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href="/atom.xml" title="Magicland" type="application/atom+xml"><link rel="alternate" href="/rss2.xml" title="Magicland" type="application/rss+xml"><title>尝试在博客中添加简易文章推荐功能 - Magicland</title><meta name="keywords" content="数据挖掘, 推荐系统,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc name="description" content="详解在静态博客中利用TF-IDF算法实现文章推荐功能的过程，包括文本分词、词频-逆文档频率计算、向量空间模型构建及余弦相似度匹配，解决传统标签推荐依赖人工标注的局限，提供技术实现思路。 - MHuiG - Magicland"><meta property="og:type" content="article"><meta property="og:title" content="尝试在博客中添加简易文章推荐功能"><meta property="og:url" content="https://blog.mhuig.top/p/175a1706/"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="详解在静态博客中利用TF-IDF算法实现文章推荐功能的过程，包括文本分词、词频-逆文档频率计算、向量空间模型构建及余弦相似度匹配，解决传统标签推荐依赖人工标注的局限，提供技术实现思路。"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2022-09-15T11:13:00.000Z"><meta 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active-action="action-pagesabout"><i class="fa-duotone fa-user-tie color-about fa-fw"></i><p>关于</p></a><a href="https://www.travellings.cn/go-by-clouds.html" target="_blank" active-action="action-https:wwwtravellingscngo-by-cloudshtml" rel="external nofollow noopener noreferrer"><i class="fa-duotone fa-subway color-travellings fa-fw"></i><p>Travelling</p></a></div></div></div></div><div id="scroll-down" style="display:none"><i class="fa fa-chevron-down scroll-down-effects"></i></div></div></div><div id="safearea"><div class="body-wrapper"><div id="l_main" class><article itemscope itemtype="http://schema.org/Article" class="article post white-box reveal md shadow floatable blur article-type-post" id="post" itemprop="blogPost"><link itemprop="mainEntityOfPage" href="https://blog.mhuig.top/p/175a1706/"><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="Magicland"></span><span hidden itemprop="post" itemscope itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="尝试在博客中添加简易文章推荐功能"><meta itemprop="description" content="详解在静态博客中利用TF-IDF算法实现文章推荐功能的过程，包括文本分词、词频-逆文档频率计算、向量空间模型构建及余弦相似度匹配，解决传统标签推荐依赖人工标注的局限，提供技术实现思路。"></span><div class="headimg-div"><div class="headimg-a"> <img itemprop="image" class="headimg lazyload" alt="尝试在博客中添加简易文章推荐功能" src="https://static.mhuig.top/npm/mhgoos@0.0.1663317354137/tf-idf.png" class="lazyload" data-srcset="https://static.mhuig.top/npm/mhgoos@0.0.1663317354137/tf-idf.png" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw=="></div></div><div class="article-meta" id="top"><h1 class="hidden" itemprop="name headline"> 尝试在博客中添加简易文章推荐功能</h1><div class="new-meta-box"><div class="new-meta-item author" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a itemprop="url" class="author" target="_blank" href="https://mhuig.top/" rel="nofollow noopener" alt="author"><img itemprop="image" src="/lib/avatar/avatar-16.webp" class="lazyload" data-srcset="/lib/avatar/avatar-16.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="author"><p itemprop="name">MHuiG</p></a></div><div class="new-meta-item wordcount"><a class="notlink"><i class="fa-duotone fa-keyboard fa-fw" aria-hidden="true"></i><p>字数：2.7k 字</p></a></div><div class="new-meta-item readtime"><a class="notlink"><i class="fa-duotone fa-hourglass-half fa-fw" aria-hidden="true"></i><p>时长：10 分钟</p></a></div></div></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p>把文档转换成 “向量”，并且尝试用线性代数等数学工具来解决信息检索这类问题，至少可以追溯到 20 世纪 70 年代. 想到在 Hexo 这种静态博客系统，特别是 GitHub Page 这种 Serverless 服务中使用 Word2Vec 等深度学习方法似乎并不现实，所以这里使用的是一个非常简单经典的 TF-IDF 算法.</p><p>TF-IDF 算法原来是搜索引擎中的核心部分，谷歌百度已经使用 TF-IDF 作为内容排名因素很长一段时间.</p><p>现在的搜索引擎一般用如下的算法计算网站页面得分：score(页面得分) = TFIDF分 * x + 链接分 * y + 用户体验分 * z（其中x+y+z=100%），TFIDF分值百度大约占 40% 左右的权重，谷歌更是达到了50%.</p><p>这是百度的计算方法： score(页面得分) = 40%的内容质量相关性（TFIDF）+ 40% 的用户体验分 + 20% 的链接分（域名+外链）.</p><p>搜索引擎使用 TF-IDF 来计算内容质量相关性，我们这边也可以用它来计算文章内容相关性，然后实现简易文章推荐功能。</p><p>当然也可以使用文章的 tags 标签最为推荐依据，但是这样过于依赖人工标注数据，顺便最近发现在 Hexo 中分类和标签只能在 <code>source/_posts</code> 文件夹中使用才能渲染出来。</p><div class="story post-story"><h2 id="构建向量空间模型"><a href="#构建向量空间模型" class="headerlink" title="构建向量空间模型"></a>构建向量空间模型</h2><p>将文档转化为向量来表示，这样文档与文档之间就可以定量的去度量他们之间的关系，挖掘文档之间的联系. 上面提到的 Word2Vec 是把单词转化为向量来表示，这样词与词之间就可以定量的去度量他们之间的关系，挖掘词之间的联系.</p><p>比如将词的维度降维到 2 维，用下图的词向量表示词:</p><p><img src="https://static.mhuig.top/npm/mhgoos@0.0.1663246894190/1.png" class="lazyload" data-srcset="https://static.mhuig.top/npm/mhgoos@0.0.1663246894190/1.png" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="单词转化为向量"></p><p><img src="https://static.mhuig.top/npm/mhgoos@0.0.1663284804787/2.png" class="lazyload" data-srcset="https://static.mhuig.top/npm/mhgoos@0.0.1663284804787/2.png" 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class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A: 我这里有苹果和西瓜</span><br><span class="line"></span><br><span class="line">句子B: 我喜欢吃西瓜 不喜欢吃苹果</span><br><span class="line"></span><br><span class="line">句子C: 我喜欢吃蔬菜</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第二步：分词"><a href="#第二步：分词" class="headerlink" title="第二步：分词"></a>第二步：分词</h2><p>使用分词工具进行分词。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">[ <span class="string">'我'</span>, <span class="string">'这里'</span>, <span class="string">'有'</span>, <span class="string">'苹果'</span>, <span class="string">'和'</span>, <span class="string">'西瓜'</span> ]</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">[</span><br><span class="line"> <span class="string">'我'</span>, <span class="string">'喜欢'</span>,</span><br><span class="line"> <span class="string">'吃'</span>, <span class="string">'西瓜'</span>,</span><br><span class="line"> <span class="string">'不'</span>, <span class="string">'喜欢'</span>,</span><br><span class="line"> <span class="string">'吃'</span>, <span class="string">'苹果'</span></span><br><span class="line">]</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">[ <span class="string">'我'</span>, <span class="string">'喜欢'</span>, <span class="string">'吃'</span>, <span class="string">'蔬菜'</span> ]</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第三步：去重收集所有句子中所有的词"><a href="#第三步：去重收集所有句子中所有的词" class="headerlink" title="第三步：去重收集所有句子中所有的词"></a>第三步：去重收集所有句子中所有的词</h2><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">[</span><br><span class="line"> <span class="string">'我'</span>, <span class="string">'这里'</span>, <span class="string">'有'</span>,</span><br><span class="line"> <span class="string">'苹果'</span>, <span class="string">'和'</span>, <span class="string">'西瓜'</span>,</span><br><span class="line"> <span class="string">'喜欢'</span>, <span class="string">'吃'</span>, <span class="string">'不'</span>,</span><br><span class="line"> <span class="string">'蔬菜'</span></span><br><span class="line">]</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第四步：计算词频"><a href="#第四步：计算词频" class="headerlink" title="第四步：计算词频"></a>第四步：计算词频</h2><p>词频背后的假设是文章的重要程度即文章的相关度与单词在文档中出现的次数成正比。