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牛顿法和高斯-牛顿法 - 知乎.html
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<meta http-equiv="content-type" content="text/html; charset=UTF-8"><meta charset="utf-8"><title>牛顿法和高斯-牛顿法 - 知乎</title><meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=1"><meta name="renderer" content="webkit"><meta name="force-rendering" content="webkit"><meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"><meta name="google-site-verification" content="FTeR0c8arOPKh8c5DYh_9uu98_zJbaWw53J-Sch9MTg"><meta data-react-helmet="true" name="keywords" content="凸优化,最优化"><meta data-react-helmet="true" name="description" content="看代码的过程中遇到了高斯-牛顿法，感慨于自己作为调包侠，对各种最优化方法知之甚少，于是学习了一下这个算法。费曼技巧推崇以教促学，遂在此写一个简单的教程，希望整理思路以备复习的同时，也能帮助到和我一样…"><meta data-react-helmet="true" property="og:title" content="牛顿法和高斯-牛顿法"><meta data-react-helmet="true" property="og:url" content="https://zhuanlan.zhihu.com/p/103724149"><meta data-react-helmet="true" property="og:description" content="看代码的过程中遇到了高斯-牛顿法，感慨于自己作为调包侠，对各种最优化方法知之甚少，于是学习了一下这个算法。费曼技巧推崇以教促学，遂在此写一个简单的教程，希望整理思路以备复习的同时，也能帮助到和我一样…"><meta data-react-helmet="true" property="og:image" content="https://pic4.zhimg.com/v2-b8699cab4602e742602c49daf7ddedf4_720w.jpg?source=172ae18b"><meta data-react-helmet="true" property="og:type" content="article"><meta data-react-helmet="true" property="og:site_name" content="知乎专栏"><link data-react-helmet="true" rel="apple-touch-icon" href="https://static.zhihu.com/heifetz/assets/apple-touch-icon-152.a53ae37b.png"><link data-react-helmet="true" rel="apple-touch-icon" href="https://static.zhihu.com/heifetz/assets/apple-touch-icon-152.a53ae37b.png" sizes="152x152"><link data-react-helmet="true" rel="apple-touch-icon" href="https://static.zhihu.com/heifetz/assets/apple-touch-icon-120.bbce8f18.png" sizes="120x120"><link data-react-helmet="true" rel="apple-touch-icon" href="https://static.zhihu.com/heifetz/assets/apple-touch-icon-76.cbade8f9.png" sizes="76x76"><link data-react-helmet="true" rel="apple-touch-icon" href="https://static.zhihu.com/heifetz/assets/apple-touch-icon-60.8f6c52aa.png" sizes="60x60"><link rel="shortcut icon" type="image/x-icon" href="https://static.zhihu.com/heifetz/favicon.ico"><link rel="search" type="application/opensearchdescription+xml" href="https://static.zhihu.com/heifetz/search.xml" title="知乎"><link rel="dns-prefetch" href="https://static.zhimg.com/"><link rel="dns-prefetch" href="https://pic1.zhimg.com/"><link rel="dns-prefetch" href="https://pic2.zhimg.com/"><link rel="dns-prefetch" href="https://pic3.zhimg.com/"><link rel="dns-prefetch" href="https://pic4.zhimg.com/"><style>
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itemprop="zhihu:followerCount"><span class="UserLink AuthorInfo-avatarWrapper"><div class="Popover"><div id="Popover6-toggle" aria-haspopup="true" aria-expanded="false" aria-owns="Popover6-content"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/yin-xiang-nan"><img class="Avatar Avatar--round AuthorInfo-avatar" src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/bf28449bb_xs.jpg" srcset="https://pic4.zhimg.com/bf28449bb_l.jpg 2x" alt="杨指北" width="38" height="38"></a></div></div></span><div class="AuthorInfo-content"><div class="AuthorInfo-head"><span class="UserLink AuthorInfo-name"><div class="Popover"><div id="Popover7-toggle" aria-haspopup="true" aria-expanded="false" aria-owns="Popover7-content"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/yin-xiang-nan">杨指北</a></div></div></span></div><div class="AuthorInfo-detail"><div class="AuthorInfo-badge"><div class="ztext AuthorInfo-badgeText">无业游民</div></div></div></div></div></div><div class="LabelContainer-wrapper"></div><div><span class="Voters"><span><span class="UserLink"><div class="Popover"><div id="Popover11-toggle" aria-haspopup="true" aria-expanded="false" aria-owns="Popover11-content"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/da-bu-liu-wai-95">保护我方Vivian</a></div></div></span></span> 等 <button type="button" class="Button Button--plain">69 人赞同了该文章</button></span></div></header><div class="Post-RichTextContainer"><div class="RichText ztext Post-RichText"><p>看代码的过程中遇到了高斯-牛顿法，感慨于自己作为调包侠，对各种最优化方法知之甚少，于是学习了一下这个算法。费曼技巧推崇以教促学，遂在此写一个简单的教程，希望整理思路以备复习的同时，也能帮助到和我一样的初学者。</p><p>本文所涉算法的研究目标是寻找多元二阶可微函数 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_016.svg" alt="[公式]" eeimg="1" data-formula="f:\mathbb{R}^M\rightarrow\mathbb{R}"> 的极小值点，即:</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_030.svg" alt="[公式]" eeimg="1" data-formula="x^*=\underset{x\in\mathbb{R}^M}{\arg\min}f(x)\tag{1}"> </p><p>本文首先介绍牛顿法（也称作 Newton-Raphson 方法），然后介绍高斯-牛顿法。</p><h2>牛顿法</h2><p>牛顿法的推导基于二阶可微函数的泰勒展开，以一维函数为例:</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_021.svg" alt="[公式]" eeimg="1" data-formula="f(x)\approx f(x_n)+f'(x_n)(x-x_n)+\frac{1}{2}f''(x_n)(x-x_n)^2 \tag{2}"> </p><p>(2) 式是对 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_022.svg" alt="[公式]" eeimg="1" data-formula="f(x)"> 在 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_019.svg" alt="[公式]" eeimg="1" data-formula="x_n"> 处的二阶近似。如下图所示：</p><figure data-size="normal"><noscript><img src="https://pic1.zhimg.com/v2-d37a40a3a45281a740c4d3774510366c_b.jpg" data-caption="" data-size="normal" data-rawwidth="455" data-rawheight="392" class="origin_image zh-lightbox-thumb" width="455" data-original="https://pic1.zhimg.com/v2-d37a40a3a45281a740c4d3774510366c_r.jpg"/></noscript><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/v2-d37a40a3a45281a740c4d3774510366c_720w.jpg" data-caption="" data-size="normal" data-rawwidth="455" data-rawheight="392" class="origin_image zh-lightbox-thumb lazy" data-original="https://pic1.zhimg.com/v2-d37a40a3a45281a740c4d3774510366c_r.jpg" data-actualsrc="https://pic1.zhimg.com/v2-d37a40a3a45281a740c4d3774510366c_b.jpg" data-lazy-status="ok" width="455"></figure><p>蓝色曲线代表原函数 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_022.svg" alt="[公式]" eeimg="1" data-formula="f(x)"> ，绿色圆点代表当前点 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_019.svg" alt="[公式]" eeimg="1" data-formula="x_n"> ，橘黄色的抛物线就是原函数在 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_019.svg" alt="[公式]" eeimg="1" data-formula="x_n"> 附近的泰勒展开。