[1252],[872,2807,2809],{"className":2808},[977],[872,2810,2812],{"className":2811,"style":2575},[981],[872,2813],{}," 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。",[111,2816,2817,2818,3030],{},"如果把 ",[872,2819,2821,2875],{"className":2820},[875],[872,2822,2824],{"className":2823},[879],[881,2825,2826],{"xmlns":883},[885,2827,2828,2872],{},[888,2829,2830,2833,2835,2837,2839,2841,2843,2845,2847,2849,2851],{},[891,2831,2832],{},"κ",[895,2834,1291],{"stretchy":1290},[891,2836,1722],{},[895,2838,1296],{"separator":925},[891,2840,1731],{},[895,2842,1305],{"stretchy":1290},[895,2844,1308],{},[891,2846,1704],{},[895,2848,1287],{},[1315,2850,1317],{},[888,2852,2853,2855,2861,2863,2866,2870],{},[895,2854,1291],{"fence":925},[899,2856,2857,2859],{},[891,2858,1722],{},[891,2860,1335],{"mathvariant":1283},[891,2862,1731],{},[891,2864,2865],{"mathvariant":1283},"\u002F",[1337,2867,2868],{},[891,2869,915],{},[895,2871,1305],{"fence":925},[917,2873,2874],{"encoding":919},"\\kappa(q, k) = \\exp\\!\\left(q^{\\top}k \u002F \\sqrt{d}\\right)",[872,2876,2878,2914],{"className":2877,"ariaHidden":925},[924],[872,2879,2881,2884,2887,2890,2893,2896,2899,2902,2905,2908,2911],{"className":2880},[929],[872,2882],{"className":2883,"style":1360},[933],[872,2885,2832],{"className":2886},[938,939],[872,2888,1291],{"className":2889},[1372],[872,2891,1722],{"className":2892,"style":1248},[938,939],[872,2894,1296],{"className":2895},[1379],[872,2897],{"className":2898,"style":1383},[943],[872,2900,1731],{"className":2901,"style":2157},[938,939],[872,2903,1305],{"className":2904},[1399],[872,2906],{"className":2907,"style":944},[943],[872,2909,1308],{"className":2910},[948],[872,2912],{"className":2913,"style":944},[943],[872,2915,2917,2921,2924,2927,2930],{"className":2916},[929],[872,2918],{"className":2919,"style":2920},[933],"height:1.8em;vertical-align:-0.65em;",[872,2922,1704],{"className":2923},[1364],[872,2925],{"className":2926,"style":1425},[943],[872,2928],{"className":2929,"style":1383},[943],[872,2931,2933,2939,2971,2974,2977,3024],{"className":2932},[1432],[872,2934,2936],{"className":2935,"style":1437},[1372,1436],[872,2937,1291],{"className":2938},[1441,1991],[872,2940,2942,2945],{"className":2941},[938],[872,2943,1722],{"className":2944,"style":1248},[938,939],[872,2946,2948],{"className":2947},[969],[872,2949,2951],{"className":2950},[973],[872,2952,2954],{"className":2953},[977],[872,2955,2957],{"className":2956,"style":958},[981],[872,2958,2959,2962],{"style":984},[872,2960],{"className":2961,"style":989},[988],[872,2963,2965],{"className":2964},[993,994,995,996],[872,2966,2968],{"className":2967},[938,996],[872,2969,1335],{"className":2970},[938,996],[872,2972,1731],{"className":2973,"style":2157},[938,939],[872,2975,2865],{"className":2976},[938],[872,2978,2980],{"className":2979},[938,1475],[872,2981,2983,3016],{"className":2982},[973,1225],[872,2984,2986,3013],{"className":2985},[977],[872,2987,2989,3001],{"className":2988,"style":1485},[981],[872,2990,2992,2995],{"className":2991,"style":1490},[1489],[872,2993],{"className":2994,"style":1468},[988],[872,2996,2998],{"className":2997,"style":1497},[938],[872,2999,915],{"className":3000},[938,939],[872,3002,3003,3006],{"style":1503},[872,3004],{"className":3005,"style":1468},[988],[872,3007,3009],{"className":3008,"style":1511},[1510],[1513,3010,3011],{"xmlns":1515,"width":1516,"height":1517,"viewBox":1518,"preserveAspectRatio":1519},[1521,3012],{"d":1523},[872,3014,1253],{"className":3015},[1252],[872,3017,3019],{"className":3018},[977],[872,3020,3022],{"className":3021,"style":1533},[981],[872,3023],{},[872,3025,3027],{"className":3026,"style":1437},[1399,1436],[872,3028,1305],{"className":3029},[1441,1991]," 當成一個核函數，上面這一坨就是",[872,3032,3034],{"className":3033},[1267],[872,3035,3037,3127],{"className":3036},[875],[872,3038,3040],{"className":3039},[879],[881,3041,3042],{"xmlns":883,"display":1276},[885,3043,3044,3124],{},[888,3045,3046,3052,3054,3122],{},[1140,3047,3048,3050],{},[891,3049,1674],{},[891,3051,1638],{},[895,3053,1308],{},[1323,3055,3056,3094],{},[888,3057,3058,3065,3067,3069,3075,3077,3083,3085,3088],{},[3059,3060,3061,3063],"munder",{},[895,3062,1688],{},[891,3064,1693],{},[891,3066,2832],{},[895,3068,1291],{"stretchy":1290},[1140,3070,3071,3073],{},[891,3072,1722],{},[891,3074,1638],{},[895,3076,1296],{"separator":925},[1140,3078,3079,3081],{},[891,3080,1731],{},[891,3082,1693],{},[895,3084,1305],{"stretchy":1290},[1315,3086,3087],{}," ",[1140,3089,3090,3092],{},[891,3091,1146],{},[891,3093,1693],{},[888,3095,3096,3102,3104,3106,3112,3114,3120],{},[3059,3097,3098,3100],{},[895,3099,1688],{},[891,3101,1693],{},[891,3103,2832],{},[895,3105,1291],{"stretchy":1290},[1140,3107,3108,3110],{},[891,3109,1722],{},[891,3111,1638],{},[895,3113,1296],{"separator":925},[1140,3115,3116,3118],{},[891,3117,1731],{},[891,3119,1693],{},[895,3121,1305],{"stretchy":1290},[895,3123,1296],{"separator":925},[917,3125,3126],{"encoding":919},"o_i = \\frac{\\sum_j \\kappa(q_i, k_j)\\, v_j}{\\sum_j \\kappa(q_i, k_j)} ,",[872,3128,3130,3185],{"className":3129,"ariaHidden":925},[924],[872,3131,3133,3136,3176,3179,3182],{"className":3132},[929],[872,3134],{"className":3135,"style":1810},[933],[872,3137,3139,3142],{"className":3138},[938],[872,3140,1674],{"className":3141},[938,939],[872,3143,3145],{"className":3144},[969],[872,3146,3148,3168],{"className":3147},[973,1225],[872,3149,3151,3165],{"className":3150},[977],[872,3152,3154],{"className":3153,"style":1829},[981],[872,3155,3156,3159],{"style":1832},[872,3157],{"className":3158,"style":989},[988],[872,3160,3162],{"className":3161},[993,994,995,996],[872,3163,1638],{"className":3164},[938,939,996],[872,3166,1253],{"className":3167},[1252],[872,3169,3171],{"className":3170},[977],[872,3172,3174],{"className":3173,"style":1851},[981],[872,3175],{},[872,3177],{"className":3178,"style":944},[943],[872,3180,1308],{"className":3181},[948],[872,3183],{"className":3184,"style":944},[943],[872,3186,3188,3192,3572],{"className":3187},[929],[872,3189],{"className":3190,"style":3191},[933],"height:2.6976em;vertical-align:-1.1218em;",[872,3193,3195,3198,3569],{"className":3194},[938],[872,3196],{"className":3197},[1372,1448],[872,3199,3201],{"className":3200},[1323],[872,3202,3204,3560],{"className":3203},[973,1225],[872,3205,3207,3557],{"className":3206},[977],[872,3208,3211,3359,3367],{"className":3209,"style":3210},[981],"height:1.5758em;",[872,3212,3214,3217],{"style":3213},"top:-2.314em;",[872,3215],{"className":3216,"style":1468},[988],[872,3218,3220,3261,3264,3267,3270,3310,3313,3316,3356],{"className":3219},[938],[872,3221,3223,3226],{"className":3222},[1364],[872,3224,1688],{"className":3225,"style":1907},[1364,1905,1906],[872,3227,3229],{"className":3228},[969],[872,3230,3232,3253],{"className":3231},[973,1225],[872,3233,3235,3250],{"className":3234},[977],[872,3236,3239],{"className":3237,"style":3238},[981],"height:0.162em;",[872,3240,3241,3244],{"style":1923},[872,3242],{"className":3243,"style":989},[988],[872,3245,3247],{"className":3246},[993,994,995,996],[872,3248,1693],{"className":3249,"style":1936},[938,939,996],[872,3251,1253],{"className":3252},[1252],[872,3254,3256],{"className":3255},[977],[872,3257,3259],{"className":3258,"style":1967},[981],[872,3260],{},[872,3262],{"className":3263,"style":1383},[943],[872,3265,2832],{"className":3266},[938,939],[872,3268,1291],{"className":3269},[1372],[872,3271,3273,3276],{"className":3272},[938],[872,3274,1722],{"className":3275,"style":1248},[938,939],[872,3277,3279],{"className":3278},[969],[872,3280,3282,3302],{"className":3281},[973,1225],[872,3283,3285,3299],{"className":3284},[977],[872,3286,3288],{"className":3287,"style":1829},[981],[872,3289,3290,3293],{"style":2556},[872,3291],{"className":3292,"style":989},[988],[872,3294,3296],{"className":3295},[993,994,995,996],[872,3297,1638],{"className":3298},[938,939,996],[872,3300,1253],{"className":3301},[1252],[872,3303,3305],{"className":3304},[977],[872,3306,3308],{"className":3307,"style":1851},[981],[872,3309],{},[872,3311,1296],{"className":3312},[1379],[872,3314],{"className":3315,"style":1383},[943],[872,3317,3319,3322],{"className":3318},[938],[872,3320,1731],{"className":3321,"style":2157},[938,939],[872,3323,3325],{"className":3324},[969],[872,3326,3328,3348],{"className":3327},[973,1225],[872,3329,3331,3345],{"className":3330},[977],[872,3332,3334],{"className":3333,"style":1829},[981],[872,3335,3336,3339],{"style":2793},[872,3337],{"className":3338,"style":989},[988],[872,3340,3342],{"className":3341},[993,994,995,996],[872,3343,1693],{"className":3344,"style":1936},[938,939,996],[872,3346,1253],{"className":3347},[1252],[872,3349,3351],{"className":3350},[977],[872,3352,3354],{"className":3353,"style":2575},[981],[872,3355],{},[872,3357,1305],{"className":3358},[1399],[872,3360,3361,3364],{"style":1538},[872,3362],{"className":3363,"style":1468},[988],[872,3365],{"className":3366,"style":1546},[1545],[872,3368,3370,3373],{"style":3369},"top:-3.8258em;",[872,3371],{"className":3372,"style":1468},[988],[872,3374,3376,3416,3419,3422,3425,3465,3468,3471,3511,3514,3517],{"className":3375},[938],[872,3377,3379,3382],{"className":3378},[1364],[872,3380,1688],{"className":3381,"style":1907},[1364,1905,1906],[872,3383,3385],{"className":3384},[969],[872,3386,3388,3408],{"className":3387},[973,1225],[872,3389,3391,3405],{"className":3390},[977],[872,3392,3394],{"className":3393,"style":3238},[981],[872,3395,3396,3399],{"style":1923},[872,3397],{"className":3398,"style":989},[988],[872,3400,3402],{"className":3401},[993,994,995,996],[872,3403,1693],{"className":3404,"style":1936},[938,939,996],[872,3406,1253],{"className":3407},[1252],[872,3409,3411],{"className":3410},[977],[872,3412,3414],{"className":3413,"style":1967},[981],[872,3415],{},[872,3417],{"className":3418,"style":1383},[943],[872,3420,2832],{"className":3421},[938,939],[872,3423,1291],{"className":3424},[1372],[872,3426,3428,3431],{"className":3427},[938],[872,3429,1722],{"className":3430,"style":1248},[938,939],[872,3432,3434],{"className":3433},[969],[872,3435,3437,3457],{"className":3436},[973,1225],[872,3438,3440,3454],{"className":3439},[977],[872,3441,3443],{"className":3442,"style":1829},[981],[872,3444,3445,3448],{"style":2556},[872,3446],{"className":3447,"style":989},[988],[872,3449,3451],{"className":3450},[993,994,995,996],[872,3452,1638],{"className":3453},[938,939,996],[872,3455,1253],{"className":3456},[1252],[872,3458,3460],{"className":3459},[977],[872,3461,3463],{"className":3462,"style":1851},[981],[872,3464],{},[872,3466,1296],{"className":3467},[1379],[872,3469],{"className":3470,"style":1383},[943],[872,3472,3474,3477],{"className":3473},[938],[872,3475,1731],{"className":3476,"style":2157},[938,939],[872,3478,3480],{"className":3479},[969],[872,3481,3483,3503],{"className":3482},[973,1225],[872,3484,3486,3500],{"className":3485},[977],[872,3487,3489],{"className