文章的关键词应当比文章中的其他词出现的次数多。</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 49.146ex;"><svg style="vertical-align: -0.566ex; min-width: 49.146ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 21722.6 1000"><g stroke="currentColor" fill="currentColor" 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631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 1,</span><br><span class="line"> <span class="string">'这里'</span>: 1,</span><br><span class="line"> <span class="string">'有'</span>: 1,</span><br><span class="line"> <span class="string">'苹果'</span>: 1,</span><br><span class="line"> <span class="string">'和'</span>: 1,</span><br><span class="line"> <span class="string">'西瓜'</span>: 1,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 1,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 1,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 1,</span><br><span class="line"> <span class="string">'喜欢'</span>: 2,</span><br><span class="line"> <span class="string">'吃'</span>: 2,</span><br><span class="line"> <span class="string">'不'</span>: 1,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 1,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 1,</span><br><span class="line"> <span class="string">'吃'</span>: 1,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 1</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第四步：词频标准化"><a href="#第四步：词频标准化" class="headerlink" title="第四步：词频标准化"></a>第四步：词频标准化</h2><p>文章的篇幅有长有短，为了便于不同文章的比较，进行"词频"标准化。词频标准化的目的是把所有的词频在同一维度上分析。</p><p>词频标准化有两种方案：</p><p>方案一：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 50.142ex;"><svg style="vertical-align: -2.077ex; min-width: 50.142ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.285ex" role="img" focusable="false" data-mjx-viewbox="0 -1418 22162.6 2336"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1418)"><g data-mml-node="math" data-latex="
词频(TF) = \frac{某个词在文章中出现的次数}{文章的总词数}
"><g data-mml-node="mtable" data-latex="词频(TF) = \frac{某个词在文章中出现的次数}{文章的总词数}" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1418) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveaspectratio="xMidYMid" viewbox="9003.3 -1418 1 2336"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,-8)"><g data-mml-node="mtd"><g data-mml-node="mi" data-latex="词"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">词</text></g><g data-mml-node="mi" data-latex="频" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">频</text></g><g data-mml-node="mo" data-latex="(" transform="translate(2000,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mi" data-latex="T" transform="translate(2389,0)"><path data-c="1D447" d="M344 631C344 628 343 621 340 611L208 83C204 68 200 59 197 55C188 44 154 39 94 39C63 39 49 42 49 16C49 5 56 0 69 0C120 0 192 4 235 3L317 2C331 2 386 0 403 0C420 0 428 8 428 24C428 34 416 39 391 39C339 39 309 41 300 46C297 48 295 52 295 58L430 603C433 617 436 626 439 629C443 635 467 638 511 638C566 638 604 634 623 625C642 616 652 595 652 561C652 543 649 517 644 483C642 475 641 469 641 464C641 453 646 447 657 447C666 447 672 456 675 473L702 645C703 650 704 656 704 662C704 672 694 677 673 677L125 677C99 677 97 674 89 655L30 481C27 472 25 466 24 462C24 452 29 447 40 447C48 447 55 455 60 470C85 543 110 588 133 607C159 628 209 638 282 638L321 638C332 638 344 639 344 631Z"/></g><g data-mml-node="mi" data-latex="F" transform="translate(3093,0)"><path data-c="1D439" d="M199 656C199 646 210 641 231 641C271 641 291 637 291 628C291 625 289 618 286 605L156 82C150 60 140 47 126 42C119 40 101 39 70 39C48 39 38 37 38 16C38 5 44 0 57 0L187 3L335 0C351 0 359 8 359 23C359 40 348 39 322 39C278 39 253 41 247 46C244 47 243 51 243 56L307 321L400 321C446 321 478 319 478 281C478 268 476 252 471 233C470 230 469 226 468 222C468 211 473 206 484 206C489 207 496 214 503 229L557 444C559 452 560 458 560 461C557 470 551 475 544 475C536 475 530 467 526 452C516 414 502 390 486 378C470 366 442 360 402 360L317 360L379 606C387 640 387 641 429 641L559 641C656 641 700 626 700 537C700 517 699 499 697 484C696 473 695 467 695 466C695 455 700 450 710 450C720 450 726 459 728 478L748 649C751 677 744 680 717 680L233 680C210 680 199 679 199 656Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(3844,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 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data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">出</text></g><g data-mml-node="mi" data-latex="现" transform="translate(8000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">现</text></g><g data-mml-node="mi" data-latex="的" transform="translate(9000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">的</text></g><g data-mml-node="mi" data-latex="次" transform="translate(10000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">次</text></g><g data-mml-node="mi" data-latex="数" transform="translate(11000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g></g><g data-mml-node="mrow" data-latex="文章的总词数" transform="translate(3220,-710)"><g data-mml-node="mi" data-latex="文"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">文</text></g><g data-mml-node="mi" data-latex="章" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">章</text></g><g data-mml-node="mi" data-latex="的" transform="translate(2000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">的</text></g><g data-mml-node="mi" data-latex="总" transform="translate(3000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">总</text></g><g data-mml-node="mi" data-latex="词" transform="translate(4000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">词</text></g><g data-mml-node="mi" data-latex="数" transform="translate(5000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g></g><rect width="12200" height="60" x="120" y="220"/></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="1278 -1418 1 2336"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:3" transform="translate(0,740)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{3}" transform="translate(389,0)"><path data-c="33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 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min-width: 52.404ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.285ex" role="img" focusable="false" data-mjx-viewbox="0 -1418 23162.6 2336"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1418)"><g data-mml-node="math" data-latex="