不难发现，在 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_019.svg" alt="[公式]" eeimg="1" data-formula="x_n"> 附近，抛物线和原函数曲线基本重合。也就是说，可以用抛物线方程代替原函数方程，而抛物线导函数的零点即为 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_012.svg" alt="[公式]" eeimg="1" data-formula="x_{n+1}"> ：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_004.svg" alt="[公式]" eeimg="1" data-formula="f'(x)\approx f''(x_n)(x-x_n)+f'(x_n)=0\tag{3}"> </p><p>解得：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_017.svg" alt="[公式]" eeimg="1" data-formula="x_{n+1} = x_n-\frac{f'(x_n)}{f''(x_n)}\tag{4}"> </p><p>(4) 即为一维函数的牛顿法迭代公式。</p><p>对于高维函数，推导过程基于多元函数的泰勒展开： </p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_033.svg" alt="[公式]" eeimg="1" data-formula="f(x)\approx f(x_n)+\nabla f(x_n)^T\cdot(x-x_n)+\frac{1}{2}(x-x_n)^T\cdot H(x_n)\cdot (x-x_n)\tag{5}"></p><p>(5) 是用高维二次曲面在 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_019.svg" alt="[公式]" eeimg="1" data-formula="x_n"> 处逼近原函数，如下图所示（<a href="https://link.zhihu.com/?target=https%3A//www.cs.ccu.edu.tw/~wtchu/courses/2014s_OPT/Lectures/Chapter%25209%2520Newton%2527s%2520Method.pdf" class=" wrap external" target="_blank" rel="nofollow noreferrer" data-za-detail-view-id="1043">图片来源</a>）：</p><figure data-size="normal"><noscript><img src="https://pic3.zhimg.com/v2-f134427002d9434f5bce9f4541768142_b.jpg" data-caption="" data-size="normal" data-rawwidth="426" data-rawheight="272" class="origin_image zh-lightbox-thumb" width="426" data-original="https://pic3.zhimg.com/v2-f134427002d9434f5bce9f4541768142_r.jpg"/></noscript><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/v2-f134427002d9434f5bce9f4541768142_720w.jpg" data-caption="" data-size="normal" data-rawwidth="426" data-rawheight="272" class="origin_image zh-lightbox-thumb lazy" data-original="https://pic3.zhimg.com/v2-f134427002d9434f5bce9f4541768142_r.jpg" data-actualsrc="https://pic3.zhimg.com/v2-f134427002d9434f5bce9f4541768142_b.jpg" data-lazy-status="ok" width="426"></figure><p>令 (5) 对 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_007.svg" alt="[公式]" eeimg="1" data-formula="x"> 的导函数为 0：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_008.svg" alt="[公式]" eeimg="1" data-formula="\nabla f(x) = H(x_n)(x-x_n)+\nabla f(x_n)=0\tag{6}"> </p><p>解得：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_024.svg" alt="[公式]" eeimg="1" data-formula="x_{n+1}=x_{n}-H(x_n)^{-1}\nabla f(x_n)\tag{7}"> </p><p>牛顿法的优点是收敛速度快，缺点是需要求矩阵的逆，计算量比较大。此外，如果矩阵非正定（在一维情况下表现为泰勒展开的二阶导数小于0），极值点为极大值，而非极小值。如果初始位置离最优点太远，也会导致迭代过程中目标函数不严格递减的情况。</p><h2>高斯-牛顿法</h2><p>高斯-牛顿法是牛顿法的特例，用于解非线性最小二乘问题。</p><p>假设观测到 N 个数据点 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_034.svg" alt="[公式]" eeimg="1" data-formula="\{(x_1, y_1), (x_2, y_2), ..., (x_N, y_N)\}"> ，其中 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_027.svg" alt="[公式]" eeimg="1" data-formula="x\in\mathbb{R}^M"> 。希望找到包含 M 个参数的非线性函数 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_002.svg" alt="[公式]" eeimg="1" data-formula="f(x, a_1, a_2, ..., a_M)">，拟合上述N个数据点。 为了方便书写，记： <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_009.svg" alt="[公式]" eeimg="1" data-formula="f_1(\mathbf{a})=f(x_1, a_1, ... a_M)">。则最小二乘的目标函数为：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_038.svg" alt="[公式]" eeimg="1" data-formula="\varepsilon(\mathbf{a})= \sum_{i=1}^N \|f_i(\mathbf{a})-y_i\|^2\tag{8}"> </p><p>我们需要找到 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_023.svg" alt="[公式]" eeimg="1" data-formula="\mathbf{a}=[a_1, a_2, ..., a_M]^T"> ，使得 (8) 的值最小。</p><p>将 (8) 对 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_037.svg" alt="[公式]" eeimg="1" data-formula="\mathbf{a}_j"> 求导：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_028.svg" alt="[公式]" eeimg="1" data-formula="\frac{\partial\varepsilon(\mathbf{a})}{a_j}=\sum_{i=1}^N2(f_i(\mathbf{a})-y_i)\cdot \frac{\partial f_i(\mathbf{a})}{\partial a_j}\tag{9}"> </p><p>令：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_010.svg" alt="[公式]" eeimg="1" data-formula="J=\begin{bmatrix} \frac{\partial f_1(\mathbf{a})}{\partial a_1} & \frac{\partial f_1(\mathbf{a})}{\partial a_2} & ... & \frac{\partial f_1(\mathbf{a})}{\partial a_M} \\ \frac{\partial f_2(\mathbf{a})}{\partial a_1} & \frac{\partial f_2(\mathbf{a})}{\partial a_2} & ... & \frac{\partial f_2(\mathbf{a})}{\partial a_M} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_N(\mathbf{a})}{\partial a_1} & \frac{\partial f_N(\mathbf{a})}{\partial a_2} & ... & \frac{\partial f_N(\mathbf{a})}{\partial a_M} \\ \end{bmatrix}"> , <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation.svg" alt="[公式]" eeimg="1" data-formula="\mathbf{r}= \begin{bmatrix} f_1(\mathbf{a})-y_1 \\ f_2(\mathbf{a})-y_2 \\ \vdots f_N(\mathbf{a})-y_N \\ \end{bmatrix}"></p><p>(9) 可以写成向量形式：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_014.svg" alt="[公式]" eeimg="1" data-formula="\nabla\varepsilon(\mathbf{a})=2J^T\mathbf{r}\tag{10}"> </p><p>接下来求海森矩阵第 k 行 j 列的元素：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_025.svg" alt="[公式]" eeimg="1" data-formula="\begin{align*} \frac{\partial^2\varepsilon(\mathbf{a})}{\partial a_k \partial a_j}&=\frac{\partial}{\partial a_k}\frac{\partial \varepsilon(\mathbf{a})}{\partial a_j} \\ &=\frac{\partial}{\partial a_k}\left(\sum_{i=1}^N2(f_i(\mathbf{a})-y_i)\cdot\frac{\partial f_i(\mathbf{a})}{\partial a_j}\right) \\ &=2\sum_{i=1}^N\left(\frac{\partial