":3488,"style":1829},[981],[872,3490,3491,3494],{"style":2793},[872,3492],{"className":3493,"style":989},[988],[872,3495,3497],{"className":3496},[993,994,995,996],[872,3498,1693],{"className":3499,"style":1936},[938,939,996],[872,3501,1253],{"className":3502},[1252],[872,3504,3506],{"className":3505},[977],[872,3507,3509],{"className":3508,"style":2575},[981],[872,3510],{},[872,3512,1305],{"className":3513},[1399],[872,3515],{"className":3516,"style":1383},[943],[872,3518,3520,3523],{"className":3519},[938],[872,3521,1146],{"className":3522,"style":1248},[938,939],[872,3524,3526],{"className":3525},[969],[872,3527,3529,3549],{"className":3528},[973,1225],[872,3530,3532,3546],{"className":3531},[977],[872,3533,3535],{"className":3534,"style":1829},[981],[872,3536,3537,3540],{"style":2556},[872,3538],{"className":3539,"style":989},[988],[872,3541,3543],{"className":3542},[993,994,995,996],[872,3544,1693],{"className":3545,"style":1936},[938,939,996],[872,3547,1253],{"className":3548},[1252],[872,3550,3552],{"className":3551},[977],[872,3553,3555],{"className":3554,"style":2575},[981],[872,3556],{},[872,3558,1253],{"className":3559},[1252],[872,3561,3563],{"className":3562},[977],[872,3564,3567],{"className":3565,"style":3566},[981],"height:1.1218em;",[872,3568],{},[872,3570],{"className":3571},[1399,1448],[872,3573,1296],{"className":3574},[1379],[111,3576,3577],{},"——非參數回歸裡的 Nadaraya–Watson 估計。一模一樣。",[111,3579,3580],{},"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。",[129,3582,3583],{"id":3583},"知道這件事有什麼用",[111,3585,3586,3587,148],{},"不是為了數學上的優雅。是因為它會告訴你",[145,3588,3589],{},"哪裡可以砍",[111,3591,3592,3593,3623,3624,3774],{},"如果 ",[872,3594,3596,3610],{"className":3595},[875],[872,3597,3599],{"className":3598},[879],[881,3600,3601],{"xmlns":883},[885,3602,3603,3607],{},[888,3604,3605],{},[891,3606,2832],{},[917,3608,3609],{"encoding":919},"\\kappa",[872,3611,3613],{"className":3612,"ariaHidden":925},[924],[872,3614,3616,3620],{"className":3615},[929],[872,3617],{"className":3618,"style":3619},[933],"height:0.4306em;",[872,3621,2832],{"className":3622},[938,939]," 可以寫成有限維特徵映射的內積，也就是 ",[872,3625,3627,3675],{"className":3626},[875],[872,3628,3630],{"className":3629},[879],[881,3631,3632],{"xmlns":883},[885,3633,3634,3672],{},[888,3635,3636,3638,3640,3642,3644,3646,3648,3651,3654,3656,3658,3664,3666,3668,3670],{},[891,3637,2832],{},[895,3639,1291],{"stretchy":1290},[891,3641,1722],{},[895,3643,1296],{"separator":925},[891,3645,1731],{},[895,3647,1305],{"stretchy":1290},[895,3649,3650],{},"≈",[891,3652,3653],{},"ϕ",[895,3655,1291],{"stretchy":1290},[891,3657,1722],{},[899,3659,3660,3662],{},[895,3661,1305],{"stretchy":1290},[891,3663,1335],{"mathvariant":1283},[891,3665,3653],{},[895,3667,1291],{"stretchy":1290},[891,3669,1731],{},[895,3671,1305],{"stretchy":1290},[917,3673,3674],{"encoding":919},"\\kappa(q,k) \\approx \\phi(q)^{\\top}\\phi(k)",[872,3676,3678,3714],{"className":3677,"ariaHidden":925},[924],[872,3679,3681,3684,3687,3690,3693,3696,3699,3702,3705,3708,3711],{"className":3680},[929],[872,3682],{"className":3683,"style":1360},[933],[872,3685,2832],{"className":3686},[938,939],[872,3688,1291],{"className":3689},[1372],[872,3691,1722],{"className":3692,"style":1248},[938,939],[872,3694,1296],{"className":3695},[1379],[872,3697],{"className":3698,"style":1383},[943],[872,3700,1731],{"className":3701,"style":2157},[938,939],[872,3703,1305],{"className":3704},[1399],[872,3706],{"className":3707,"style":944},[943],[872,3709,3650],{"className":3710},[948],[872,3712],{"className":3713,"style":944},[943],[872,3715,3717,3721,3724,3727,3730,3762,3765,3768,3771],{"className":3716},[929],[872,3718],{"className":3719,"style":3720},[933],"height:1.0991em;vertical-align:-0.25em;",[872,3722,3653],{"className":3723},[938,939],[872,3725,1291],{"className":3726},[1372],[872,3728,1722],{"className":3729,"style":1248},[938,939],[872,3731,3733,3736],{"className":3732},[1399],[872,3734,1305],{"className":3735},[1399],[872,3737,3739],{"className":3738},[969],[872,3740,3742],{"className":3741},[973],[872,3743,3745],{"className":3744},[977],[872,3746,3748],{"className":3747,"style":958},[981],[872,3749,3750,3753],{"style":984},[872,3751],{"className":3752,"style":989},[988],[872,3754,3756],{"className":3755},[993,994,995,996],[872,3757,3759],{"className":3758},[938,996],[872,3760,1335],{"className":3761},[938,996],[872,3763,3653],{"className":3764},[938,939],[872,3766,1291],{"className":3767},[1372],[872,3769,1731],{"className":3770,"style":2157},[938,939],[872,3772,1305],{"className":3773},[1399],"，那麼",[872,3776,3778],{"className":3777},[1267],[872,3779,3781,3885],{"className":3780},[875],[872,3782,3784],{"className":3783},[879],[881,3785,3786],{"xmlns":883,"display":1276},[885,3787,3788,3882],{},[888,3789,3790,3796,3798,3880],{},[1140,3791,3792,3794],{},[891,3793,1674],{},[891,3795,1638],{},[895,3797,3650],{},[1323,3799,3800,3844],{},[888,3801,3802,3804,3806,3812,3818,3824,3826,3828,3834,3836],{},[891,3803,3653],{},[895,3805,1291],{"stretchy":1290},[1140,3807,3808,3810],{},[891,3809,1722],{},[891,3811,1638],{},[899,3813,3814,3816],{},[895,3815,1305],{"stretchy":1290},[891,3817,1335],{"mathvariant":1283},[3059,3819,3820,3822],{},[895,3821,1688],{},[891,3823,1693],{},[891,3825,3653],{},[895,3827,1291],{"stretchy":1290},[1140,3829,3830,3832],{},[891,3831,1731],{},[891,3833,1693],{},[895,3835,1305],{"stretchy":1290},[1718,3837,3838,3840,3842],{},[891,3839,1146],{},[891,3841,1693],{},