{"id":6012,"date":"2025-06-26T14:13:13","date_gmt":"2025-06-26T06:13:13","guid":{"rendered":"https:\/\/nullthought.net\/?p=6012"},"modified":"2025-06-26T14:13:15","modified_gmt":"2025-06-26T06:13:15","slug":"%e5%86%85%e7%a7%af%ef%bc%88inner-product%ef%bc%89%e5%ba%94%e7%94%a8%e7%9a%84%e4%b8%80%e4%b8%aa%e7%ae%80%e5%8d%95%e5%bd%a2%e8%b1%a1%e5%ae%9e%e4%be%8b%ef%bc%9a%e7%94%a8%e6%88%b7%e5%90%91%e9%87%8f","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=6012","title":{"rendered":"\u5185\u79ef\uff08Inner Product\uff09\u5e94\u7528\u7684\u4e00\u4e2a\u7b80\u5355\u5f62\u8c61\u5b9e\u4f8b\uff1a\u7528\u6237\u5411\u91cf\u4e0e\u7269\u54c1\u5411\u91cf\u7684\u5185\u79ef\u53cd\u6620\u201c\u559c\u597d\u7a0b\u5ea6\u201d"},"content":{"rendered":"\n<p>\u5185\u79ef\uff08Inner Product\uff09\u662f\u7ebf\u6027\u4ee3\u6570\u4e2d\u4e00\u4e2a\u57fa\u672c\u800c\u6838\u5fc3\u7684\u6982\u5ff5\uff0c\u662f\u8861\u91cf\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u76f8\u4f3c\u7a0b\u5ea6\u6216\u201c\u5bf9\u9f50\u7a0b\u5ea6\u201d\u7684\u6570\u5b66\u5de5\u5177\u3002\u5728\u4e0d\u540c\u7684\u8bed\u5883\u4e0b\uff0c\u5185\u79ef\u4e5f\u88ab\u79f0\u4e3a\u70b9\u79ef\uff08Dot Product\uff09\u3001\u6570\u91cf\u79ef\uff08Scalar Product\uff09\uff0c\u5728\u4fe1\u606f\u68c0\u7d22\u3001\u673a\u5668\u5b66\u4e60\u4e2d\u5e38\u7528\u6765\u8861\u91cf\u4e24\u4e2a\u5411\u91cf\u7684\u76f8\u4f3c\u6027\u3002\u4ee5\u4e0b\u662f\u5bf9\u5185\u79ef\u7684\u8be6\u7ec6\u89e3\u91ca\uff1a<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e00\u3001\u51e0\u4f55\u76f4\u89c2\u89e3\u91ca<\/h4>\n\n\n\n<p>\u8bbe\u6709\u4e24\u4e2a\u5411\u91cf\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>a=(a<sub>1,<\/sub>a<sub>2<\/sub>,\u2026,a<sub>n<\/sub>)<\/li>\n\n\n\n<li>b=(b<sub>1<\/sub>,b<sub>2<\/sub>,\u2026,b<sub>n<\/sub>)<\/li>\n<\/ul>\n\n\n\n<p>\u5b83\u4eec\u7684<strong>\u5185\u79ef\u5b9a\u4e49\u4e3a<\/strong>\uff1a <\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"206\" height=\"86\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/06\/image-2.png\" alt=\"\" class=\"wp-image-6013\"\/><\/figure>\n\n\n\n<p>\u8fd9\u662f\u6700\u5e38\u7528\u7684 <strong>\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4<\/strong> \u4e2d\u7684\u5185\u79ef\u5f62\u5f0f\uff0c\u4e5f\u79f0\u4e3a<strong>\u6807\u51c6\u5185\u79ef<\/strong>\u3002<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">\u51e0\u4f55\u610f\u4e49\uff1a<\/h6>\n\n\n\n<p>\u5185\u79ef\u53ef\u4ee5\u8868\u8fbe\u4e3a\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"292\" height=\"66\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/06\/image-3.png\" alt=\"\" class=\"wp-image-6014\"\/><\/figure>\n\n\n\n<p>\u5176\u4e2d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u2225a\u2225 \u548c \u2225b\u2225 \u662f\u5411\u91cf\u7684\u6b27\u51e0\u91cc\u5f97\u8303\u6570\uff08\u957f\u5ea6\uff09<\/li>\n\n\n\n<li>\u03b8 \u662f\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u5939\u89d2<\/li>\n<\/ul>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u5f53\u4e24\u4e2a\u5411\u91cf\u65b9\u5411\u5b8c\u5168\u4e00\u81f4\uff08\u540c\u5411\uff09\uff0c\u5939\u89d2\u4e3a 0\u00b0\uff0ccos\u03b8=1\uff0c\u5185\u79ef\u6700\u5927\uff1b<br>\u5f53\u5b83\u4eec\u6b63\u4ea4\uff08\u5782\u76f4\uff09\uff0ccos\u03b8=0\uff0c\u5185\u79ef\u4e3a0\uff1b<br>\u5f53\u5b83\u4eec\u65b9\u5411\u76f8\u53cd\uff08\u53cd\u5411\uff09\uff0ccos\u03b8=-1\uff0c\u5185\u79ef\u4e3a\u8d1f\u6570\u3002<\/p>\n<\/blockquote>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e8c\u3001\u5728\u673a\u5668\u5b66\u4e60\u4e0e\u4fe1\u606f\u68c0\u7d22\u4e2d\u7684\u5e94\u7528<\/h4>\n\n\n\n<p>\u5728\u4fe1\u606f\u68c0\u7d22\uff08IR\uff09\u548c\u63a8\u8350\u7cfb\u7edf\u4e2d\uff0c<strong>\u5185\u79ef\u5e38\u7528\u6765\u8861\u91cf\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u7684\u201c\u76f8\u5173\u6027\u201d\u6216\u201c\u76f8\u4f3c\u5ea6\u201d<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u6587\u672c\u68c0\u7d22<\/strong>\u4e2d\uff0cquery \u5411\u91cf\u548c document \u5411\u91cf\u7684\u5185\u79ef\u8d8a\u5927\uff0c\u8868\u793a\u5185\u5bb9\u8d8a\u76f8\u5173\uff1b<\/li>\n\n\n\n<li><strong>\u63a8\u8350\u7cfb\u7edf<\/strong>\u4e2d\uff0c\u7528\u6237\u5411\u91cf\u4e0e\u7269\u54c1\u5411\u91cf\u7684\u5185\u79ef\u8d8a\u5927\uff0c\u8868\u793a\u7528\u6237\u53ef\u80fd\u8d8a\u559c\u6b22\u8be5\u7269\u54c1\uff1b<\/li>\n\n\n\n<li><strong>\u795e\u7ecf\u7f51\u7edc<\/strong>\u4e2d\uff0cattention\u673a\u5236\uff08\u5982Transformer\uff09\u5c31\u5927\u91cf\u4f7f\u7528\u4e86\u5185\u79ef\uff08Q\u00b7K^T\uff09\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u7528 BERT \u6a21\u578b\u5c06\u53e5\u5b50\u7f16\u7801\u4e3a\u5411\u91cf\uff0c\u90a3\u4e48\u6bd4\u8f83\u4e24\u4e2a\u53e5\u5b50\u7684\u8bed\u4e49\u662f\u5426\u76f8\u8fd1\uff0c\u53ef\u4ee5\u901a\u8fc7\u5b83\u4eec\u5411\u91cf\u7684\u5185\u79ef\u6765\u5224\u65ad\u3002<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">\u4f18\u70b9\uff1a<\/h6>\n\n\n\n<p>\u8ba1\u7b97\u7b80\u5355\uff0c\u6613\u4e8e\u5728\u5927\u89c4\u6a21\u5411\u91cf\u5e93\u4e2d\u4f7f\u7528\uff1b<\/p>\n\n\n\n<p>\u4e0e\u5411\u91cf\u8303\u6570\u7ed3\u5408\u540e\uff0c\u53ef\u8f6c\u5316\u4e3a<strong>\u4f59\u5f26\u76f8\u4f3c\u5ea6\uff08cosine