f_i(\mathbf{a})}{\partial a_k}\frac{\partial f_i(\mathbf{a})}{\partial a_j}+(f_i(\mathbf{a})-y_i)\cdot \frac{\partial^2f_i(\mathbf{a})}{\partial a_k\partial a_j}\right) \end{align*}\tag{11}"> </p><p>令 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_020.svg" alt="[公式]" eeimg="1" data-formula="H"> 为 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_031.svg" alt="[公式]" eeimg="1" data-formula="\varepsilon(\mathbf{a})"> 的海森矩阵，由 (11) 可知：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_026.svg" alt="[公式]" eeimg="1" data-formula="H = 2\cdot (J^TJ + S)\tag{12}"> </p><p>其中：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_035.svg" alt="[公式]" eeimg="1" data-formula="S_{k,j}=\sum_{i=1}^N(f_i(\mathbf{a})-y_i)\cdot \frac{\partial^2f_i(\mathbf{a})}{\partial a_k\partial a_j}"> </p><p>把 (10) 和 (12) 带入 (7) 得：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_036.svg" alt="[公式]" eeimg="1" data-formula="\begin{align*} \mathbf{a}_{n+1}&=\mathbf{a}_n-H^{-1}\nabla\varepsilon(\mathbf{a}) \\ &=\mathbf{a}_n-(J^TJ+S)^{-1}\cdot J^T\mathbf{r} \end{align*}\tag{13}"> </p><p>很多时候 (13) 中的 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_005.svg" alt="[公式]" eeimg="1" data-formula="S"> 可以忽略，最终高斯-牛顿法的迭代公式为：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_015.svg" alt="[公式]" eeimg="1" data-formula="\mathbf{a}_{n+1}=\mathbf{a}_n-(J^TJ)^{-1}\cdot J^T\mathbf{r}\tag{14}"> </p><p class="ztext-empty-paragraph"><br></p><p>除了借助牛顿法之外，高斯牛顿法还可以直接通过多元函数的一阶泰勒展开推导：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_038.svg" alt="[公式]" eeimg="1" data-formula="\varepsilon(\mathbf{a})= \sum_{i=1}^N \|f_i(\mathbf{a})-y_i\|^2\tag{8}"> </p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_028.svg" alt="[公式]" eeimg="1" data-formula="\frac{\partial\varepsilon(\mathbf{a})}{a_j}=\sum_{i=1}^N2(f_i(\mathbf{a})-y_i)\cdot \frac{\partial f_i(\mathbf{a})}{\partial a_j}\tag{9}"> </p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_029.svg" alt="[公式]" eeimg="1" data-formula="\nabla\varepsilon(\mathbf{a})=2J^T \begin{bmatrix} (f_1(\mathbf{a})-y_1)\\ \vdots \\ (f_N(\mathbf{a})-y_N) \end{bmatrix}=2J^T\cdot (\vec{f}(\mathbf{a})-\vec{y})\tag{10}"> </p><p>我们希望找到 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_013.svg" alt="[公式]" eeimg="1" data-formula="\mathbf{a}_{n+1}"> 使得 (10) 为 0 。</p><p>根据多元函数的一阶泰勒展开： <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_003.svg" alt="[公式]" eeimg="1" data-formula="\vec{f}(\mathbf{a}_{n+1}) \approx \vec{f}(\mathbf{a}_{n})+J(\mathbf{a}_n)\cdot(\mathbf{a}_{n+1}-\mathbf{a}_n)"> ，(10) 可以变为（注意 (15) 的第一行括号外的 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_018.svg" alt="[公式]" eeimg="1" data-formula="J^T"> 仍然用 <img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_006.svg" alt="[公式]" eeimg="1" data-formula="J(\mathbf{a}_n)^T"> 近似）：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_032.svg" alt="[公式]" eeimg="1" data-formula="\begin{align*} \nabla\varepsilon(\mathbf{a}_{n+1})&=2J^T(\vec{f}(\mathbf{a}_{n+1})-\vec{y})\\ & \approx 2J^T(\vec{f}(\mathbf{a}_n)+J\cdot(\mathbf{a}_{n+1}-\mathbf{a}_n)-\vec{y})=0 \end{align*}\tag{15}"> </p><p>即：</p><p><img src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/equation_011.svg" alt="[公式]" eeimg="1" data-formula="\begin{align} J^TJ(\mathbf{a}_{n+1}-\mathbf{a}_n)&=-J^T(\vec{f}(\mathbf{a}_n)-\vec{y})\\ \mathbf{a}_{n+1}&=\mathbf{a}_n-(J^TJ)^{-1}J^T\cdot \mathbf{r} \end{align}\tag{16}"> </p><p>(16) 与 (14) 相同。</p><p class="ztext-empty-paragraph"><br></p><p class="ztext-empty-paragraph"><br></p><p>参考资料：</p><p>[1] <a href="https://link.zhihu.com/?target=http%3A//fourier.eng.hmc.edu/e176/lectures/NM/node36.html" class=" external" target="_blank" rel="nofollow noreferrer" data-za-detail-view-id="1043"><span class="invisible">http://</span><span class="visible">fourier.eng.hmc.edu/e17</span><span class="invisible">6/lectures/NM/node36.html</span><span class="ellipsis"></span></a></p><p>[2] <a href="https://link.zhihu.com/?target=https%3A//www.cs.ccu.edu.tw/~wtchu/courses/2014s_OPT/Lectures/Chapter%25209%2520Newton%2527s%2520Method.pdf" class=" external" target="_blank" rel="nofollow noreferrer" data-za-detail-view-id="1043"><span class="invisible">https://www.</span><span class="visible">cs.ccu.edu.tw/~wtchu/co</span><span class="invisible">urses/2014s_OPT/Lectures/Chapter%209%20Newton%27s%20Method.pdf</span><span class="ellipsis"></span></a></p><p>[3] <a href="https://link.zhihu.com/?target=https%3A//github.com/MarekKowalski/FaceSwap/tree/master/FaceSwap" class=" external" target="_blank" rel="nofollow noreferrer" data-za-detail-view-id="1043"><span class="invisible">https://</span><span class="visible">github.com/MarekKowalsk</span><span class="invisible">i/FaceSwap/tree/master/FaceSwap</span><span class="ellipsis"></span></a></p><p></p></div></div><div class="ContentItem-time">编辑于 2020-02-27</div><div class="Post-topicsAndReviewer"><div class="TopicList Post-Topics"><div class="Tag Topic" data-za-detail-view-path-module="TopicItem" data-za-extra-module="{"card":{"content":{"type":"Topic","token":"19602355"}}}"><span class="Tag-content"><a class="TopicLink" href="https://www.zhihu.com/topic/19602355" target="_blank"><div class="Popover"><div 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src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/5b34a6bb69c8aea509e62cfba14059b9_s.jpg" srcset="https://pic2.zhimg.com/5b34a6bb69c8aea509e62cfba14059b9_xs.jpg?source=06d4cd63 2x" alt="明天会更好" width="24" height="24"></a></div></div></span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/huang-liang-61">明天会更好</a></span><span class="CommentItemV2-time">2020-03-30</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext"><p>写的太好了。