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<h2 class="wrapper title">cs224n 解答拾遗：word embedding 之SVD分解</h2>
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<span>Author：</span><span>flypython</span> | <span>Date: </span><span>2020-01-02</span> | <span>Category：</span><span><a href="/fly/自然语言处理/" title="自然语言处理">自然语言处理</a><a href="/fly/自然语言处理/cs224n/" title="cs224n">cs224n</a></span>
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<blockquote>
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<p>老师，请问共现矩阵奇异值分解后为什么是用U的行来做word embedding呢?</p>
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</blockquote>
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<p>cs224n课程第一周里，提到了2种生成word embedding的方法。<br>一种是基于word2vec的神经网络生成词向量的方法。<br>另一种是基于统计的词向量生成的方法：使用共现矩阵，再进行SVD分解。</p>
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<p><img src="https://raw.githubusercontent.com/jcjview/jcjview.github.io/master/img/count_based.png" alt></p>
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<p>首先要说明，这2种方法输出的word embedding 都是distributed representation(稠密表达)。</p>
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<p>其次值得说的是，这2种方法各有优缺点（可能面试里会问），在glove方法中，综合进行了两种算法，达到最优的效果。</p>
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<p>那么，我们这里详细探讨一下利用SVD生成词向量的过程，解答上面提到的问题。</p>
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<blockquote>
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<p>共现矩阵奇异值分解后为什么是用U的行来做word embedding</p>
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</blockquote>
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<h3 id="1-生成共现矩阵（Window-based-Co-occurrence-Matrix）"><a href="#1-生成共现矩阵（Window-based-Co-occurrence-Matrix）" class="headerlink" title="1.生成共现矩阵（Window based Co-occurrence Matrix）"></a>1.生成共现矩阵（Window based Co-occurrence Matrix）</h3><p>构造一个矩阵X，这个矩阵的大小是 $V*V$ ，这里V是词表的长度。这个矩阵称之为共现矩阵。</p>
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<p>我们要统计，每个中心词在左右k个窗口的范围内，上下文词出现的词频。于是这里的共现矩阵，第一个维度（列），代表这个中心词，第二个维度（行）代表这个中心词对应的上下文词。<br>$X={x_{ij}}$为第i个词作为中心词时，对应第j个词作为其上下文词时候的词频。</p>
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<p>我们还是用课上的例子，有3句话：</p>
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<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">I enjoy flying。</span><br><span class="line">I like NLP。</span><br><span class="line">I like deep learning。</span><br></pre></td></tr></table></figure>
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<p>假设这里k=1，统计的共现矩阵X为：<br><img src="https://raw.githubusercontent.com/jcjview/jcjview.github.io/master/img/Window_based_Co-occurrence_Matrix1.png" alt></p>
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<p>可以看出，这里的共现矩阵X为对称矩阵，也就是对中心词like，enjoy的出现次数是等于对于中心词enjoy，like的出现次数。</p>
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<h3 id="2-SVD分解的过程"><a href="#2-SVD分解的过程" class="headerlink" title="2.SVD分解的过程"></a>2.SVD分解的过程</h3><p>假设X的size是$V<em>V$,那么U的矩阵是$V</em>K$,奇异值矩阵S是$K<em>K$,而$V^T$的矩阵size 是$K</em>V$.<br>这里V是词表长度，上面解释了，K是指最后需要输出的词向量维度，一般我们取100-300之间的一个值。</p>
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<p>SVD分解公式可以写作<br>$X=USV^T$<br>写成分量式为：</p>
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<p>$x_{ij}= \sum_{k=1}^n u_{ij}s_kv_{jk}$</p>
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<p>那么从这个公式看出，对于每个样本word $x_{ij}$ 与$u_{ij}$是对于的，而不是与$v^T_j$对应，$v^T_j$与每个维度对应。<br><img src="https://raw.githubusercontent.com/jcjview/jcjview.github.io/master/img/svd1.png" alt></p>
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<h4 id="这里需要解释一下我们是如何得到SVD分解公式的："><a href="#这里需要解释一下我们是如何得到SVD分解公式的：" class="headerlink" title="这里需要解释一下我们是如何得到SVD分解公式的："></a>这里需要解释一下我们是如何得到SVD分解公式的：</h4><p>由线性代数基本知识可知，对于任意矩阵X（size $V*V$ ）来说，我们可以进行奇异值分解，</p>
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<p>$X=U \Lambda V^T$，</p>
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<p>这里， $U, V, \Lambda$都是size $V*V$ 的对角矩阵，并且每个值都是非负实数，按大小从大到小排列。</p>
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<p><img src="https://raw.githubusercontent.com/jcjview/jcjview.github.io/master/img/svd2.png" alt></p>
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<p>那么我们对$\Lambda$进行降维，只保留前K个值，这样U和V也会跟着变，从维度上看，矩阵U去掉了后面的一些列，变成了$V<em>K$;矩阵$V^T$，去掉了前面的一些行，变成了$K</em>V$（实质上做了空间变换，而不是简单的去除一些数据）。也就是上面的图：<br><img src="https://raw.githubusercontent.com/jcjview/jcjview.github.io/master/img/svd1.png" alt></p>
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<p>这里就解释了，为什么是用U的行来做word embedding。</p>
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<h3 id="3-如果U对应的词向量，那么V对应的是什么呢？"><a href="#3-如果U对应的词向量，那么V对应的是什么呢？" class="headerlink" title="3.如果U对应的词向量，那么V对应的是什么呢？"></a>3.如果U对应的词向量，那么V对应的是什么呢？</h3><p>还有一个问题，如果U对应的词向量，那么V对应的是什么呢？对于词共现矩阵来说，V的意义还不是很清晰，我们考虑这样一个矩阵：</p>
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<p>词-文档矩阵 Word-Document Matrix，行是所有的词，维度为V，列是每个文档，维度为M。</p>
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<p>对这样一个矩阵进行SVD分解，并降维到K个奇异值上取值，</p>
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<p><img src="https://raw.githubusercontent.com/jcjview/jcjview.github.io/master/img/lsa1.png" alt></p>
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<p>SVD分解公式可以写作<br>$X=U\Sigma V^T$</p>
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<p>U的矩阵size$V*K$,</p>
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<p>$V^T$的矩阵size 是$K*V$</p>
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<p>写成分量式为：</p>
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<p>$x_{ij}= \sum_{k=1}^n u_{ij}\sigma_kv_{jk}$</p>
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<p>$u_{ij}$ 对应的第i个词和第j个主题的相关度，$u_{i}是词的主题向量$。</p>
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<p>$v_{jk}$ 对应第j个文档和第k个主题的相关度，<strong>所以$v_j$是第j个文档的文档向量。</strong></p>
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<p>这样就说明了V在SVD分解的意义。</p>
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