[891,3843,1335],{"mathvariant":1283},[888,3845,3846,3848,3850,3856,3862,3868,3870,3872,3878],{},[891,3847,3653],{},[895,3849,1291],{"stretchy":1290},[1140,3851,3852,3854],{},[891,3853,1722],{},[891,3855,1638],{},[899,3857,3858,3860],{},[895,3859,1305],{"stretchy":1290},[891,3861,1335],{"mathvariant":1283},[3059,3863,3864,3866],{},[895,3865,1688],{},[891,3867,1693],{},[891,3869,3653],{},[895,3871,1291],{"stretchy":1290},[1140,3873,3874,3876],{},[891,3875,1731],{},[891,3877,1693],{},[895,3879,1305],{"stretchy":1290},[891,3881,1347],{"mathvariant":1283},[917,3883,3884],{"encoding":919},"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)} .",[872,3886,3888,3944],{"className":3887,"ariaHidden":925},[924],[872,3889,3891,3895,3935,3938,3941],{"className":3890},[929],[872,3892],{"className":3893,"style":3894},[933],"height:0.6331em;vertical-align:-0.15em;",[872,3896,3898,3901],{"className":3897},[938],[872,3899,1674],{"className":3900},[938,939],[872,3902,3904],{"className":3903},[969],[872,3905,3907,3927],{"className":3906},[973,1225],[872,3908,3910,3924],{"className":3909},[977],[872,3911,3913],{"className":3912,"style":1829},[981],[872,3914,3915,3918],{"style":1832},[872,3916],{"className":3917,"style":989},[988],[872,3919,3921],{"className":3920},[993,994,995,996],[872,3922,1638],{"className":3923},[938,939,996],[872,3925,1253],{"className":3926},[1252],[872,3928,3930],{"className":3929},[977],[872,3931,3933],{"className":3932,"style":1851},[981],[872,3934],{},[872,3936],{"className":3937,"style":944},[943],[872,3939,3650],{"className":3940},[948],[872,3942],{"className":3943,"style":944},[943],[872,3945,3947,3951,4412],{"className":3946},[929],[872,3948],{"className":3949,"style":3950},[933],"height:2.7967em;vertical-align:-1.1218em;",[872,3952,3954,3957,4409],{"className":3953},[938],[872,3955],{"className":3956},[1372,1448],[872,3958,3960],{"className":3959},[1323],[872,3961,3963,4401],{"className":3962},[973,1225],[872,3964,3966,4398],{"className":3965},[977],[872,3967,3970,4153,4161],{"className":3968,"style":3969},[981],"height:1.6749em;",[872,3971,3972,3975],{"style":3213},[872,3973],{"className":3974,"style":1468},[988],[872,3976,3978,3981,3984,4024,4058,4061,4101,4104,4107,4110,4150],{"className":3977},[938],[872,3979,3653],{"className":3980},[938,939],[872,3982,1291],{"className":3983},[1372],[872,3985,3987,3990],{"className":3986},[938],[872,3988,1722],{"className":3989,"style":1248},[938,939],[872,3991,3993],{"className":3992},[969],[872,3994,3996,4016],{"className":3995},[973,1225],[872,3997,3999,4013],{"className":3998},[977],[872,4000,4002],{"className":4001,"style":1829},[981],[872,4003,4004,4007],{"style":2556},[872,4005],{"className":4006,"style":989},[988],[872,4008,4010],{"className":4009},[993,994,995,996],[872,4011,1638],{"className":4012},[938,939,996],[872,4014,1253],{"className":4015},[1252],[872,4017,4019],{"className":4018},[977],[872,4020,4022],{"className":4021,"style":1851},[981],[872,4023],{},[872,4025,4027,4030],{"className":4026},[1399],[872,4028,1305],{"className":4029},[1399],[872,4031,4033],{"className":4032},[969],[872,4034,4036],{"className":4035},[973],[872,4037,4039],{"className":4038},[977],[872,4040,4043],{"className":4041,"style":4042},[981],"height:0.7751em;",[872,4044,4046,4049],{"style":4045},"top:-2.989em;margin-right:0.05em;",[872,4047],{"className":4048,"style":989},[988],[872,4050,4052],{"className":4051},[993,994,995,996],[872,4053,4055],{"className":4054},[938,996],[872,4056,1335],{"className":4057},[938,996],[872,4059],{"className":4060,"style":1383},[943],[872,4062,4064,4067],{"className":4063},[1364],[872,4065,1688],{"className":4066,"style":1907},[1364,1905,1906],[872,4068,4070],{"className":4069},[969],[872,4071,4073,4093],{"className":4072},[973,1225],[872,4074,4076,4090],{"className":4075},[977],[872,4077,4079],{"className":4078,"style":3238},[981],[872,4080,4081,4084],{"style":1923},[872,4082],{"className":4083,"style":989},[988],[872,4085,4087],{"className":4086},[993,994,995,996],[872,4088,1693],{"className":4089,"style":1936},[938,939,996],[872,4091,1253],{"className":4092},[1252],[872,4094,4096],{"className":4095},[977],[872,4097,4099],{"className":4098,"style":1967},[981],[872,4100],{},[872,4102],{"className":4103,"style":1383},[943],[872,4105,3653],{"className":4106},[938,939],[872,4108,1291],{"className":4109},[1372],[872,4111,4113,4116],{"className":4112},[938],[872,4114,1731],{"className":4115,"style":2157},[938,939],[872,4117,4119],{"className":4118},[969],[872,4120,4122,4142],{"className":4121},[973,1225],[872,4123,4125,4139],{"className":4124},[977],[872,4126,4128],{"className":4127,"style":1829},[981],[872,4129,4130,4133],{"style":2793},[872,4131],{"className":4132,"style":989},[988],[872,4134,4136],{"className":4135},[993,994,995,996],[872,4137,1693],{"className":4138,"style":1936},[938,939,996],[872,4140,1253],{"className":4141},[1252],[872,4143,4145],{"className":4144},[977],[872,4146,4148],{"className":4147,"style":2575},[981],[872,4149],{},[872,4151,1305],{"className":4152},[1399],[872,4154,4155,4158],{"style":1538},[872,4156],{"className":4157,"style":1468},[988],[872,4159],{"className":4160,"style":1546},[1545],[872,4162,4163,4166],{"style":3369},[872,4164],{"className":4165,"style":1468},[988],[872,4167,4169,4172,4175,4215,4247,4250,4290,4293,4296,4299,4339,4342],{"className":4168},[938],[872,4170,3653],{"className":4171},[938,939],[872,4173,1291],{"className":4174},[1372],[872,4176,4178,4181