和网上其他资料不同，这个没有错误</p></div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 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ztext"><p>谢谢鼓励</p></div></div><div class="CommentItemV2-footer"><button type="button" disabled="disabled" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" fill-rule="evenodd"></path></svg></span>赞</button><button type="button" class="Button Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Recommend" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M6.433 14.285l-.918 5.355c-.219 1.274.532 1.82 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href="https://www.zhihu.com/people/ppxia-36">阿枣</a></span><span class="CommentItemV2-time">2020-05-01</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext">简单的说 高斯牛顿就是相当于将牛顿法求海赛矩阵的过程简化了？</div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" fill-rule="evenodd"></path></svg></span>赞</button><button type="button" class="Button CommentItemV2-hoverBtn 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class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/keratin-51">smellyCat</a></span><span class="CommentItemV2-time">2020-05-18</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext"><p>太棒了！感谢分享！！</p></div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" 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0 3.031-.3 4.382-.84l.42.42 2.125 2.118s.782.571 1.314 0-.074-1.305-.074-1.305L3.955 3.183s-.76-.742-1.385-.19c-.626.554 0 1.35 0 1.35zm4.963 4.96l1.55 1.552c-.05.21-.08.43-.08.65 0 1.66 1.341 3.001 3.002 3.001.22 0 .44-.03.65-.08l1.551 1.551c-.67.33-1.41.53-2.2.53a5.004 5.004 0 0 1-5.003-5.002c0-.79.2-1.53.53-2.201zm4.312-.78l3.151 3.152.02-.16c0-1.66-1.34-3.001-3.001-3.001l-.17.01z" opacity=".9" fill-rule="evenodd"></path></svg></span>折叠</button><button type="button" class="Button CommentItemV2-hoverBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="transform: rotate(180deg); margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 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srcset="https://pic2.zhimg.com/da8e974dc_xs.jpg?source=06d4cd63 2x" alt="幽雎" width="24" height="24"></a></div></div></span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/qiao-dong-jia-99">幽雎</a></span><span class="CommentItemV2-time">2020-05-18</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext">必须赞。</div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" fill-rule="evenodd"></path></svg></span>赞</button><button type="button" class="Button CommentItemV2-hoverBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Reply" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M22.959 17.22c-1.686-3.552-5.128-8.062-11.636-8.65-.539-.053-1.376-.436-1.376-1.561V4.678c0-.521-.635-.915-1.116-.521L1.469 10.67a1.506 1.506 0 0 0-.1 2.08s6.99 6.818 7.443 7.114c.453.295 1.136.124 1.135-.501V17a1.525 1.525 0 0 1 1.532-1.466c1.186-.139 7.597-.077 10.33 2.396 0 0 .396.257.536.257.892 0 .614-.967.614-.967z" fill-rule="evenodd"></path></svg></span>回复</button><button type="button" class="Button CommentItemV2-hoverBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Recommend" style="margin-right: 5px;" fill="currentColor" 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src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/v2-05bb4dae18c541bcdbd2e5d9b7979e78_s.jpg" srcset="https://pic2.zhimg.com/v2-05bb4dae18c541bcdbd2e5d9b7979e78_xs.jpg?source=06d4cd63 2x" alt="百把人" width="24" height="24"></a></div></div></span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/ru-shou-44">百把人</a></span><span class="CommentItemV2-time">2020-06-23</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext"><p>感谢，讲得太清楚了，拯救了我的考试周XD</p></div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 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aria-owns="Popover23-content"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/lu-yang-24-1"><img class="Avatar UserLink-avatar" src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/da8e974dc_s_002.jpg" srcset="https://pic1.zhimg.com/da8e974dc_xs.jpg?source=06d4cd63 2x" alt="阮不是无情的人" width="24" height="24"></a></div></div></span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/lu-yang-24-1">阮不是无情的人</a></span><span class="CommentItemV2-time">2020-07-13</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext">非常受用，让我可以区别开牛顿法跟高斯牛顿法的不同，还有共同之处。再次感谢</div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; 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class="Popover"><div id="Popover24-toggle" aria-haspopup="true" aria-expanded="false" aria-owns="Popover24-content"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/lu-yang-24-1"><img class="Avatar UserLink-avatar" src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/da8e974dc_s.jpg" srcset="https://pic2.zhimg.com/da8e974dc_xs.jpg?source=06d4cd63 2x" alt="阮不是无情的人" width="24" height="24"></a></div></div></span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/lu-yang-24-1">阮不是无情的人</a></span><span class="CommentItemV2-time">2020-07-14</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext"><p>有个问题就是，我现在想求一个函数的最小值，而不是优化参数。那这时候没有实验数据，我应该怎么办?是不是让实验数据的y等于0，让r = f(a)?</p></div></div><div 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class="CommentItemV2-reply">回复</span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/lu-yang-24-1">阮不是无情的人</a></span><span class="CommentItemV2-time">2020-07-14</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext"><p>高斯牛顿法只能优化公式 (8) 这种形式。求一般函数的最小值用牛顿法就好了。