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众所周知，SVD(奇异值分解)可以用于最小二乘法求齐次线性方程组$A\vec,">
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<p>原创发表于 <a href="https://blog.davidz.cn">DavidZ Blog</a>，遵循 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode" target="_blank" rel="noopener">CC 4.0 BY-NC-SA</a> 版权协议，转载请附上原文出处链接及本声明。</p>
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<p>众所周知，SVD(奇异值分解)可以用于最小二乘法求齐次线性方程组$A\vec{x}=\vec{0}$的解。我看了很多资料，大多使用数学公式推导，得出结论。但是，曾经线性代数差点挂科的我，总觉得有些蹊跷。想了两天，终于有了一些感性的认知（不一定是对的😂），赶紧记录下来。</p>
<h2 id="矩阵有何意义"><a class="header-anchor" href="#矩阵有何意义"></a>矩阵有何意义</h2>
<p>按照我的理解，一个矩阵的实际意义是对应一个线性变换，这个变换可以理解为瞬间运动。例如，一个旋转矩阵，</p>
<p>$$<br>
A=\begin{bmatrix}<br>
cos\theta &amp; sin\theta \\\<br>
-sin\theta &amp; cos\theta<br>
\end{bmatrix}<br>
$$</p>
<p>它的意思是，把一个向量顺时针旋转$\theta$。也就是说，给定一个$\vec{v_1}=[-1, 1]^T$, 那么变换的结果就是$\vec{v_2}=A\vec{v_1}=[1, 1]^T$.</p>
<p>除了旋转，矩阵还可以表示包括缩放，投影在内的所有<strong>线性变换</strong>。</p>
<p>十分推荐大家去看 3Blue1Brown 的 <em>线性代数的本质</em>，B 站有<a href="https://www.bilibili.com/video/BV1ys411472E" target="_blank" rel="noopener">官方翻译版</a>，它完全颠覆了我对线性代数的认知。</p>
<h2 id="SVD到底干了什么"><a class="header-anchor" href="#SVD到底干了什么"></a>SVD到底干了什么</h2>
<p>$$<br>
A = U\Sigma V^T<br>
$$</p>
<p>SVD把矩阵$A(m\times n)$分解成了,</p>
<ul>
<li>$U(m\times m)$: 左奇异矩阵</li>
<li>$\Sigma(m\times n)$: 奇异值矩阵</li>
<li>$V(n\times n)$: 右奇异矩阵</li>
</ul>
<p>重点来了，SVD的意思就是，把一个本来由矩阵$A$表示的变换，转化成一个由$U,\Sigma,V$表示的变换。这个变换是，把一个向量，从以$V$为基向量的空间线性变换到成以$U$为基向量的空间中去（$\Sigma$的意义可以说是缩放，待证实，暂时忽略）。这样我们就可以更深入的理解这个变换了。</p>
<p>例如旋转$\vec{v_1}$90度得到$\vec{v_2}$，</p>
<p><img src="//davidz.cn/static/blog/2020-06-04-SVD-for-homogeneous-linear-equation/rotate90.svg" alt="rotate 90"></p>
<p>其中，</p>
<p>$$<br>
U=\begin{bmatrix}<br>
0 &amp; 1 \\\<br>
1 &amp; 0<br>
\end{bmatrix}<br>
$$</p>
<p>$$<br>
\Sigma=\begin{bmatrix}<br>
1 &amp; 1<br>
\end{bmatrix}<br>
$$</p>
<p>$$<br>
V=\begin{bmatrix}<br>
-1 &amp; 0 \\\<br>
0 &amp; 1<br>
\end{bmatrix}<br>
$$</p>
<p>也就是说，矩阵$A$可以被理解为，我们把一个向量$\vec{v_1}$，从以$\vec{e_1}=[-1, 0]^T,\vec{e_2}=[0, 1]^T$为基向量的空间线性变换到了以$\vec{e_1}=[0, 1]^T,\vec{e_2}=[1, 0]^T$为基向量的空间中。这个变换表现为旋转了90度。</p>
<h2 id="所以如何理解"><a class="header-anchor" href="#所以如何理解"></a>所以如何理解</h2>
<p>说回求齐次线性方程组$Ax=0$的解来。</p>
<p>按照矩阵的意义，我们这里要求的是，已知一个线性变换$A$，给定一个$\vec{x}$，使得线性变换后的结果为$\vec{0}$。</p>
<p>此时非常重要的是，如果$x=\vec{0}$，那一定成立，但是我们想找一个非平凡的解。</p>
<p>我们暂时不关心这个解是否存在，也就是说如果不存在，我们就找个最接近的（最小二乘法思想），我们直接使用SVD分解矩阵$A$，得到对应的$U,\Sigma,V$。</p>
<p>按照SVD的作用，我们现在可以说，矩阵$A$这个线性变换，把一个$\vec{x}$，从以$V$为基向量的空间线性变换到了以$U$为基向量的空间中，而我们想找，在以$V$为基向量的空间中，哪个向量会在投影后趋近于或者等于$\vec{0}$。更重要的是，我们只在乎这个向量的方向，而不在乎他的大小，因为它等于$\vec{0}$是个平凡解，这就像最小二乘法中，我们规定$|\vec{x}|=1$。</p>
<p>这时，答案就开始变得清晰了，因为我们想找的$\vec{x}$，应该就是$V$这组基向量中特异值$\sigma$最小的那一个$\vec{e_{min}}$，也就是说$\vec{x}=\vec{e_{min}}$。此时有两种情况，</p>
<ol>
<li>$\sigma=0$， 那么$\vec{x}$投影后的就是$\vec{0}$。</li>
<li>$\sigma\neq0$，那么$\vec{x}$投影后是使$A\vec{x}$最小的解。因为如果$\vec{x}\neq\vec{e_{min}}$，也就是说它偏离了$\vec{e_{min}}$，那么它一定由$\vec{e_{min}}$和另一个基向量线性组合，而无论怎么组合，$\sigma_{combine}\geq\sigma_{min}$。</li>
</ol>
<p>因此，我们求解$A\vec{x}=0$的过程就是，</p>
<ol>
<li>$U,\Sigma,V^T=SVD(A)$</li>
<li>$\vec{x}=V[:, -1]$</li>
</ol>
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<h4>目录</h4>
<ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#矩阵有何意义"><span class="toc-number">1.</span> <span class="toc-text">¶矩阵有何意义</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#SVD到底干了什么"><span class="toc-number">2.</span> <span class="toc-text">¶SVD到底干了什么</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#所以如何理解"><span class="toc-number">3.</span> <span class="toc-text">¶所以如何理解</span></a></li></ol>
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<script>
$(document).ready(function () {