],{"className":4177},[938],[872,4179,1722],{"className":4180,"style":1248},[938,939],[872,4182,4184],{"className":4183},[969],[872,4185,4187,4207],{"className":4186},[973,1225],[872,4188,4190,4204],{"className":4189},[977],[872,4191,4193],{"className":4192,"style":1829},[981],[872,4194,4195,4198],{"style":2556},[872,4196],{"className":4197,"style":989},[988],[872,4199,4201],{"className":4200},[993,994,995,996],[872,4202,1638],{"className":4203},[938,939,996],[872,4205,1253],{"className":4206},[1252],[872,4208,4210],{"className":4209},[977],[872,4211,4213],{"className":4212,"style":1851},[981],[872,4214],{},[872,4216,4218,4221],{"className":4217},[1399],[872,4219,1305],{"className":4220},[1399],[872,4222,4224],{"className":4223},[969],[872,4225,4227],{"className":4226},[973],[872,4228,4230],{"className":4229},[977],[872,4231,4233],{"className":4232,"style":958},[981],[872,4234,4235,4238],{"style":984},[872,4236],{"className":4237,"style":989},[988],[872,4239,4241],{"className":4240},[993,994,995,996],[872,4242,4244],{"className":4243},[938,996],[872,4245,1335],{"className":4246},[938,996],[872,4248],{"className":4249,"style":1383},[943],[872,4251,4253,4256],{"className":4252},[1364],[872,4254,1688],{"className":4255,"style":1907},[1364,1905,1906],[872,4257,4259],{"className":4258},[969],[872,4260,4262,4282],{"className":4261},[973,1225],[872,4263,4265,4279],{"className":4264},[977],[872,4266,4268],{"className":4267,"style":3238},[981],[872,4269,4270,4273],{"style":1923},[872,4271],{"className":4272,"style":989},[988],[872,4274,4276],{"className":4275},[993,994,995,996],[872,4277,1693],{"className":4278,"style":1936},[938,939,996],[872,4280,1253],{"className":4281},[1252],[872,4283,4285],{"className":4284},[977],[872,4286,4288],{"className":4287,"style":1967},[981],[872,4289],{},[872,4291],{"className":4292,"style":1383},[943],[872,4294,3653],{"className":4295},[938,939],[872,4297,1291],{"className":4298},[1372],[872,4300,4302,4305],{"className":4301},[938],[872,4303,1731],{"className":4304,"style":2157},[938,939],[872,4306,4308],{"className":4307},[969],[872,4309,4311,4331],{"className":4310},[973,1225],[872,4312,4314,4328],{"className":4313},[977],[872,4315,4317],{"className":4316,"style":1829},[981],[872,4318,4319,4322],{"style":2793},[872,4320],{"className":4321,"style":989},[988],[872,4323,4325],{"className":4324},[993,994,995,996],[872,4326,1693],{"className":4327,"style":1936},[938,939,996],[872,4329,1253],{"className":4330},[1252],[872,4332,4334],{"className":4333},[977],[872,4335,4337],{"className":4336,"style":2575},[981],[872,4338],{},[872,4340,1305],{"className":4341},[1399],[872,4343,4345,4348],{"className":4344},[938],[872,4346,1146],{"className":4347,"style":1248},[938,939],[872,4349,4351],{"className":4350},[969],[872,4352,4354,4389],{"className":4353},[973,1225],[872,4355,4357,4386],{"className":4356},[977],[872,4358,4360,4372],{"className":4359,"style":958},[981],[872,4361,4363,4366],{"style":4362},"top:-2.4413em;margin-left:-0.0359em;margin-right:0.05em;",[872,4364],{"className":4365,"style":989},[988],[872,4367,4369],{"className":4368},[993,994,995,996],[872,4370,1693],{"className":4371,"style":1936},[938,939,996],[872,4373,4374,4377],{"style":984},[872,4375],{"className":4376,"style":989},[988],[872,4378,4380],{"className":4379},[993,994,995,996],[872,4381,4383],{"className":4382},[938,996],[872,4384,1335],{"className":4385},[938,996],[872,4387,1253],{"className":4388},[1252],[872,4390,4392],{"className":4391},[977],[872,4393,4396],{"className":4394,"style":4395},[981],"height:0.3948em;",[872,4397],{},[872,4399,1253],{"className":4400},[1252],[872,4402,4404],{"className":4403},[977],[872,4405,4407],{"className":4406,"style":3566},[981],[872,4408],{},[872,4410],{"className":4411},[1399,1448],[872,4413,1347],{"className":4414},[938],[111,4416,4417,4418,4467,4468,4535],{},"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 ",[872,4419,4421,4444],{"className":4420},[875],[872,4422,4424],{"className":4423},[879],[881,4425,4426],{"xmlns":883},[885,4427,4428,4441],{},[888,4429,4430,4433,4435,4437,4439],{},[891,4431,4432],{},"O",[895,4434,1291],{"stretchy":1290},[891,4436,909],{},[891,4438,1036],{},[895,4440,1305],{"stretchy":1290},[917,4442,4443],{"encoding":919},"O(nm)",[872,4445,4447],{"className":4446,"ariaHidden":925},[924],[872,4448,4450,4453,4457,4460,4464],{"className":4449},[929],[872,4451],{"className":4452,"style":1360},[933],[872,4454,4432],{"className":4455,"style":4456},[938,939],"margin-right:0.0278em;",[872,4458,1291],{"className":4459},[1372],[872,4461,4463],{"className":4462},[938,939],"nm",[872,4465,1305],{"className":4466},[1399]," 掉到 ",[872,4469,4471,4496],{"className":4470},[875],[872,4472,4474],{"className":4473},[879],[881,4475,4476],{"xmlns":883},[885,4477,4478,4493],{},[888,4479,4480,4482,4484,4486,4489,4491],{},[891,4481,4432],{},[895,4483,1291],{"stretchy":1290},[891,4485,909],{},[895,4487,4488],{},"+",[891,4490,1036],{},[895,4492,1305],{"stretchy":1290},[917,4494,4495],{"encoding":919},"O(n + m)",[872,4497,4499,4523],{"className":4498,"ariaHidden":925},[924],[872,4500,4502,4505,4508,4511,4514,4517,4520],{"className":4501},[929],[872,4503],{"className":4504,"style":1360},[933],[872,4506,4432],{"className":4507,"style":4456},[938,939],[872,4509,1291],{"className":4510},[1372],[872,4512,909],{"className":4513},[938,939],[872,4515],{"className":4516,"style":1162},[943],[872,4518,4488],{"className":4519},[1006],[872,4521],{"className":4522,"style":1162},[943],[872,4524,4526,4529,4532],{"className":4525},[929],[872,4527],{"className":4528,"style":1360},[933],[872,4530,1036],{"className":4531},[938,939],[872,4533,1305],{"className":4534},[1399],"：",[872,4537,4539],{"className":4538},[1267],[872,4540,4542,4647],{"className":4541},[875],[872,4543,4545],{"className":4544},[879],[881,4546,4547],{"xmlns":883,"display":1276},[885,4548,4549,4644],{},[4550,4551,4555,4594],"mtable",{"rowspacing":4552,"columnalign":4553,"columnspacing":4554},"0.25em","right left","0em",[4556,4557,4558,4568],"mtr",{},[4559,4560,4561],"mtd",{},[4562,4563,4565],"mstyle",{"scriptlevel":4564,"displaystyle":925},"0",[1315,4566,4567],{},"標準 attention",[4559,4569,4570],{},[4562,4571,4572],{"scriptlevel":4564,"displaystyle":925},[888,4573,4574,4576,4579,4582,4584,4586,4588,4590,4592],{},[888,4575],{},[895,4577,4578],{},":",[943,4580],{"width":4581},"1em",[891,4583,4432],{},[895,4585,1291],{"stretchy":1290},[891,4587,909],{},[891,4589,1036],{},[891,4591,915],{},[895,4593,1305],{"stretchy":1290},[4556,4595,4596,4603],{},[4559,4597,4598],{},[4562,4599,4600],{"scriptlevel":4564,"displaystyle":925},[1315,4601,4602],{},"線性 attention",[4559,4604,4605],{},[4562,4606,4607],{"scriptlevel":4564,"displaystyle":925},[888,4608,4609,4611,4613,4615,4617,4619],{},[888,4610],{},[895,4612,4578],{},[943,4614],{"width":4581},[891,4616,4432],{},[1315,4618,1317],{},[888,4620,4621,4623,4625,4632,4634,4636,4642],{},[895,4622,1291],{"fence":925},[891,4624,909],{},[899,4626,4627,4629],{},[891,4628,915],{},[1697,4630,4631],{},"2",[895,4633,4488],{},[891,4635,1036],{},[899,4637,4638,4640],{},[891,4639,915],{},[1697,4641,4631],{},[895,4643,1305],{"fence":925},[917,4645,4646],{"encoding":919},"\\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}",[872,4648,4650],{"className":4649,"ariaHidden":925},[924],[872,4651,4653,4657],{"className":4652},[929],[872,4654],{"className":4655,"style":4656},[933],"height:2.7241em;vertical-align:-1.1121em;",[872,4658,4660],{"className":4659},[938],[872,4661,4663,4730],{"className":4662},[4550],[872,4664,4667],{"className":4665},[4666],"col-align-r",[872,4668,4670,4721],{"className":4669},[973,1225],[872,4671,4673,4718],{"className":4672},[977],[872,4674,4677,4699],{"className":4675,"style":4676},[981],"height:1.6121em;",[872,4678,4680,4683],{"style":4679},"top:-3.7721em;",[872,4681],{"className":4682,"style":1468},[988],[872,4684,4686],{"className":4685},[938],[872,4687,4690,4695],{"className":4688},[938,4689],"text",[872,4691,4694],{"className":4692},[938,4693],"cjk_fallback","標準",[872,4696,4698],{"className":4697},[938]," attention",[872,4700,4702,4705],{"style":4701},"top:-2.2479em;",[872,4703],{"className":4704,"style":1468},[988],[872,4706,4708],{"className":4707},[938],[872,4709,4711,4715],{"className":4710},[938,4689],[872,4712,4714],{"className":4713},[938,4693],"線性",[872,4716,4698],{"className":4717},[938],[872,4719,1253],{"className":4720},[1252],[872,4722,4724],{"className":4723},[977],[872,4725,4728],{"className":4726,"style":4727},[981],"height:1.1121em;",[872,4729],{},[872,4731,4734],{"className":4732},[4733],"col-align-l",[872,4735,4737,4907],{"className":4736},[973,1225],[872,4738,4740,4904],{"className":4739},[977],[872,4741,4743,4782],{"className":4742,"style":4676},[981],[872,4744,4745,4748],{"style":4679},[872,4746],{"className":4747,"style":1468},[988],[872,4749,4751,4754,4757,4760,4764,4767,4770,4773,4776,4779],{"className":4750},[938],[872,4752],{"className":4753},[938],[872,4755],{"className":4756,"style":944},[943],[872,4758,4578],{"className":4759},[948],[872,4761],{"className":4762,"style":4763},[943],"margin-right:1em;",[872,4765],{"className":4766,"style":944},[943],[872,4768,4432],{"className":4769,"style":4456},[938,939],[872,4771,1291],{"className":4772},[1372],[872,4774,4463],{"className":4775},[938,939],[872,4777,915],{"className":4778},[938,939],[872,4780,1305],{"className":4781},[1399],[872,4783,4784,4787],{"style":4701},[872,4785],{"className":4786,"style":1468},[988],[872,4788,4790,4793,4796,4799,4802,4805,4808,4811,4814],{"className":4789},[938],[872,4791],{"className":4792},[938],[872,4794],{"className":4795,"style":944},[943],[872,4797,4578],{"className":4798},[948],[872,4800],{"className":4801,"style":4763},[943],[872,4803],{"className":4804,"style":944},[943],[872,4806,4432],{"className":4807,"style":4456},[938,939],[872,4809],{"className":4810,"style":1425},[943],[872,4812],{"className":4813,"style":1383},[943],[872,4815,4817,4823,4826,4857,4860,4863,4866,4869,4898],{"className":4816},[1432],[872,4818,4820],{"className":4819,"style":1437},[1372,1436],[872,4821,1291],{"className":4822},[1441,1244],[872,4824,909],{"className":4825},[938,939],[872,4827,4829,4832],{"className":4828},[938],[872,4830,915],{"className":4831},[938,939],[872,4833,4835],{"className":4834},[969],[872,4836,4838],{"className":4837},[973],[872,4839,4841],{"className":4840},[977],[872,4842,4845],{"className":4843,"style":4844},[981],"height:0.8641em;",[872,4846,4848,4851],{"style":4847},"top:-3.113em;margin-right:0.05em;",[872,4849],{"className":4850,"style":989},[988],[872,4852,4854],{"className":4853},[993,994,995,996],[872,4855,4631],{"className":4856},[938,996],[872,4858],{"className":4859,"style":1162},[943],[872,4861,4488],{"className":4862},[1006],[872,4864],{"className":4865,"style":1162},[943],[872,4867,1036],{"className":4868},[938,939],[872,4870,4872,4875],{"className":4871},[938],[872,4873,915],{"className":4874},[938,939],[872,4876,4878],{"className":4877},[969],[872,4879,4881],{"className":4880},[973],[872,4882,4884],{"className":4883},[977],[872,4885,4887],{"className":4886,"style":4844},[981],[872,4888,4889,4892],{"style":4847},[872,4890],{"className":4891,"style":989},[988],[872,4893,4895],{"className":4894},[993,994,995,996],[872,4896,4631],{"className":4897},[938,996],[872,4899,4901],{"className":4900,"style":1437},[1399,1436],[872,4902,1305],{"className":4903},[1441,1244],[872,4905,1253],{"className":4906},[1252],[872,4908,4910],{"className":4909},[977],[872,4911,4913],{"className":4912,"style":4727},[981],[872,4914],{},[111,4916,4917],{},"這就是 