词频(TF) = \frac{某个词在文章中出现的次数}{该文出现最多的词出现的次数}
"><g data-mml-node="mtable" data-latex="词频(TF) = \frac{某个词在文章中出现的次数}{该文出现最多的词出现的次数}" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 1418) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveaspectratio="xMidYMid" viewbox="9503.3 -1418 1 2336"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr" transform="translate(0,-8)"><g data-mml-node="mtd"><g data-mml-node="mi" data-latex="词"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">词</text></g><g data-mml-node="mi" data-latex="频" transform="translate(1000,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">频</text></g><g data-mml-node="mo" data-latex="(" transform="translate(2000,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 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47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></mjx-container> 这个值来代替原来的 TF 取值。这样的计算保持了一个平衡，既有区分度，但也不至于完全线性增长。</p><p>这里使用比较简单的方案一作为栗子：</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'这里'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'有'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'苹果'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'和'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0.16666666666666666,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 0.125,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0.125,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0.125,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0.25,</span><br><span class="line"> <span class="string">'吃'</span>: 0.25,</span><br><span class="line"> <span class="string">'不'</span>: 0.125,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: 0.25,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0.25,</span><br><span class="line"> <span class="string">'吃'</span>: 0.25,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.25</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第五步：计算逆文档频率"><a href="#第五步：计算逆文档频率" class="headerlink" title="第五步：计算逆文档频率"></a>第五步：计算逆文档频率</h2><p>仅有词频(TF)不能比较完整地描述文档的相关度。因为语言的因素，有一些单词可能会比较自然地在很多文档中反复出现，比如这里中的 “我”、“和” 等等。这些词大多起到了链接语句的作用，是保持语言连贯不可或缺的部分。</p><p>很明显，如果有太多文档都涵盖了某个单词，这个单词也就越不重要，或者说是这个单词就越没有信息量。因此，我们需要对词频(TF)的值进行修正，而 逆文档频率(IDF)的想法是用文档频率(DF)的倒数来进行修正。</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 57.833ex;"><svg style="vertical-align: -2.077ex; min-width: 57.833ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.285ex" role="img" focusable="false" data-mjx-viewbox="0 -1418 25562 2336"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1418)"><g data-mml-node="math" data-latex="
逆文档频率(IDF) = \log(\frac{语料库的文档总数}{包含该词的文档数+1})
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-0.2876820724517809,</span><br><span class="line"> <span class="string">'这里'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'有'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.4054651081081644</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.2876820724517809,</span><br><span class="line"> <span class="string">'这里'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'有'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.4054651081081644</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.2876820724517809,</span><br><span class="line"> <span class="string">'这里'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'有'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0.4054651081081644,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.4054651081081644</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第六步：计算TF-IDF"><a href="#第六步：计算TF-IDF" class="headerlink" title="第六步：计算TF-IDF"></a>第六步：计算TF-IDF</h2><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 53.286ex;"><svg style="vertical-align: -0.566ex; min-width: 53.286ex;" 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683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>可以看到，TF-IDF与一个词在文档中的出现次数成正比，与该词在整个文档库中的出现次数成反比。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.047947012075296815,</span><br><span class="line"> <span class="string">'这里'</span>: 0.06757751801802739,</span><br><span class="line"> <span class="string">'有'</span>: 0.06757751801802739,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0.06757751801802739,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.03596025905647261,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0.05068313851352055,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">{</span><br><span class="line"> <span class="string">'我'</span>: -0.07192051811294523,</span><br><span class="line"> <span class="string">'这里'</span>: 0,</span><br><span class="line"> <span class="string">'有'</span>: 0,</span><br><span class="line"> <span class="string">'苹果'</span>: 0,</span><br><span class="line"> <span class="string">'和'</span>: 0,</span><br><span class="line"> <span class="string">'西瓜'</span>: 0,</span><br><span class="line"> <span class="string">'喜欢'</span>: 0,</span><br><span class="line"> <span class="string">'吃'</span>: 0,</span><br><span class="line"> <span class="string">'不'</span>: 0,</span><br><span class="line"> <span class="string">'蔬菜'</span>: 0.1013662770270411</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第七步：列出文档向量"><a href="#第七步：列出文档向量" class="headerlink" title="第七步：列出文档向量"></a>第七步：列出文档向量</h2><p>这里准确来说应该是 TF-IDF 特征向量，这是一个稀疏矩阵。下面把它作为描述文章特征的向量。</p><p>说明一点，这里的文章的特征不是由某几个关键词的 TF-IDF 值决定的，而是由所有的词的 TF-IDF 值共同决定的。