计算目标函数在 x_i 处的二阶泰勒展开的极小值点，将其作为 x_i+1，然后反复迭代</p></div></div><div class="CommentItemV2-footer"><button type="button" disabled="disabled" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" fill-rule="evenodd"></path></svg></span>赞</button><button type="button" class="Button Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Recommend" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M6.433 14.285l-.918 5.355c-.219 1.274.532 1.82 1.676 1.218l4.81-2.528 4.807 2.528c1.145.601 1.896.056 1.677-1.218l-.918-5.355 3.89-3.792c.925-.903.64-1.785-.64-1.97l-2.843-.414c-.538.583-4.195 4.547-5.757 6.12-.865.822-1.572.015-1.572.015l-1.92-1.906s-.405-.623.213-1.204c.62-.58 1.174-.193 1.174-.193l1.32 1.14 4.008-4.34-2.404-4.872c-.572-1.159-1.5-1.159-2.072 0L8.56 7.741l-5.376.781c-1.279.186-1.566 1.068-.64 1.971l3.89 3.792z" fill-rule="evenodd"></path></svg></span>推荐</button><button type="button" class="Button Button--plain"><span style="display: inline-flex; align-items: 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src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/da8e974dc_s_003.jpg" srcset="https://pic4.zhimg.com/da8e974dc_xs.jpg?source=06d4cd63 2x" alt="阮不是无情的人" width="24" height="24"></a></div></div></span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/lu-yang-24-1">阮不是无情的人</a></span><span class="CommentItemV2-reply">回复</span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/yin-xiang-nan">杨指北</a></span><span class="CommentItemV2-roleInfo"> (作者) </span><span class="CommentItemV2-time">2020-07-14</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext">非
常感谢回复。我们现在任务其实是利用levenberg-marquardt来求一个函数的最小值，LM方法就是高斯牛顿的改良。我发现r=f(a)-
y，如果此时假设y=0，此时r就是f(a)。利用这点是可以求最小值的。但是我们主要是不太明白其中的道理，为什么把y=0就可以变成求函数最小值问题
了。</div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" fill-rule="evenodd"></path></svg></span>赞</button><button type="button" class="Button CommentItemV2-hoverBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Reply" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M22.959 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data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/chen-guang-29-74">晨光</a></span><span class="CommentItemV2-time">2020-07-28</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext"><p>写的真好，真的学习了，对我帮助很大，感谢版主</p><p></p></div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" 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srcset="https://pic2.zhimg.com/v2-932c5653487733aff0990fa07373cd81_xs.jpg?source=06d4cd63 2x" alt="加加林" width="24" height="24"></a></div></div></span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/jia-jia-lin-68-55">加加林</a></span><span class="CommentItemV2-time">2020-08-10</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext"><p>写的真好，感谢楼主</p></div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" fill-rule="evenodd"></path></svg></span>赞</button><button type="button" class="Button CommentItemV2-hoverBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Reply" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M22.959 17.22c-1.686-3.552-5.128-8.062-11.636-8.65-.539-.053-1.376-.436-1.376-1.561V4.678c0-.521-.635-.915-1.116-.521L1.469 10.67a1.506 1.506 0 0 0-.1 2.08s6.99 6.818 7.443 7.114c.453.295 1.136.124 1.135-.501V17a1.525 1.525 0 0 1 1.532-1.466c1.186-.139 7.597-.077 10.33 2.396 0 0 .396.257.536.257.892 0 .614-.967.614-.967z" fill-rule="evenodd"></path></svg></span>回复</button><button type="button" class="Button CommentItemV2-hoverBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Recommend" 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UserLink-avatar" src="%E7%89%9B%E9%A1%BF%E6%B3%95%E5%92%8C%E9%AB%98%E6%96%AF-%E7%89%9B%E9%A1%BF%E6%B3%95%20-%20%E7%9F%A5%E4%B9%8E_files/da8e974dc_s_002.jpg" srcset="https://pic1.zhimg.com/da8e974dc_xs.jpg?source=06d4cd63 2x" alt="迷路的野猪" width="24" height="24"></a></div></div></span><span class="UserLink"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/mi-lu-de-ye-zhu-45">迷路的野猪</a></span><span class="CommentItemV2-time">2020-08-28</span></div><div class="CommentItemV2-metaSibling"><div class="CommentRichText CommentItemV2-content"><div class="RichText ztext"><p>应该只有残差r接近于零或者接近线性函数从而接近与零时，二阶信息项才可以忽略。通常称为“小残量问题”，否则高斯牛顿法不收敛。</p></div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path 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-H\g里H求逆一样不稳定，尤其对于病态问题。另外如果初始值离最优解太远的话，收敛很慢。<br>我个人的经验，一般最小二乘问题用Levenberg-Marquardt算法，这是一个类似信赖域的算法，迭代步s = -(H + λI)\g，在H对角元素上添加阻尼系数使算法介于最速下降和牛顿法之间。<br>如果要对变量做不等式约束（边界约束），个人认为最靠谱的是LM结合对数障碍函数法，病态问题边界约束无压力。其它什么信赖域反射之类的，只能实现满秩问题的不等式约束，不能应用在病态问题中。<br>对于变量线性等式约束，看下Convex Optimization一书，里面的可行牛顿方向法（全名忘了，feasible Newton什么什么的）可用，稍微做下改变，用到LM里，那么你就可以稳定地实现满足变量线性等式约束及边界约束的最小二乘了。</div></div><div class="CommentItemV2-footer"><button type="button" class="Button CommentItemV2-likeBtn Button--plain"><span style="display: inline-flex; align-items: center;"><svg class="Zi Zi--Like" style="margin-right: 5px;" fill="currentColor" viewBox="0 0 24 24" width="16" height="16"><path d="M14.445 9h5.387s2.997.154 1.95 3.669c-.168.51-2.346 6.911-2.346 6.911s-.763 1.416-2.86 1.416H8.989c-1.498 0-2.005-.896-1.989-2v-7.998c0-.987.336-2.032 1.114-2.639 4.45-3.773 3.436-4.597 4.45-5.83.985-1.13 3.2-.5 3.037 2.362C15.201 7.397 14.445 9 14.445 9zM3 9h2a1 1 0 0 1 1 1v10a1 1 0 0 1-1 1H3a1 1 0 0 1-1-1V10a1 1 0 0 1 1-1z" 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2019」即可获得"}}},"questions":{},"answers":{},"articles":{"103724149":{"trackUrl":["https:\u002F\u002Fsugar.zhihu.com\u002Fplutus_adreaper\u002Fpage_monitor_log?si=__SESSIONID__&ti=__ATOKEN__&at=view&pf=__OS__&ed=BiBUKF0xBSkqGGFQA2B_AHGhHcvz4B4D&idfa=__IDFA__&imei=__IMEI__&androidid=__ANDROIDID__&oaid=__OAID__&ci=__CREATIVEID__&zid=__ZONEID__"],"id":103724149,"title":"牛顿法和高斯-牛顿法","type":"article","articleType":"normal","excerptTitle":"","url":"https:\u002F\u002Fzhuanlan.zhihu.com\u002Fp\u002F103724149","imageUrl":"https:\u002F\u002Fpic4.zhimg.com\u002Fv2-b8699cab4602e742602c49daf7ddedf4_720w.jpg?source=172ae18b","titleImage":"https:\u002F\u002Fpic4.zhimg.com\u002Fv2-b8699cab4602e742602c49daf7ddedf4_720w.jpg?source=172ae18b","excerpt":"\u003Cimg src=\"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-688d9da69e93d086c10407eec06c3ab8_200x112.png\" data-caption=\"\" data-size=\"normal\" data-rawwidth=\"455\" data-rawheight=\"392\" data-watermark=\"watermark\" data-original-src=\"v2-688d9da69e93d086c10407eec06c3ab8\" data-watermark-src=\"v2-d37a40a3a45281a740c4d3774510366c\" data-private-watermark-src=\"\" data-original=\"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-688d9da69e93d086c10407eec06c3ab8_r.png\" class=\"origin_image inline-img zh-lightbox-thumb\"\u002F\u003E看代码的过程中遇到了高斯-牛顿法，感慨于自己作为调包侠，对各种最优化方法知之甚少，于是学习了一下这个算法。