linear attention 那一整家族在做的事。",[129,4919,4920],{"id":4920},"代價在分母",[111,4922,4923,4924],{},"這裡有個很容易被忽略的地方：",[145,4925,4926],{},"誤差最大的地方不是分子，是分母。",[111,4928,4929,4930,5162,5163,5352],{},"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，",[872,4931,4933,4979],{"className":4932},[875],[872,4934,4936],{"className":4935},[879],[881,4937,4938],{"xmlns":883},[885,4939,4940,4976],{},[888,4941,4942,4944,4946,4952,4958,4964,4966,4968,4974],{},[891,4943,3653],{},[895,4945,1291],{"stretchy":1290},[1140,4947,4948,4950],{},[891,4949,1722],{},[891,4951,1638],{},[899,4953,4954,4956],{},[895,4955,1305],{"stretchy":1290},[891,4957,1335],{"mathvariant":1283},[1140,4959,4960,4962],{},[895,4961,1688],{},[891,4963,1693],{},[891,4965,3653],{},[895,4967,1291],{"stretchy":1290},[1140,4969,4970,4972],{},[891,4971,1731],{},[891,4973,1693],{},[895,4975,1305],{"stretchy":1290},[917,4977,4978],{"encoding":919},"\\phi(q_i)^{\\top}\\sum_j \\phi(k_j)",[872,4980,4982],{"className":4981,"ariaHidden":925},[924],[872,4983,4985,4989,4992,4995,5035,5067,5070,5110,5113,5116,5119,5159],{"className":4984},[929],[872,4986],{"className":4987,"style":4988},[933],"height:1.2849em;vertical-align:-0.4358em;",[872,4990,3653],{"className":4991},[938,939],[872,4993,1291],{"className":4994},[1372],[872,4996,4998,5001],{"className":4997},[938],[872,4999,1722],{"className":5000,"style":1248},[938,939],[872,5002,5004],{"className":5003},[969],[872,5005,5007,5027],{"className":5006},[973,1225],[872,5008,5010,5024],{"className":5009},[977],[872,5011,5013],{"className":5012,"style":1829},[981],[872,5014,5015,5018],{"style":2556},[872,5016],{"className":5017,"style":989},[988],[872,5019,5021],{"className":5020},[993,994,995,996],[872,5022,1638],{"className":5023},[938,939,996],[872,5025,1253],{"className":5026},[1252],[872,5028,5030],{"className":5029},[977],[872,5031,5033],{"className":5032,"style":1851},[981],[872,5034],{},[872,5036,5038,5041],{"className":5037},[1399],[872,5039,1305],{"className":5040},[1399],[872,5042,5044],{"className":5043},[969],[872,5045,5047],{"className":5046},[973],[872,5048,5050],{"className":5049},[977],[872,5051,5053],{"className":5052,"style":958},[981],[872,5054,5055,5058],{"style":984},[872,5056],{"className":5057,"style":989},[988],[872,5059,5061],{"className":5060},[993,994,995,996],[872,5062,5064],{"className":5063},[938,996],[872,5065,1335],{"className":5066},[938,996],[872,5068],{"className":5069,"style":1383},[943],[872,5071,5073,5076],{"className":5072},[1364],[872,5074,1688],{"className":5075,"style":1907},[1364,1905,1906],[872,5077,5079],{"className":5078},[969],[872,5080,5082,5102],{"className":5081},[973,1225],[872,5083,5085,5099],{"className":5084},[977],[872,5086,5088],{"className":5087,"style":3238},[981],[872,5089,5090,5093],{"style":1923},[872,5091],{"className":5092,"style":989},[988],[872,5094,5096],{"className":5095},[993,994,995,996],[872,5097,1693],{"className":5098,"style":1936},[938,939,996],[872,5100,1253],{"className":5101},[1252],[872,5103,5105],{"className":5104},[977],[872,5106,5108],{"className":5107,"style":1967},[981],[872,5109],{},[872,5111],{"className":5112,"style":1383},[943],[872,5114,3653],{"className":5115},[938,939],[872,5117,1291],{"className":5118},[1372],[872,5120,5122,5125],{"className":5121},[938],[872,5123,1731],{"className":5124,"style":2157},[938,939],[872,5126,5128],{"className":5127},[969],[872,5129,5131,5151],{"className":5130},[973,1225],[872,5132,5134,5148],{"className":5133},[977],[872,5135,5137],{"className":5136,"style":1829},[981],[872,5138,5139,5142],{"style":2793},[872,5140],{"className":5141,"style":989},[988],[872,5143,5145],{"className":5144},[993,994,995,996],[872,5146,1693],{"className":5147,"style":1936},[938,939,996],[872,5149,1253],{"className":5150},[1252],[872,5152,5154],{"className":5153},[977],[872,5155,5157],{"className":5156,"style":2575},[981],[872,5158],{},[872,5160,1305],{"className":5161},[1399]," 和真正的 ",[872,5164,5166,5204],{"className":5165},[875],[872,5167,5169],{"className":5168},[879],[881,5170,5171],{"xmlns":883},[885,5172,5173,5201],{},[888,5174,5175,5181,5183,5185,5191,5193,5199],{},[1140,5176,5177,5179],{},[895,5178,1688],{},[891,5180,1693],{},[891,5182,2832],{},[895,5184,1291],{"stretchy":1290},[1140,5186,5187,5189],{},[891,5188,1722],{},[891,5190,1638],{},[895,5192,1296],{"separator":925},[1140,5194,5195,5197],{},[891,5196,1731],{},[891,5198,1693],{},[895,5200,1305],{"stretchy":1290},[917,5202,5203],{"encoding":919},"\\sum_j \\kappa(q_i,k_j)",[872,5205,5207],{"className":5206,"ariaHidden":925},[924],[872,5208,5210,5214,5254,5257,5260,5263,5303,5306,5309,5349],{"className":5209},[929],[872,5211],{"className":5212,"style":5213},[933],"height:1.1858em;vertical-align:-0.4358em;",[872,5215,5217,5220],{"className":5216},[1364],[872,5218,1688],{"className":5219,"style":1907},[1364,1905,1906],[872,5221,5223],{"className":5222},[969],[872,5224,5226,5246],{"className":5225},[973,1225],[872,5227,5229,5243],{"className":5228},[977],[872,5230,5232],{"className":5231,"style":3238},[981],[872,5233,5234,5237],{"style":1923},[872,5235],{"className":5236,"style":989},[988],[872,5238,5240],{"className":5239},[993,994,995,996],[872,5241,1693],{"className":5242,"style":1936},[938,939,996],[872,5244,1253],{"className":5245},[1252],[872,5247,5249],{"className":5248},[977],[872,5250,5252],{"className":5251,"style":1967},[981],[872,5253],{},[872,5255],{"className":5256,"style":1383},[943],[872,5258,2832],{"className":5259},[938,939],[872,5261,1291],{"className":5262},[1372],[872,5264,5266,5269],{"className":5265},[938],[872,5267,1722],{"className":5268,"style":1248},[938,939],[872,5270,5272],{"className":5271},[969],[872,5273,5275,5295],{"className":5274},[973,1225],[872,5276,5278,5292],{"className":5277},[977],[872,5279,5281],{"className":5280,"style":1829},[981],[872,5282,5283,5286],{"style":2556},[872,5284],{"className":5285,"style":989},[988],[872,5287,5289],{"className":5288},[993,994,995,996],[872,5290,1638],{"className":5291},[938,939,996],[872,5293,1253],{"className":5294},[1252],[872,5296,5298],{"className":5297},[977],[872,5299,5301],{"className":5300,"style":1851},[981],[872,5302],{},[872,5304,1296],{"className":5305},[1379],[872,5307],{"className":5308,"style":1383},[943],[872,5310,5312,5315],{"className":5311},[938],[872,5313,1731],{"className":5314,"style":2157},[938,939],[872,5316,5318],{"className":5317},[969],[872,5319,5321,5341],{"className":5320},[973,1225],[872,5322,5324,5338],{"className":5323},[977],[872,5325,5327],{"className":5326,"style":1829},[981],[872,5328,5329,5332],{"style":2793},[872,5330],{"className":5331,"style":989},[988],[872,5333,5335],{"className":5334},[993,994,995,996],[872,5336,1693],{"className":5337,"style":1936},[938,939,996],[872,5339,1253],{"className":5340},[1252],[872,5342,5344],{"className":5343},[977],[872,5345,5347],{"className":5346,"style":2575},[981],[872,5348],{},[872,5350,1305],{"className":5351},[1399]," 差得越多，輸出就越不可信。",[111,5354,5355],{},"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。",[129,5357,5358],{"id":5358},"實務上的三個檢查點",[213,5360,5361,5371],{},[216,5362,5363],{},[219,5364,5365,5368],{},[222,5366,5367],{},"檢查",[222,5369,5370],{},"為什麼",[232,5372,5373,5381,5460],{},[219,5374,5375,5378],{},[237,5376,5377],{},"分母的相對誤差",[237,5379,5380],{},"比分子更致命，因為它會整體縮放輸出",[219,5382,5383,5457],{},[237,5384,5385,5386],{},"特徵維度 ",[872,5387,5389,5407],{"className":5388},[875],[872,5390,5392],{"className":5391},[879],[881,5393,5394],{"xmlns":883},[885,5395,5396,5404],{},[888,5397,5398],{},[1140,5399,5400,5402],{},[891,5401,915],{},[891,5403,3653],{},[917,5405,5406],{"encoding":919},"d_\\phi",[872,5408,5410],{"className":5409,"ariaHidden":925},[924],[872,5411,5413,5416],{"className":5412},[929],[872,5414],{"className":5415,"style":2772},[933],[872,5417,5419,5422],{"className":5418},[938],[872,5420,915],{"className":5421},[938,939],[872,5423,5425],{"className":5424},[969],[872,5426,5428,5449],{"className":5427},[973,1225],[872,5429,5431,5446],{"className":5430},[977],[872,5432,5435],{"className":5433,"style":5434},[981],"height:0.3361em;",[872,5436,5437,5440],{"style":1832},[872,5438],{"className":5439,"style":989},[988],[872,5441,5443],{"className":5442},[993,994,995,996],[872,5444,3653],{"className":5445},[938,939,996],[872,5447,1253],{"className":5448},[1252],[872,5450,5452],{"className":5451},[977],[872,5453,5455],{"className":5454,"style":2575},[981],[872,5456],{},[237,5458,5459],{},"太少則近似粗糙，太多則省不下計算",[219,5461,5462,5465],{},[237,5463,5464],{},"序列長度分佈",[237,5466,5467],{},"短序列幾乎沒有收益，長序列才值得換",[5469,5470,5474],"pre",{"className":5471,"code":5472,"language":5473,"meta":593,"style":593},"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",[509,5475,5476,5483,5488,5494],{"__ignoreMap":593},[872,5477,5480],{"class":5478,"line":5479},"line",1,[872,5481,5482],{},"# 概念示意：先把 K、V 的累加量算好\n",[872,5484,5485],{"class":5478,"line":594},[872,5486,5487],{},"kv = einsum(\"m d, m e -> d e\", phi(K), V)   # O(m d^2)\n",[872,5489,5491],{"class":5478,"line":5490},3,[872,5492,5493],{},"z  = phi(K).sum(dim=0)                       # O(m d)\n",[872,5495,5497],{"class":5478,"line":5496},4,[872,5498,5499],{},"out = einsum(\"n d, d e -> n e\", phi(Q), kv) \u002F einsum(\"n d, d -> n\", phi(Q), z)\n",[129,5501,5502],{"id":5502},"給下一個要調這個模型的人",[111,5504,5505],{},"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。",[111,5507,5508,5509,5512],{},"因為換成 linear attention 不是「更快的 attention」，而是",[145,5510,5511],{},"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。",[111,5514,5515],{},"知道自己在付什麼，比知道自己在用什麼重要。",[5517,5518,5519],"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":593,"searchDepth":594,"depth":594,"links":5521},[5522,5523,5524,5525,5526,5527],{"id":858,"depth":594,"text":858},{"id":867,"depth":594,"text":867},{"id":3583,"depth":594,"text":3583},{"id":4920,"depth":594,"text":4920},{"id":5358,"depth":594,"text":5358},{"id":5502,"depth":594,"text":5502},"\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。",{},"\u002Fimages\u002Fblog\u002Fattention-kernel.jpg","\u002Fblog\u002Fzh-tw\u002Fattention-as-kernel",{"title":853,"description":5531},"attention-as-kernel","blog\u002Fzh-TW\u002Fattention-as-kernel",[5539,5540,5541],"transformer","數學","推論","2026-10-22","4vG7yvfd7IC1RrjTl2r8g8HG6o7V05WdgDW8PBTgxes",1792820245141]