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">句子A:</span><br><span class="line"></span><br><span class="line">[</span><br><span class="line"> -0.047947012075296815,</span><br><span class="line"> 0.06757751801802739,</span><br><span class="line"> 0.06757751801802739,</span><br><span class="line"> 0,</span><br><span class="line"> 0.06757751801802739,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0</span><br><span class="line">]</span><br><span class="line"></span><br><span class="line">句子B:</span><br><span class="line"></span><br><span class="line">[</span><br><span class="line"> -0.03596025905647261,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0.05068313851352055,</span><br><span class="line"> 0</span><br><span class="line">]</span><br><span class="line"></span><br><span class="line">句子C:</span><br><span class="line"></span><br><span class="line">[</span><br><span class="line"> -0.07192051811294523,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0,</span><br><span class="line"> 0.1013662770270411</span><br><span class="line">]</span><br></pre></td></tr></tbody></table></figure><p>可以对文档向量进行标准化，使得这些向量能够不受向量里有效元素多少的影响，也就是不同的文档可能有不同的长度。把向量都标准化为一个单位向量的长度。这个时候再进行点积运算，就相当于在原来的向量上进行余弦相似度的运算。</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 33.501ex;"><svg style="vertical-align: -0.949ex; min-width: 33.501ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="3.029ex" role="img" focusable="false" data-mjx-viewbox="0 -919.5 14807.6 1339"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -919.5)"><g data-mml-node="math" data-latex="
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data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p></div><div class="story post-story"><h2 id="第八步：计算余弦相似度"><a href="#第八步：计算余弦相似度" class="headerlink" title="第八步：计算余弦相似度"></a>第八步：计算余弦相似度</h2><p>到这里，如何寻找两篇相似文章的问题转变成了如何计算这两个向量的相似程度的问题。</p><p>什么叫做向量相似？</p><p>一般地，如果两个向量平行或重合，我们认为这两个向量相似度为 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewbox="0 -666 500 666"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g></g></g></svg></mjx-container>。如果这两个向量垂直，或者说正交，我们认为这两个向量的相似度为 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 500 688"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="0"><g data-mml-node="mn" data-latex="0"><path data-c="30" d="M249-22C390-22 460 92 460 320C460 473 428 575 365 625C330 652 291 666 250 666C109 666 39 551 39 320C39 136 88-22 249-22M361 524C368 489 371 425 371 332C371 240 367 172 360 128C347 48 310 8 249 8C226 8 203 17 182 34C155 57 139 104 132 176C129 201 128 253 128 332C128 419 131 480 136 513C145 568 163 603 191 618C213 630 232 636 249 636C314 636 350 583 361 524Z"/></g></g></g></svg></mjx-container>。</p><p>以二维空间为例，<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.245ex" height="1.919ex" role="img" focusable="false" viewbox="0 -837 550.5 848"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\vec{a}"><g data-mml-node="TeXAtom" data-latex="\vec{a}" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi" data-latex="a"><path data-c="1D44E" d="M498 144C498 153 493 158 482 158C474 158 468 151 465 137C445 58 421 18 394 18C377 18 368 32 368 60C368 73 372 97 381 132L438 357C443 376 445 387 445 392C445 412 434 422 412 422C391 422 377 410 370 387C349 424 319 442 281 442C216 442 159 409 109 343C63 281 40 217 40 150C40 63 91-11 175-11C218-11 260 12 300 58C311 20 345-11 392-11C461-11 482 70 498 144M341 374C350 353 355 339 355 330C355 326 354 321 353 314L304 122C301 111 294 99 285 87C248 41 212 18 177 18C138 18 118 48 118 107C118 131 124 167 136 215C157 300 187 357 224 388C244 405 263 413 282 413C309 413 329 400 341 374Z"/></g><g data-mml-node="mo" transform="translate(550.5,26)"><path data-c="20D7" d="M-69 599C-60 602-56 607-56 616C-56 625-60 630-69 633C-105 645-131 667-147 700C-150 707-156 711-163 711C-176 711-182 705-182 693C-182 678-169 658-143 634L-453 634C-466 634-472 628-472 616C-472 604-466 598-453 598L-143 598C-169 574-182 554-182 539C-182 527-176 521-163 521C-156 521-150 525-147 532C-131 565-105 587-69 599Z"/></g></g></g></g></g></svg></mjx-container> 和 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.058ex" height="2.489ex" role="img" focusable="false" viewbox="0 -1089 467.5 1100"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\vec{b}"><g data-mml-node="TeXAtom" data-latex="\vec{b}" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi" data-latex="b"><path data-c="1D44F" d="M281 445C245 445 209 428 174 394L243 679C241 688 238 694 226 694C193 694 120 685 107 684C92 682 84 675 84 660C84 650 93 645 112 645C131 645 157 646 157 632C157 628 152 608 143 571L63 249C52 207 47 173 47 148C47 62 93-11 175-11C240-11 297 22 347 89C392 151 415 216 415 283C415 370 364 445 281 445M279 415C318 415 337 385 337 326C337 299 331 263 319 218C297 133 268 75 233 44C213 27 194 19 175 19C134 19 114 51 114 115C114 137 119 170 129 214L151 304C154 319 159 330 165 338C204 389 242 415 279 415Z"/></g><g data-mml-node="mo" transform="translate(467.5,278)"><path data-c="20D7" d="M-69 599C-60 602-56 607-56 616C-56 625-60 630-69 633C-105 645-131 667-147 700C-150 707-156 711-163 711C-176 711-182 705-182 693C-182 678-169 658-143 634L-453 634C-466 634-472 628-472 616C-472 604-466 598-453 598L-143 598C-169 574-182 554-182 539C-182 527-176 521-163 521C-156 521-150 525-147 532C-131 565-105 587-69 599Z"/></g></g></g></g></g></svg></mjx-container> 是两个向量，我们要计算它们的夹角 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.061ex" height="1.62ex" role="img" focusable="false" viewbox="0 -705 469 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="θ"><g data-mml-node="mi" data-latex="θ"><path data-c="1D703" d="M164-11C209-11 256 16 302 70C382 163 455 335 455 498C455 539 448 582 434 625C416 678 382 705 333 705C288 705 243 678 196 624C116 532 42 360 42 196C42 94 75-11 164-11M332 675C365 675 382 637 382 560C382 517 372 452 352 367L155 367C180 460 204 529 229 574C266 641 301 675 332 675M145 327L341 327C322 246 299 181 274 132C235 57 199 19 164 19C131 19 115 58 115 135C115 185 125 249 145 327Z"/></g></g></g></svg></mjx-container> 。余弦定理告诉我们，可以用下面的公式求得：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 29.522ex;"><svg style="vertical-align: -1.931ex; min-width: 29.522ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="4.993ex" role="img" focusable="false" data-mjx-viewbox="0 -1353.5 13048.8 2206.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1353.5)"><g data-mml-node="math" data-latex="