similarity\uff09<\/strong>\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e09\u3001\u5177\u4f53\u6570\u503c\u4f8b\u5b50\uff1a\u8bf4\u660e<strong>\u7528\u6237\u5411\u91cf\u4e0e\u7269\u54c1\u5411\u91cf\u7684\u5185\u79ef\u5982\u4f55\u53cd\u6620\u201c\u559c\u597d\u7a0b\u5ea6\u201d<\/strong><\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">\ud83e\udde0 \u573a\u666f\u8bbe\u5b9a\uff1a\u63a8\u8350\u7cfb\u7edf\u4e2d\u7684\u5185\u79ef\u76f8\u4f3c\u5ea6<\/h5>\n\n\n\n<p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u975e\u5e38\u7b80\u5316\u7684\u63a8\u8350\u7cfb\u7edf\uff0c\u7528\u6237\u548c\u5546\u54c1\u90fd\u88ab\u8868\u793a\u4e3a\u957f\u5ea6\u4e3a 3 \u7684\u5411\u91cf\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7528\u6237\u7684\u504f\u597d\u5411\u91cf\uff08User Embedding\uff09\u8868\u793a\u4ed6\u5bf9\u5c5e\u6027\u3010\u751c\u5ea6\u3001\u8f9b\u8fa3\u3001\u70ed\u5ea6\u3011\u7684\u504f\u597d\u7a0b\u5ea6\uff1b<\/li>\n\n\n\n<li>\u5546\u54c1\u7684\u7279\u5f81\u5411\u91cf\uff08Item Embedding\uff09\u8868\u793a\u8be5\u5546\u54c1\u5728\u8fd9\u4e09\u79cd\u5c5e\u6027\u4e0a\u7684\u5b9e\u9645\u8868\u73b0\u3002<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">\ud83d\udc64 \u7528\u6237\u5411\u91cf<\/h5>\n\n\n\n<p>\u5047\u8bbe\u4e00\u4e2a\u7528\u6237\u504f\u597d\u5982\u4e0b\uff1au=[0.9, 0.1, 0.5]<\/p>\n\n\n\n<p>\u542b\u4e49\u89e3\u91ca\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u559c\u6b22\u751c\u7684\u7a0b\u5ea6\u662f <strong>0.9<\/strong><\/li>\n\n\n\n<li>\u559c\u6b22\u8fa3\u7684\u7a0b\u5ea6\u662f <strong>0.1<\/strong><\/li>\n\n\n\n<li>\u559c\u6b22\u70ed\u98df\u7684\u7a0b\u5ea6\u662f <strong>0.5<\/strong><\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">\ud83c\udf5c \u5546\u54c1A\u5411\u91cf\uff08\u8fa3\u6912\u70ed\u6c64\uff09\uff1a<\/h5>\n\n\n\n<p>i<sub>A<\/sub>=[0.2, 0.9, 0.8]<\/p>\n\n\n\n<p>\u542b\u4e49\u89e3\u91ca\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5fae\u751c<\/li>\n\n\n\n<li>\u975e\u5e38\u8fa3<\/li>\n\n\n\n<li>\u5f88\u70ed<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">\ud83c\udf70 \u5546\u54c1B\u5411\u91cf\uff08\u51b0\u9547\u751c\u70b9\uff09\uff1a<\/h5>\n\n\n\n<p>i<sub>B<\/sub>=[0.95, 0.05, 0.1]<\/p>\n\n\n\n<p>\u542b\u4e49\u89e3\u91ca\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5f88\u751c<\/li>\n\n\n\n<li>\u4e0d\u8fa3<\/li>\n\n\n\n<li>\u51b7\u7684<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">\ud83d\udd22 \u8ba1\u7b97\u7528\u6237\u4e0e\u6bcf\u4e2a\u5546\u54c1\u7684\u5185\u79ef<\/h5>\n\n\n\n<p>\u6211\u4eec\u7528\u5411\u91cf\u5185\u79ef\u6765\u8ba1\u7b97\u201c\u5339\u914d\u5ea6\u201d\uff1a<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">\u5bf9\u5546\u54c1A\uff1a<\/h6>\n\n\n\n<p>\u27e8u,i<sub>A<\/sub>\u27e9=(0.9\u22c50.2)+(0.1\u22c50.9)+(0.5\u22c50.8)=0.18+0.09+0.40=0.67<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">\u5bf9\u5546\u54c1B\uff1a<\/h6>\n\n\n\n<p>\u27e8u,i<sub>B<\/sub>\u27e9=(0.9\u22c50.95)+(0.1\u22c50.05)+(0.5\u22c50.1)=0.855+0.005+0.05=0.91<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u2705 \u7ed3\u679c\u5206\u6790<\/h5>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u5546\u54c1B\uff08\u51b0\u9547\u751c\u70b9\uff09\u5339\u914d\u5ea6 = 0.91<\/strong><\/li>\n\n\n\n<li><strong>\u5546\u54c1A\uff08\u8fa3\u6912\u70ed\u6c64\uff09\u5339\u914d\u5ea6 = 0.67<\/strong><\/li>\n<\/ul>\n\n\n\n<p>\u6240\u4ee5\uff0c\u7cfb\u7edf\u66f4\u503e\u5411\u4e8e\u63a8\u8350\u5546\u54c1B\u7ed9\u7528\u6237\uff0c\u56e0\u4e3a\u5b83\u4e0e\u7528\u6237\u504f\u597d\u5411\u91cf\u7684\u5185\u79ef\u66f4\u5927\uff0c\u5373\u5339\u914d\u5ea6\u66f4\u9ad8\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u2705 \u5ef6\u4f38\u8bf4\u660e<\/h5>\n\n\n\n<p>\u5728\u5b9e\u9645\u7cfb\u7edf\u4e2d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7528\u6237\u5411\u91cf\u53ef\u80fd\u662f\u901a\u8fc7\u884c\u4e3a\uff08\u70b9\u51fb\u3001\u8bc4\u5206\u3001\u6d4f\u89c8\u65f6\u95f4\uff09\u5b66\u4e60\u5f97\u5230\uff1b<\/li>\n\n\n\n<li>\u5546\u54c1\u5411\u91cf\u5219\u53ef\u80fd\u6765\u81ea\u6587\u672c\u63cf\u8ff0\u3001\u56fe\u50cf\u3001\u5386\u53f2\u6807\u7b7e\u7b49\u7f16\u7801\uff1b<\/li>\n\n\n\n<li>\u5185\u79ef\u53ef\u4ee5\u7528\u6765\u5feb\u901f\u8fdb\u884c\u6279\u91cf\u63a8\u8350\uff08\u4f8b\u5982\u7528\u77e9\u9635\u4e58\u6cd5\u5b9e\u73b0 top-k \u63a8\u8350\uff09\uff1b<\/li>\n\n\n\n<li>\u6709\u65f6\u4e5f\u4f1a\u5bf9\u5411\u91cf\u8fdb\u884c\u5f52\u4e00\u5316\uff0c\u4f7f\u5185\u79ef\u53d8\u6210\u201c\u4f59\u5f26\u76f8\u4f3c\u5ea6\u201d\u3002<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u5185\u79ef\uff08Inner Product\uff09\u662f\u7ebf\u6027\u4ee3\u6570\u4e2d\u4e00\u4e2a\u57fa\u672c\u800c\u6838\u5fc3\u7684\u6982\u5ff5\uff0c\u662f\u8861\u91cf\u4e24\u4e2a\u5411\u91cf\u4e4b\u95f4\u76f8\u4f3c\u7a0b\u5ea6\u6216\u201c\u5bf9\u9f50\u7a0b\u5ea6\u201d\u7684 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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