if ($('span[id^="busuanzi_"]').length) {
initialBusuanzi();
}
});
</script>
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<link rel="stylesheet" href="//cdn.bootcss.com/photoswipe/4.1.3/photoswipe.min.css">
<link rel="stylesheet" href="//cdn.bootcss.com/photoswipe/4.1.3/default-skin/default-skin.min.css">
<script src="//cdn.bootcss.com/photoswipe/4.1.3/photoswipe.min.js"></script>
<script src="//cdn.bootcss.com/photoswipe/4.1.3/photoswipe-ui-default.min.js"></script>
<!-- Root element of PhotoSwipe. Must have class pswp. -->
<div class="pswp" tabindex="-1" role="dialog" aria-hidden="true">
<!-- Background of PhotoSwipe.
It's a separate element as animating opacity is faster than rgba(). -->
<div class="pswp__bg"></div>
<!-- Slides wrapper with overflow:hidden. -->
<div class="pswp__scroll-wrap">
<!-- Container that holds slides.
PhotoSwipe keeps only 3 of them in the DOM to save memory.
Don't modify these 3 pswp__item elements, data is added later on. -->
<div class="pswp__container">
<div class="pswp__item"></div>
<div class="pswp__item"></div>
<div class="pswp__item"></div>
</div>
<!-- Default (PhotoSwipeUI_Default) interface on top of sliding area. Can be changed. -->
<div class="pswp__ui pswp__ui--hidden">
<div class="pswp__top-bar">
<!-- Controls are self-explanatory. Order can be changed. -->
<div class="pswp__counter"></div>
<button class="pswp__button pswp__button--close" title="Close (Esc)"></button>
<button class="pswp__button pswp__button--share" title="Share"></button>
<button class="pswp__button pswp__button--fs" title="Toggle fullscreen"></button>
<button class="pswp__button pswp__button--zoom" title="Zoom in/out"></button>
<!-- Preloader demo http://codepen.io/dimsemenov/pen/yyBWoR -->
<!-- element will get class pswp__preloader--active when preloader is running -->
<div class="pswp__preloader">
<div class="pswp__preloader__icn">
<div class="pswp__preloader__cut">
<div class="pswp__preloader__donut"></div>
</div>
</div>
</div>
</div>
<div class="pswp__share-modal pswp__share-modal--hidden pswp__single-tap">
<div class="pswp__share-tooltip"></div>
</div>
<button class="pswp__button pswp__button--arrow--left" title="Previous (arrow left)">
</button>
<button class="pswp__button pswp__button--arrow--right" title="Next (arrow right)">
</button>
<div class="pswp__caption">
<div class="pswp__caption__center"></div>
</div>
</div>
</div>
</div>
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<!-- Global site tag (gtag.js) - Google Analytics -->
<script async src="//www.googletagmanager.com/gtag/js?id=UA-157733505-1"></script>
<script>
window.dataLayer = window.dataLayer || [];
function gtag() {
dataLayer.push(arguments);
}
gtag('js', new Date());
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gtag('config', 'UA-157733505-1');
</script>
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<script>
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function initialTyped() {
var typedTextEl = $('.typed-text');
if (typedTextEl && typedTextEl.length > 0) {
var typed = new Typed('.typed-text', {
strings: ['凡事都要留几分', '颜值是第一生产力'],
typeSpeed: 90,
loop: true,
loopCount: Infinity,
backSpeed: 20,
});
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}
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}
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Site updated: 2020-07-17 11:27:07
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if ($('.article-header') && $('.article-header').length) {
$(document).ready(function () {
initialTyped();
});
}
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</script>
</html>
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