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data-mml-node="mtd" id="mjx-eqn:8" transform="translate(0,591.5)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{8}" transform="translate(389,0)"><path data-c="38" d="M250 666C201 666 158 650 122 618C86 586 68 546 68 497C68 456 83 419 113 386C120 378 142 361 179 335C88 288 42 227 42 153C42 100 64 57 107 24C147-7 194-22 249-22C305-22 354-4 395 32C436 68 457 115 457 170C457 216 441 257 408 294C396 307 365 330 316 361C393 402 431 454 431 515C431 606 344 666 250 666M379 515C379 463 348 418 286 381L167 459C136 479 120 505 120 536C120 596 185 633 249 633C318 633 379 584 379 515M250 14C168 14 99 73 99 153C99 220 136 274 210 315L328 240C376 209 400 174 400 134C400 62 325 14 250 14Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>假定<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewbox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g></g></g></svg></mjx-container>和<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewbox="0 -683 759 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="B"><g data-mml-node="mi" data-latex="B"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659C203 643 223 644 235 644C265 643 283 641 288 640C293 639 295 636 295 631C295 629 294 623 291 613L159 82C154 60 144 47 129 42C122 40 104 39 73 39C52 39 42 31 42 15C42 5 52 0 73 0L426 0C491 0 551 20 608 59C671 102 703 155 703 217C703 296 638 344 567 358C655 379 756 446 756 543M554 644C623 644 658 612 658 547C658 496 638 454 596 420C554 386 507 370 456 370L317 370L377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228C603 176 583 131 542 94C501 57 455 39 402 39L267 39C257 39 250 39 246 40C240 40 237 42 237 45C237 46 237 49 238 53C269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"/></g></g></g></svg></mjx-container>是两个<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewbox="0 -442 600 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="n"><g data-mml-node="mi" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18C427 18 420 28 420 47C420 61 426 84 438 115C478 224 498 296 498 333C498 403 451 442 381 442C322 442 271 416 230 363C222 407 187 442 136 442C80 442 58 390 44 345C34 313 29 294 29 287C29 278 34 273 45 273C50 273 53 274 56 276C61 285 64 292 65 299C83 375 106 413 133 413C151 413 160 399 160 371C160 358 155 331 144 290L87 63C84 50 78 24 78 19C78-1 89-11 111-11C130-11 144-1 151 19C153 24 159 49 170 92L191 181L221 295C232 318 249 341 270 365C299 397 335 413 378 413C411 413 427 391 427 348C427 310 406 234 363 120C356 102 353 87 353 74C353 25 390-11 438-11C482-11 517 14 542 64C561 104 571 131 571 144C571 153 566 158 555 158C552 158 537 149 537 137Z"/></g></g></g></svg></mjx-container>维向量，我们把这些向量放到<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.357ex" height="1.025ex" role="img" focusable="false" viewbox="0 -442 600 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="n"><g data-mml-node="mi" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18C427 18 420 28 420 47C420 61 426 84 438 115C478 224 498 296 498 333C498 403 451 442 381 442C322 442 271 416 230 363C222 407 187 442 136 442C80 442 58 390 44 345C34 313 29 294 29 287C29 278 34 273 45 273C50 273 53 274 56 276C61 285 64 292 65 299C83 375 106 413 133 413C151 413 160 399 160 371C160 358 155 331 144 290L87 63C84 50 78 24 78 19C78-1 89-11 111-11C130-11 144-1 151 19C153 24 159 49 170 92L191 181L221 295C232 318 249 341 270 365C299 397 335 413 378 413C411 413 427 391 427 348C427 310 406 234 363 120C356 102 353 87 353 74C353 25 390-11 438-11C482-11 517 14 542 64C561 104 571 131 571 144C571 153 566 158 555 158C552 158 537 149 537 137Z"/></g></g></g></svg></mjx-container>维欧几里得空间中讨论，向量之间的距离使用欧几里得距离。</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 26.576ex;"><svg style="vertical-align: -2.095ex; min-width: 26.576ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="5.321ex" role="img" focusable="false" data-mjx-viewbox="0 -1426 11746.7 2352"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1426)"><g data-mml-node="math" data-latex="
\cos \theta=\frac{A \cdot B}{|A| \times |B|}
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217L590 16C595 11 600 9 607 9C620 9 630 19 630 32Z"/></g><g data-mml-node="mo" data-latex="|" transform="translate(2528.4,0)"><path data-c="7C" d="M139-250C152-250 159-242 159-226L159 726C159 742 152 750 139 750C126 750 119 742 119 726L119-226C119-242 126-250 139-250Z"/></g><g data-mml-node="mi" data-latex="B" transform="translate(2806.4,0)"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659C203 643 223 644 235 644C265 643 283 641 288 640C293 639 295 636 295 631C295 629 294 623 291 613L159 82C154 60 144 47 129 42C122 40 104 39 73 39C52 39 42 31 42 15C42 5 52 0 73 0L426 0C491 0 551 20 608 59C671 102 703 155 703 217C703 296 638 344 567 358C655 379 756 446 756 543M554 644C623 644 658 612 658 547C658 496 638 454 596 420C554 386 507 370 456 370L317 370L377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228C603 176 583 131 542 94C501 57 455 39 402 39L267 39C257 39 250 39 246 40C240 40 237 42 237 45C237 46 237 49 238 53C269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"/></g><g data-mml-node="mo" data-latex="|" transform="translate(3565.4,0)"><path data-c="7C" d="M139-250C152-250 159-242 159-226L159 726C159 742 152 750 139 750C126 750 119 742 119 726L119-226C119-242 126-250 139-250Z"/></g></g><rect width="4043.4" height="60" x="120" y="220"/></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="1278 -1426 1 2352"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:9" transform="translate(0,782)"><text data-id-align="true"/><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 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572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g></g></g></svg></mjx-container>是 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="14.873ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 6574 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="[A1, A2, ..., An] "><g data-mml-node="mo" data-latex="["><path data-c="5B" d="M233-202L159-202L159 702L233 702C248 702 256 710 256 726C256 742 248 750 233 750L114 750L114-250L233-250C248-250 256-242 256-226C256-214 245-202 233-202Z"/></g><g data-mml-node="mi" data-latex="A" transform="translate(278,0)"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(1028,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mo" data-latex="," transform="translate(1528,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50C86 20 109-5 139-5C153-5 165-1 174 8L175 0C175-63 154-117 112-160C105-168 101-174 