费曼技巧推崇以教促学，遂在此写一个简单的教程，希望整理思路以备复习的同时，也能帮助到和我一样的初学者。 本文所涉算法的研究目标是寻找多…","created":1580082348,"updated":1582815643,"author":{"isFollowed":false,"avatarUrlTemplate":"https:\u002F\u002Fpic1.zhimg.com\u002Fbf28449bb.jpg?source=172ae18b","uid":"28236180357120","userType":"people","isFollowing":false,"urlToken":"yin-xiang-nan","id":"2bcb30744c0f014ce6702f4d94922945","description":"","name":"杨指北","isAdvertiser":false,"headline":"无业游民","gender":1,"url":"\u002Fpeople\u002F2bcb30744c0f014ce6702f4d94922945","avatarUrl":"https:\u002F\u002Fpic1.zhimg.com\u002Fbf28449bb_l.jpg?source=172ae18b","isOrg":false,"type":"people","vipInfo":{"isVip":true,"vipIcon":{"url":"https:\u002F\u002Fpic2.zhimg.com\u002Fv2-4812630bc27d642f7cafcd6cdeca3d7a_r.png?source=172ae18b","nightModeUrl":"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-c9686ff064ea3579730756ac6c289978_r.png?source=172ae18b"}},"levelInfo":{"exp":150802,"level":8,"nicknameColor":{"color":"","nightModeColor":""},"levelIcon":"https:\u002F\u002Fpic3.zhimg.com\u002Fv2-ce2d9b858d70d9e107f14c57a0ceb563_l.png","iconInfo":{"url":"https:\u002F\u002Fpic3.zhimg.com\u002Fv2-ce2d9b858d70d9e107f14c57a0ceb563_l.png","nightModeUrl":"https:\u002F\u002Fpic2.zhimg.com\u002Fv2-2c1a38e54e53ce62f8b7bc346a043c61_l.png","width":93,"height":51}},"badge":[],"badgeV2":{"title":"","mergedBadges":[],"detailBadges":[],"icon":"","nightIcon":""},"exposedMedal":{"medalId":"1124316222665379841","medalName":"我的知乎 2019","avatarUrl":"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-2592b0b52e1fac99f69b38e00252413b_r.png?source=172ae18b","miniAvatarUrl":"https:\u002F\u002Fpic4.zhimg.com\u002Fv2-ad363cc3088dc8de7544fd08b1c4987a_l.png?source=172ae18b","description":"参与「我的知乎 2019」即可获得"}},"commentPermission":"all","copyrightPermission":"need_review","state":"published","imageWidth":426,"imageHeight":272,"content":"\u003Cp\u003E看代码的过程中遇到了高斯-牛顿法，感慨于自己作为调包侠，对各种最优化方法知之甚少，于是学习了一下这个算法。费曼技巧推崇以教促学，遂在此写一个简单的教程，希望整理思路以备复习的同时，也能帮助到和我一样的初学者。\u003C\u002Fp\u003E\u003Cp\u003E本文所涉算法的研究目标是寻找多元二阶可微函数 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%3A%5Cmathbb%7BR%7D%5EM%5Crightarrow%5Cmathbb%7BR%7D\" alt=\"f:\\mathbb{R}^M\\rightarrow\\mathbb{R}\" eeimg=\"1\"\u002F\u003E 的极小值点，即:\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x%5E%2A%3D%5Cunderset%7Bx%5Cin%5Cmathbb%7BR%7D%5EM%7D%7B%5Carg%5Cmin%7Df%28x%29%5Ctag%7B1%7D\" alt=\"x^*=\\underset{x\\in\\mathbb{R}^M}{\\arg\\min}f(x)\\tag{1}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E本文首先介绍牛顿法（也称作 Newton-Raphson 方法），然后介绍高斯-牛顿法。\u003C\u002Fp\u003E\u003Ch2\u003E牛顿法\u003C\u002Fh2\u003E\u003Cp\u003E牛顿法的推导基于二阶可微函数的泰勒展开，以一维函数为例:\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28x%29%5Capprox+f%28x_n%29%2Bf%27%28x_n%29%28x-x_n%29%2B%5Cfrac%7B1%7D%7B2%7Df%27%27%28x_n%29%28x-x_n%29%5E2+%5Ctag%7B2%7D\" alt=\"f(x)\\approx f(x_n)+f'(x_n)(x-x_n)+\\frac{1}{2}f''(x_n)(x-x_n)^2 \\tag{2}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E(2) 式是对 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28x%29\" alt=\"f(x)\" eeimg=\"1\"\u002F\u003E 在 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x_n\" alt=\"x_n\" eeimg=\"1\"\u002F\u003E 处的二阶近似。如下图所示：\u003C\u002Fp\u003E\u003Cfigure data-size=\"normal\"\u003E\u003Cnoscript\u003E\u003Cimg src=\"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-d37a40a3a45281a740c4d3774510366c_b.jpg\" data-caption=\"\" data-size=\"normal\" data-rawwidth=\"455\" data-rawheight=\"392\" class=\"origin_image zh-lightbox-thumb\" width=\"455\" data-original=\"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-d37a40a3a45281a740c4d3774510366c_r.jpg\"\u002F\u003E\u003C\u002Fnoscript\u003E\u003Cimg src=\"data:image\u002Fsvg+xml;utf8,<svg xmlns='http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg' width='455' height='392'><\u002Fsvg>\" data-caption=\"\" data-size=\"normal\" data-rawwidth=\"455\" data-rawheight=\"392\" class=\"origin_image zh-lightbox-thumb lazy\" width=\"455\" data-original=\"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-d37a40a3a45281a740c4d3774510366c_r.jpg\" data-actualsrc=\"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-d37a40a3a45281a740c4d3774510366c_b.jpg\"\u002F\u003E\u003C\u002Ffigure\u003E\u003Cp\u003E蓝色曲线代表原函数 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28x%29\" alt=\"f(x)\" eeimg=\"1\"\u002F\u003E ，绿色圆点代表当前点 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x_n\" alt=\"x_n\" eeimg=\"1\"\u002F\u003E ，橘黄色的抛物线就是原函数在 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x_n\" alt=\"x_n\" eeimg=\"1\"\u002F\u003E 附近的泰勒展开。不难发现，在 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x_n\" alt=\"x_n\" eeimg=\"1\"\u002F\u003E 附近，抛物线和原函数曲线基本重合。也就是说，可以用抛物线方程代替原函数方程，而抛物线导函数的零点即为 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x_%7Bn%2B1%7D\" alt=\"x_{n+1}\" eeimg=\"1\"\u002F\u003E ：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%27%28x%29%5Capprox+f%27%27%28x_n%29%28x-x_n%29%2Bf%27%28x_n%29%3D0%5Ctag%7B3%7D\" alt=\"f'(x)\\approx f''(x_n)(x-x_n)+f'(x_n)=0\\tag{3}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E解得：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x_%7Bn%2B1%7D+%3D+x_n-%5Cfrac%7Bf%27%28x_n%29%7D%7Bf%27%27%28x_n%29%7D%5Ctag%7B4%7D\" alt=\"x_{n+1} = x_n-\\frac{f'(x_n)}{f''(x_n)}\\tag{4}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E(4) 即为一维函数的牛顿法迭代公式。