101-178C101-188 105-193 114-193C123-193 135-181 152-158C186-110 203-57 203 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180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g><g data-mml-node="mo" data-latex="," transform="translate(3222.7,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50C86 20 109-5 139-5C153-5 165-1 174 8L175 0C175-63 154-117 112-160C105-168 101-174 101-178C101-188 105-193 114-193C123-193 135-181 152-158C186-110 203-57 203 0C203 53 185 106 139 106Z"/></g><g data-mml-node="mo" data-latex="." transform="translate(3667.3,0)"><path data-c="2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"/></g><g data-mml-node="mo" data-latex="." transform="translate(3945.3,0)"><path data-c="2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"/></g><g data-mml-node="mo" data-latex="." transform="translate(4223.3,0)"><path data-c="2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 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transform="translate(5696,0)"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18C427 18 420 28 420 47C420 61 426 84 438 115C478 224 498 296 498 333C498 403 451 442 381 442C322 442 271 416 230 363C222 407 187 442 136 442C80 442 58 390 44 345C34 313 29 294 29 287C29 278 34 273 45 273C50 273 53 274 56 276C61 285 64 292 65 299C83 375 106 413 133 413C151 413 160 399 160 371C160 358 155 331 144 290L87 63C84 50 78 24 78 19C78-1 89-11 111-11C130-11 144-1 151 19C153 24 159 49 170 92L191 181L221 295C232 318 249 341 270 365C299 397 335 413 378 413C411 413 427 391 427 348C427 310 406 234 363 120C356 102 353 87 353 74C353 25 390-11 438-11C482-11 517 14 542 64C561 104 571 131 571 144C571 153 566 158 555 158C552 158 537 149 537 137Z"/></g><g data-mml-node="mo" data-latex="]" transform="translate(6296,0)"><path data-c="5D" d="M45-250L164-250L164 750L45 750C30 750 22 742 22 726C22 710 30 702 45 702L119 702L119-202L45-202C30-202 22-210 22-226C22-242 30-250 45-250Z"/></g></g></g></svg></mjx-container> ，<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewbox="0 -683 759 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="B"><g data-mml-node="mi" data-latex="B"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659C203 643 223 644 235 644C265 643 283 641 288 640C293 639 295 636 295 631C295 629 294 623 291 613L159 82C154 60 144 47 129 42C122 40 104 39 73 39C52 39 42 31 42 15C42 5 52 0 73 0L426 0C491 0 551 20 608 59C671 102 703 155 703 217C703 296 638 344 567 358C655 379 756 446 756 543M554 644C623 644 658 612 658 547C658 496 638 454 596 420C554 386 507 370 456 370L317 370L377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228C603 176 583 131 542 94C501 57 455 39 402 39L267 39C257 39 250 39 246 40C240 40 237 42 237 45C237 46 237 49 238 53C269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"/></g></g></g></svg></mjx-container>是 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="14.934ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 6601 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="[B1, B2, ..., Bn] "><g data-mml-node="mo" data-latex="["><path data-c="5B" d="M233-202L159-202L159 702L233 702C248 702 256 710 256 726C256 742 248 750 233 750L114 750L114-250L233-250C248-250 256-242 256-226C256-214 245-202 233-202Z"/></g><g data-mml-node="mi" data-latex="B" transform="translate(278,0)"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659C203 643 223 644 235 644C265 643 283 641 288 640C293 639 295 636 295 631C295 629 294 623 291 613L159 82C154 60 144 47 129 42C122 40 104 39 73 39C52 39 42 31 42 15C42 5 52 0 73 0L426 0C491 0 551 20 608 59C671 102 703 155 703 217C703 296 638 344 567 358C655 379 756 446 756 543M554 644C623 644 658 612 658 547C658 496 638 454 596 420C554 386 507 370 456 370L317 370L377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228C603 176 583 131 542 94C501 57 455 39 402 39L267 39C257 39 250 39 246 40C240 40 237 42 237 45C237 46 237 49 238 53C269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(1037,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mo" data-latex="," transform="translate(1537,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50C86 20 109-5 139-5C153-5 165-1 174 8L175 0C175-63 154-117 112-160C105-168 101-174 101-178C101-188 105-193 114-193C123-193 135-181 152-158C186-110 203-57 203 0C203 53 185 106 139 106Z"/></g><g data-mml-node="mi" data-latex="B" transform="translate(1981.7,0)"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659C203 643 223 644 235 644C265 643 283 641 288 640C293 639 295 636 295 631C295 629 294 623 291 613L159 82C154 60 144 47 129 42C122 40 104 39 73 39C52 39 42 31 42 15C42 5 52 0 73 0L426 0C491 0 551 20 608 59C671 102 703 155 703 217C703 296 638 344 567 358C655 379 756 446 756 543M554 644C623 644 658 612 658 547C658 496 638 454 596 420C554 386 507 370 456 370L317 370L377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228C603 176 583 131 542 94C501 57 455 39 402 39L267 39C257 39 250 39 246 40C240 40 237 42 237 45C237 46 237 49 238 53C269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"/></g><g data-mml-node="mn" data-latex="2" transform="translate(2740.7,0)"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g><g data-mml-node="mo" data-latex="," transform="translate(3240.7,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50C86 20 109-5 139-5C153-5 165-1 174 8L175 0C175-63 154-117 112-160C105-168 101-174 101-178C101-188 105-193 114-193C123-193 135-181 152-158C186-110 203-57 203 0C203 53 185 106 139 106Z"/></g><g data-mml-node="mo" data-latex="." transform="translate(3685.3,0)"><path data-c="2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"/></g><g data-mml-node="mo" data-latex="." transform="translate(3963.3,0)"><path data-c="2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"/></g><g data-mml-node="mo" data-latex="." transform="translate(4241.3,0)"><path data-c="2E" d="M192 53C192 82 168 106 139 106C110 106 86 82 86 53C86 24 110 0 139 0C168 0 192 24 192 53Z"/></g><g data-mml-node="mo" data-latex="," transform="translate(4519.3,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50C86 20 109-5 139-5C153-5 165-1 174 8L175 0C175-63 154-117 112-160C105-168 101-174 101-178C101-188 105-193 114-193C123-193 135-181 152-158C186-110 203-57 203 0C203 53 185 106 139 106Z"/></g><g data-mml-node="mi" data-latex="B" transform="translate(4964,0)"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659C203 643 223 644 235 644C265 643 283 641 288 640C293 639 295 636 295 631C295 629 294 623 291 613L159 82C154 60 144 47 129 42C122 40 104 39 73 39C52 39 42 31 42 15C42 5 52 0 73 0L426 0C491 0 551 20 608 59C671 102 703 155 703 217C703 296 638 344 567 358C655 379 756 446 756 543M554 644C623 644 658 612 658 547C658 496 638 454 596 420C554 386 507 370 456 370L317 370L377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228C603 176 583 131 542 94C501 57 455 39 402 39L267 39C257 39 250 39 246 40C240 40 237 42 237 45C237 46 237 49 238 53C269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"/></g><g data-mml-node="mi" data-latex="n" transform="translate(5723,0)"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18C427 18 420 28 420 47C420 61 426 84 438 115C478 224 498 296 498 333C498 403 451 442 381 442C322 442 271 416 230 363C222 407 187 442 136 442C80 442 58 390 44 345C34 313 29 294 29 287C29 278 34 273 45 273C50 273 53 274 56 276C61 285 64 292 65 299C83 375 106 413 133 413C151 413 160 399 160 371C160 358 155 331 144 290L87 63C84 50 78 24 78 19C78-1 89-11 111-11C130-11 144-1 151 19C153 24 159 49 170 92L191 181L221 295C232 318 249 341 270 365C299 397 335 413 378 413C411 413 427 391 427 348C427 310 406 234 363 120C356 102 353 87 353 74C353 25 390-11 438-11C482-11 517 14 542 64C561 104 571 131 571 144C571 153 566 158 555 158C552 158 537 149 537 137Z"/></g><g data-mml-node="mo" data-latex="]" transform="translate(6323,0)"><path data-c="5D" d="M45-250L164-250L164 750L45 750C30 750 22 742 22 726C22 710 30 702 45 702L119 702L119-202L45-202C30-202 22-210 22-226C22-242 30-250 45-250Z"/></g></g></g></svg></mjx-container> ，则<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewbox="0 -716 750 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="A"><g data-mml-node="mi" data-latex="A"><path data-c="1D434" d="M137 3C155 3 217 0 236 0C251 0 258 8 258 23C258 32 252 38 239 39C210 40 196 50 196 69C196 78 205 98 224 128C251 173 270 206 283 228L526 228C526 221 527 207 530 185C537 112 541 72 541 67C541 48 519 39 474 39C455 39 446 31 446 15C446 5 452 0 464 0C488 0 564 3 588 3C608 3 679 0 699 0C714 0 721 8 721 24C721 34 712 39 694 39C666 39 649 41 644 44C639 47 635 56 634 71L574 689C571 708 572 716 551 716C539 716 529 710 522 697L178 120C148 70 108 43 59 39C43 38 35 30 35 15C35 5 41 0 52 0C68 0 121 3 137 3M492 577L522 267L307 267Z"/></g></g></g></svg></mjx-container>与<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="1.545ex" role="img" focusable="false" viewbox="0 -683 759 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="B"><g data-mml-node="mi" data-latex="B"><path data-c="1D435" d="M756 543C756 634 666 683 568 683L236 683C213 683 203 682 203 659C203 643 223 644 235 644C265 643 283 641 288 640C293 639 295 636 295 631C295 629 294 623 291 613L159 82C154 60 144 47 129 42C122 40 104 39 73 39C52 39 42 31 42 15C42 5 52 0 73 0L426 0C491 0 551 20 608 59C671 102 703 155 703 217C703 296 638 344 567 358C655 379 756 446 756 543M554 644C623 644 658 612 658 547C658 496 638 454 596 420C554 386 507 370 456 370L317 370L377 610C385 644 385 644 427 644M582 299C596 276 603 253 603 228C603 176 583 131 542 94C501 57 455 39 402 39L267 39C257 39 250 39 246 40C240 40 237 42 237 45C237 46 237 49 238 53C269 180 293 276 309 340L493 340C536 340 565 326 582 299Z"/></g></g></g></svg></mjx-container>的夹角<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.061ex" height="1.62ex" role="img" focusable="false" viewbox="0 -705 469 716"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="θ"><g data-mml-node="mi" data-latex="θ"><path data-c="1D703" d="M164-11C209-11 256 16 302 70C382 163 455 335 455 498C455 539 448 582 434 625C416 678 382 705 333 705C288 705 243 678 196 624C116 532 42 360 42 196C42 94 75-11 164-11M332 675C365 675 382 637 382 560C382 517 372 452 352 367L155 367C180 460 204 529 229 574C266 641 301 675 332 675M145 327L341 327C322 246 299 181 274 132C235 57 199 19 164 19C131 19 115 58 115 135C115 185 125 249 145 327Z"/></g></g></g></svg></mjx-container>的余弦等于：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 44.014ex;"><svg style="vertical-align: -3.24ex; min-width: 44.014ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="7.61ex" role="img" focusable="false" data-mjx-viewbox="0 -1931.9 19454.4 3363.7"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1931.9)"><g data-mml-node="math" data-latex="
\cos\theta=\frac{\sum_{i=1}^{n}(A_i \times B_i)}{\sqrt{\sum_{i=1}^{n}A_i^2} \times \sqrt{\sum_{i=1}^{n}B_i^2} }
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data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{10}" transform="translate(389,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/><path data-c="30" d="M249-22C390-22 460 92 460 320C460 473 428 575 365 625C330 652 291 666 250 666C109 666 39 551 39 320C39 136 88-22 249-22M361 524C368 489 371 425 371 332C371 240 367 172 360 128C347 48 310 8 249 8C226 8 203 17 182 34C155 57 139 104 132 176C129 201 128 253 128 332C128 419 131 480 136 513C145 568 163 603 191 618C213 630 232 636 249 636C314 636 350 583 361 524Z" transform="translate(500,0)"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(1389,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">{</span><br><span class="line"> <span class="string">'句子A'</span>: {</span><br><span class="line"> <span class="string">'句子A'</span>: 1.0000000000000002,</span><br><span class="line"> <span class="string">'句子B'</span>: 0.21934876427664535,</span><br><span class="line"> <span class="string">'句子C'</span>: 0.21934876427664535</span><br><span class="line"> },</span><br><span class="line"> <span class="string">'句子B'</span>: {</span><br><span class="line"> <span class="string">'句子A'</span>: 0.21934876427664535,</span><br><span class="line"> <span class="string">'句子B'</span>: 1,</span><br><span class="line"> <span class="string">'句子C'</span>: 0.33484380220099325</span><br><span class="line"> },</span><br><span class="line"> <span class="string">'句子C'</span>: {</span><br><span class="line"> <span class="string">'句子A'</span>: 0.21934876427664535,</span><br><span class="line"> <span class="string">'句子B'</span>: 0.33484380220099325,</span><br><span class="line"> <span class="string">'句子C'</span>: 1</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure></div><div class="story post-story"><h2 id="第九步：余弦相似度按照降序排序"><a href="#第九步：余弦相似度按照降序排序" class="headerlink" title="第九步：余弦相似度按照降序排序"></a>第九步：余弦相似度按照降序排序</h2><p>两两计算向量夹角的余弦值，然后按降序排列，取排在最前面的几个文章，就是相似度较高的文章，作为推荐结果。