\u003C\u002Fp\u003E\u003Cp\u003E对于高维函数，推导过程基于多元函数的泰勒展开： \u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28x%29%5Capprox+f%28x_n%29%2B%5Cnabla+f%28x_n%29%5ET%5Ccdot%28x-x_n%29%2B%5Cfrac%7B1%7D%7B2%7D%28x-x_n%29%5ET%5Ccdot+H%28x_n%29%5Ccdot+%28x-x_n%29%5Ctag%7B5%7D\" alt=\"f(x)\\approx f(x_n)+\\nabla f(x_n)^T\\cdot(x-x_n)+\\frac{1}{2}(x-x_n)^T\\cdot H(x_n)\\cdot (x-x_n)\\tag{5}\" eeimg=\"1\"\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E(5) 是用高维二次曲面在 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x_n\" alt=\"x_n\" eeimg=\"1\"\u002F\u003E 处逼近原函数，如下图所示（\u003Ca href=\"https:\u002F\u002Flink.zhihu.com\u002F?target=https%3A\u002F\u002Fwww.cs.ccu.edu.tw\u002F~wtchu\u002Fcourses\u002F2014s_OPT\u002FLectures\u002FChapter%25209%2520Newton%2527s%2520Method.pdf\" class=\" wrap external\" target=\"_blank\" rel=\"nofollow noreferrer\"\u003E图片来源\u003C\u002Fa\u003E）：\u003C\u002Fp\u003E\u003Cfigure data-size=\"normal\"\u003E\u003Cnoscript\u003E\u003Cimg src=\"https:\u002F\u002Fpic3.zhimg.com\u002Fv2-f134427002d9434f5bce9f4541768142_b.jpg\" data-caption=\"\" data-size=\"normal\" data-rawwidth=\"426\" data-rawheight=\"272\" class=\"origin_image zh-lightbox-thumb\" width=\"426\" data-original=\"https:\u002F\u002Fpic3.zhimg.com\u002Fv2-f134427002d9434f5bce9f4541768142_r.jpg\"\u002F\u003E\u003C\u002Fnoscript\u003E\u003Cimg src=\"data:image\u002Fsvg+xml;utf8,<svg xmlns='http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg' width='426' height='272'><\u002Fsvg>\" data-caption=\"\" data-size=\"normal\" data-rawwidth=\"426\" data-rawheight=\"272\" class=\"origin_image zh-lightbox-thumb lazy\" width=\"426\" data-original=\"https:\u002F\u002Fpic3.zhimg.com\u002Fv2-f134427002d9434f5bce9f4541768142_r.jpg\" data-actualsrc=\"https:\u002F\u002Fpic3.zhimg.com\u002Fv2-f134427002d9434f5bce9f4541768142_b.jpg\"\u002F\u003E\u003C\u002Ffigure\u003E\u003Cp\u003E令 (5) 对 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x\" alt=\"x\" eeimg=\"1\"\u002F\u003E 的导函数为 0：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cnabla+f%28x%29+%3D+H%28x_n%29%28x-x_n%29%2B%5Cnabla+f%28x_n%29%3D0%5Ctag%7B6%7D\" alt=\"\\nabla f(x) = H(x_n)(x-x_n)+\\nabla f(x_n)=0\\tag{6}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E解得：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x_%7Bn%2B1%7D%3Dx_%7Bn%7D-H%28x_n%29%5E%7B-1%7D%5Cnabla+f%28x_n%29%5Ctag%7B7%7D\" alt=\"x_{n+1}=x_{n}-H(x_n)^{-1}\\nabla f(x_n)\\tag{7}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E牛顿法的优点是收敛速度快，缺点是需要求矩阵的逆，计算量比较大。此外，如果矩阵非正定（在一维情况下表现为泰勒展开的二阶导数小于0），极值点为极大值，而非极小值。如果初始位置离最优点太远，也会导致迭代过程中目标函数不严格递减的情况。\u003C\u002Fp\u003E\u003Ch2\u003E高斯-牛顿法\u003C\u002Fh2\u003E\u003Cp\u003E高斯-牛顿法是牛顿法的特例，用于解非线性最小二乘问题。\u003C\u002Fp\u003E\u003Cp\u003E假设观测到 N 个数据点 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5C%7B%28x_1%2C+y_1%29%2C+%28x_2%2C+y_2%29%2C+...%2C+%28x_N%2C+y_N%29%5C%7D\" alt=\"\\{(x_1, y_1), (x_2, y_2), ..., (x_N, y_N)\\}\" eeimg=\"1\"\u002F\u003E ，其中 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=x%5Cin%5Cmathbb%7BR%7D%5EM\" alt=\"x\\in\\mathbb{R}^M\" eeimg=\"1\"\u002F\u003E 。希望找到包含 M 个参数的非线性函数 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28x%2C+a_1%2C+a_2%2C+...%2C+a_M%29\" alt=\"f(x, a_1, a_2, ..., a_M)\" eeimg=\"1\"\u002F\u003E，拟合上述N个数据点。 为了方便书写，记： \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f_1%28%5Cmathbf%7Ba%7D%29%3Df%28x_1%2C+a_1%2C+...+a_M%29\" alt=\"f_1(\\mathbf{a})=f(x_1, a_1, ... a_M)\" eeimg=\"1\"\u002F\u003E。则最小二乘的目标函数为：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29%3D+%5Csum_%7Bi%3D1%7D%5EN+%5C%7Cf_i%28%5Cmathbf%7Ba%7D%29-y_i%5C%7C%5E2%5Ctag%7B8%7D\" alt=\"\\varepsilon(\\mathbf{a})= \\sum_{i=1}^N \\|f_i(\\mathbf{a})-y_i\\|^2\\tag{8}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E我们需要找到 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cmathbf%7Ba%7D%3D%5Ba_1%2C+a_2%2C+...%2C+a_M%5D%5ET\" alt=\"\\mathbf{a}=[a_1, a_2, ..., a_M]^T\" eeimg=\"1\"\u002F\u003E ，使得 (8) 的值最小。\u003C\u002Fp\u003E\u003Cp\u003E将 (8) 对 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cmathbf%7Ba%7D_j\" alt=\"\\mathbf{a}_j\" eeimg=\"1\"\u002F\u003E 求导：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cfrac%7B%5Cpartial%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29%7D%7Ba_j%7D%3D%5Csum_%7Bi%3D1%7D%5EN2%28f_i%28%5Cmathbf%7Ba%7D%29-y_i%29%5Ccdot+%5Cfrac%7B%5Cpartial+f_i%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_j%7D%5Ctag%7B9%7D\" alt=\"\\frac{\\partial\\varepsilon(\\mathbf{a})}{a_j}=\\sum_{i=1}^N2(f_i(\\mathbf{a})-y_i)\\cdot \\frac{\\partial f_i(\\mathbf{a})}{\\partial a_j}\\tag{9}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E令：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=J%3D%5Cbegin%7Bbmatrix%7D+%5Cfrac%7B%5Cpartial+f_1%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_1%7D+%26+%5Cfrac%7B%5Cpartial+f_1%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_2%7D+%26+...+%26+%5Cfrac%7B%5Cpartial+f_1%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_M%7D+%5C%5C+%5Cfrac%7B%5Cpartial+f_2%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_1%7D+%26+%5Cfrac%7B%5Cpartial+f_2%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_2%7D+%26+...+%26+%5Cfrac%7B%5Cpartial+f_2%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_M%7D+%5C%5C+%5Cvdots++%26+%5Cvdots++%26+%5Cddots+%26+%5Cvdots+++%5C%5C+%5Cfrac%7B%5Cpartial+f_N%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_1%7D+%26+%5Cfrac%7B%5Cpartial+f_N%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_2%7D+%26+...