</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">{</span><br><span class="line"> <span class="string">'句子A'</span>: {</span><br><span class="line"> <span class="string">'句子A'</span>: 1.0000000000000002,</span><br><span class="line"> <span class="string">'句子B'</span>: 0.21934876427664535,</span><br><span class="line"> <span class="string">'句子C'</span>: 0.21934876427664535</span><br><span class="line"> },</span><br><span class="line"> <span class="string">'句子B'</span>: {</span><br><span class="line"> <span class="string">'句子B'</span>: 1,</span><br><span class="line"> <span class="string">'句子C'</span>: 0.33484380220099325,</span><br><span class="line"> <span class="string">'句子A'</span>: 0.21934876427664535</span><br><span class="line"> },</span><br><span class="line"> <span class="string">'句子C'</span>: {</span><br><span class="line"> <span class="string">'句子C'</span>: 1,</span><br><span class="line"> <span class="string">'句子B'</span>: 0.33484380220099325,</span><br><span class="line"> <span class="string">'句子A'</span>: 0.21934876427664535,</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><p>相同句子的余弦值是 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewbox="0 -666 500 666"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g></g></g></svg></mjx-container>.余弦值越接近<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewbox="0 -666 500 666"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g></g></g></svg></mjx-container>，就表明夹角越接近<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewbox="0 -666 500 688"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="0"><g data-mml-node="mn" data-latex="0"><path data-c="30" d="M249-22C390-22 460 92 460 320C460 473 428 575 365 625C330 652 291 666 250 666C109 666 39 551 39 320C39 136 88-22 249-22M361 524C368 489 371 425 371 332C371 240 367 172 360 128C347 48 310 8 249 8C226 8 203 17 182 34C155 57 139 104 132 176C129 201 128 253 128 332C128 419 131 480 136 513C145 568 163 603 191 618C213 630 232 636 249 636C314 636 350 583 361 524Z"/></g></g></g></svg></mjx-container>度，也就是两个向量越相似.</p></div><div class="story post-story"><h2 id="第十步：收集推荐结果"><a href="#第十步：收集推荐结果" class="headerlink" title="第十步：收集推荐结果"></a>第十步：收集推荐结果</h2><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">{</span><br><span class="line"> <span class="string">'句子A'</span>: [ <span class="string">'句子B'</span>, <span class="string">'句子C'</span> ],</span><br><span class="line"> <span class="string">'句子B'</span>: [ <span class="string">'句子C'</span>, <span class="string">'句子A'</span> ],</span><br><span class="line"> <span class="string">'句子C'</span>: [ <span class="string">'句子B'</span>, <span class="string">'句子A'</span> ]</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><p>即：</p><figure class="highlight bash"><table><tbody><tr><td class="code"><pre><span class="line">A-B</span><br><span class="line">B-C</span><br><span class="line">C-B</span><br></pre></td></tr></tbody></table></figure><p>A和C之间没什么关系。</p></div><div class="story post-story"><h2 id="源码实现"><a href="#源码实现" class="headerlink" title="源码实现"></a>源码实现</h2><div class="tag link"><a class="link-card" title="Volantis主题实现文章推荐" target="_blank" rel="external nofollow noopener noreferrer" href="/go.html?u=aHR0cHM6Ly9naXRodWIuY29tL3ZvbGFudGlzLXgvaGV4by10aGVtZS12b2xhbnRpcy9ibG9iLzY4NGM0MDMzM2IxNmI1ZDFkZjFhOWU0YTdmMWRhOTAyMTZlNWExN2Evc2NyaXB0cy9oZWxwZXJzL-aWh-eroOaOqOiNkC5qcw"><div class="left"><img 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class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%9E%84%E5%BB%BA%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4%E6%A8%A1%E5%9E%8B"><span class="toc-text">构建向量空间模型</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E4%B8%BE%E4%B8%AA%E6%A0%97%E5%AD%90"><span class="toc-text">举个栗子</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%B8%80%E6%AD%A5%EF%BC%9A%E6%95%B0%E6%8D%AE%E6%B8%85%E6%B4%97"><span class="toc-text">第一步：数据清洗</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%BA%8C%E6%AD%A5%EF%BC%9A%E5%88%86%E8%AF%8D"><span class="toc-text">第二步：分词</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%B8%89%E6%AD%A5%EF%BC%9A%E5%8E%BB%E9%87%8D%E6%94%B6%E9%9B%86%E6%89%80%E6%9C%89%E5%8F%A5%E5%AD%90%E4%B8%AD%E6%89%80%E6%9C%89%E7%9A%84%E8%AF%8D"><span class="toc-text">第三步：去重收集所有句子中所有的词</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%9B%9B%E6%AD%A5%EF%BC%9A%E8%AE%A1%E7%AE%97%E8%AF%8D%E9%A2%91"><span class="toc-text">第四步：计算词频</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%9B%9B%E6%AD%A5%EF%BC%9A%E8%AF%8D%E9%A2%91%E6%A0%87%E5%87%86%E5%8C%96"><span class="toc-text">第四步：词频标准化</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%BA%94%E6%AD%A5%EF%BC%9A%E8%AE%A1%E7%AE%97%E9%80%86%E6%96%87%E6%A1%A3%E9%A2%91%E7%8E%87"><span class="toc-text">第五步：计算逆文档频率</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%85%AD%E6%AD%A5%EF%BC%9A%E8%AE%A1%E7%AE%97TF-IDF"><span class="toc-text">第六步：计算TF-IDF</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%B8%83%E6%AD%A5%EF%BC%9A%E5%88%97%E5%87%BA%E6%96%87%E6%A1%A3%E5%90%91%E9%87%8F"><span class="toc-text">第七步：列出文档向量</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%85%AB%E6%AD%A5%EF%BC%9A%E8%AE%A1%E7%AE%97%E4%BD%99%E5%BC%A6%E7%9B%B8%E4%BC%BC%E5%BA%A6"><span class="toc-text">第八步：计算余弦相似度</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E4%B9%9D%E6%AD%A5%EF%BC%9A%E4%BD%99%E5%BC%A6%E7%9B%B8%E4%BC%BC%E5%BA%A6%E6%8C%89%E7%85%A7%E9%99%8D%E5%BA%8F%E6%8E%92%E5%BA%8F"><span class="toc-text">第九步：余弦相似度按照降序排序</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AC%AC%E5%8D%81%E6%AD%A5%EF%BC%9A%E6%94%B6%E9%9B%86%E6%8E%A8%E8%8D%90%E7%BB%93%E6%9E%9C"><span class="toc-text">第十步：收集推荐结果</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%BA%90%E7%A0%81%E5%AE%9E%E7%8E%B0"><span class="toc-text">源码实现</span></a></li></ol></div></section></div><div class="widget pjax"></div><div class="widget pjax"></div><div class="widget pjax"></div><div class="widget pjax"></div><div class="widget pjax"></div><div class="widget pjax"></div><div class="widget pjax"></div><div 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