+%26+%5Cfrac%7B%5Cpartial+f_N%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_M%7D+%5C%5C+%5Cend%7Bbmatrix%7D\" alt=\"J=\\begin{bmatrix} \\frac{\\partial f_1(\\mathbf{a})}{\\partial a_1} & \\frac{\\partial f_1(\\mathbf{a})}{\\partial a_2} & ... & \\frac{\\partial f_1(\\mathbf{a})}{\\partial a_M} \\\\ \\frac{\\partial f_2(\\mathbf{a})}{\\partial a_1} & \\frac{\\partial f_2(\\mathbf{a})}{\\partial a_2} & ... & \\frac{\\partial f_2(\\mathbf{a})}{\\partial a_M} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\frac{\\partial f_N(\\mathbf{a})}{\\partial a_1} & \\frac{\\partial f_N(\\mathbf{a})}{\\partial a_2} & ... & \\frac{\\partial f_N(\\mathbf{a})}{\\partial a_M} \\\\ \\end{bmatrix}\" eeimg=\"1\"\u002F\u003E , \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cmathbf%7Br%7D%3D+%5Cbegin%7Bbmatrix%7D+f_1%28%5Cmathbf%7Ba%7D%29-y_1+%5C%5C+f_2%28%5Cmathbf%7Ba%7D%29-y_2+%5C%5C+%5Cvdots+f_N%28%5Cmathbf%7Ba%7D%29-y_N+%5C%5C+%5Cend%7Bbmatrix%7D\" alt=\"\\mathbf{r}= \\begin{bmatrix} f_1(\\mathbf{a})-y_1 \\\\ f_2(\\mathbf{a})-y_2 \\\\ \\vdots f_N(\\mathbf{a})-y_N \\\\ \\end{bmatrix}\" eeimg=\"1\"\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E(9) 可以写成向量形式：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cnabla%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29%3D2J%5ET%5Cmathbf%7Br%7D%5Ctag%7B10%7D\" alt=\"\\nabla\\varepsilon(\\mathbf{a})=2J^T\\mathbf{r}\\tag{10}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E接下来求海森矩阵第 k 行 j 列的元素：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cbegin%7Balign%2A%7D+%5Cfrac%7B%5Cpartial%5E2%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_k+%5Cpartial+a_j%7D%26%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+a_k%7D%5Cfrac%7B%5Cpartial+%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_j%7D+%5C%5C+%26%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+a_k%7D%5Cleft%28%5Csum_%7Bi%3D1%7D%5EN2%28f_i%28%5Cmathbf%7Ba%7D%29-y_i%29%5Ccdot%5Cfrac%7B%5Cpartial+f_i%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_j%7D%5Cright%29+%5C%5C+%26%3D2%5Csum_%7Bi%3D1%7D%5EN%5Cleft%28%5Cfrac%7B%5Cpartial+f_i%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_k%7D%5Cfrac%7B%5Cpartial+f_i%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_j%7D%2B%28f_i%28%5Cmathbf%7Ba%7D%29-y_i%29%5Ccdot+%5Cfrac%7B%5Cpartial%5E2f_i%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_k%5Cpartial+a_j%7D%5Cright%29+%5Cend%7Balign%2A%7D%5Ctag%7B11%7D\" alt=\"\\begin{align*} \\frac{\\partial^2\\varepsilon(\\mathbf{a})}{\\partial a_k \\partial a_j}&=\\frac{\\partial}{\\partial a_k}\\frac{\\partial \\varepsilon(\\mathbf{a})}{\\partial a_j} \\\\ &=\\frac{\\partial}{\\partial a_k}\\left(\\sum_{i=1}^N2(f_i(\\mathbf{a})-y_i)\\cdot\\frac{\\partial f_i(\\mathbf{a})}{\\partial a_j}\\right) \\\\ &=2\\sum_{i=1}^N\\left(\\frac{\\partial f_i(\\mathbf{a})}{\\partial a_k}\\frac{\\partial f_i(\\mathbf{a})}{\\partial a_j}+(f_i(\\mathbf{a})-y_i)\\cdot \\frac{\\partial^2f_i(\\mathbf{a})}{\\partial a_k\\partial a_j}\\right) \\end{align*}\\tag{11}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E令 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=H\" alt=\"H\" eeimg=\"1\"\u002F\u003E 为 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29\" alt=\"\\varepsilon(\\mathbf{a})\" eeimg=\"1\"\u002F\u003E 的海森矩阵，由 (11) 可知：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=H+%3D+2%5Ccdot+%28J%5ETJ+%2B+S%29%5Ctag%7B12%7D\" alt=\"H = 2\\cdot (J^TJ + S)\\tag{12}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E其中：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=S_%7Bk%2Cj%7D%3D%5Csum_%7Bi%3D1%7D%5EN%28f_i%28%5Cmathbf%7Ba%7D%29-y_i%29%5Ccdot+%5Cfrac%7B%5Cpartial%5E2f_i%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_k%5Cpartial+a_j%7D\" alt=\"S_{k,j}=\\sum_{i=1}^N(f_i(\\mathbf{a})-y_i)\\cdot \\frac{\\partial^2f_i(\\mathbf{a})}{\\partial a_k\\partial a_j}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E把 (10) 和 (12) 带入 (7) 得：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cbegin%7Balign%2A%7D+%5Cmathbf%7Ba%7D_%7Bn%2B1%7D%26%3D%5Cmathbf%7Ba%7D_n-H%5E%7B-1%7D%5Cnabla%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29+%5C%5C+%26%3D%5Cmathbf%7Ba%7D_n-%28J%5ETJ%2BS%29%5E%7B-1%7D%5Ccdot+J%5ET%5Cmathbf%7Br%7D+%5Cend%7Balign%2A%7D%5Ctag%7B13%7D\" alt=\"\\begin{align*} \\mathbf{a}_{n+1}&=\\mathbf{a}_n-H^{-1}\\nabla\\varepsilon(\\mathbf{a}) \\\\ &=\\mathbf{a}_n-(J^TJ+S)^{-1}\\cdot J^T\\mathbf{r} \\end{align*}\\tag{13}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E很多时候 (13) 中的 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=S\" alt=\"S\" eeimg=\"1\"\u002F\u003E 可以忽略，最终高斯-牛顿法的迭代公式为：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cmathbf%7Ba%7D_%7Bn%2B1%7D%3D%5Cmathbf%7Ba%7D_n-%28J%5ETJ%29%5E%7B-1%7D%5Ccdot+J%5ET%5Cmathbf%7Br%7D%5Ctag%7B14%7D\" alt=\"\\mathbf{a}_{n+1}=\\mathbf{a}_n-(J^TJ)^{-1}\\cdot J^T\\mathbf{r}\\tag{14}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E除了借助牛顿法之外，高斯牛顿法还可以直接通过多元函数的一阶泰勒展开推导：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29%3D+%5Csum_%7Bi%3D1%7D%5EN+%5C%7Cf_i%28%5Cmathbf%7Ba%7D%29-y_i%5C%7C%5E2%5Ctag%7B8%7D\" alt=\"\\varepsilon(\\mathbf{a})= \\sum_{i=1}^N \\|f_i(\\mathbf{a})-y_i\\|^2\\tag{8}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cfrac%7B%5Cpartial%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29%7D%7Ba_j%7D%3D%5Csum_%7Bi%3D1%7D%5EN2%28f_i%28%5Cmathbf%7Ba%7D%29-y_i%29%5Ccdot+%5Cfrac%7B%5Cpartial+f_i%28%5Cmathbf%7Ba%7D%29%7D%7B%5Cpartial+a_j%7D%5Ctag%7B9%7D\" alt=\"\\frac{\\partial\\varepsilon(\\mathbf{a})}{a_j}=\\sum_{i=1}^N2(f_i(\\mathbf{a})-y_i)\\cdot \\frac{\\partial f_i(\\mathbf{a})}{\\partial a_j}\\tag{9}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cnabla%5Cvarepsilon%28%5Cmathbf%7Ba%7D%29%3D2J%5ET+%5Cbegin%7Bbmatrix%7D+%28f_1%28%5Cmathbf%7Ba%7D%29-y_1%29%5C%5C+%5Cvdots+%5C%5C+%28f_N%28%5Cmathbf%7Ba%7D%29-y_N%29+%5Cend%7Bbmatrix%7D%3D2J%5ET%5Ccdot+%28%5Cvec%7Bf%7D%28%5Cmathbf%7Ba%7D%29-%5Cvec%7By%7D%29%5Ctag%7B10%7D\" alt=\"\\nabla\\varepsilon(\\mathbf{a})=2J^T \\begin{bmatrix} (f_1(\\mathbf{a})-y_1)\\\\ \\vdots \\\\ (f_N(\\mathbf{a})-y_N) \\end{bmatrix}=2J^T\\cdot (\\vec{f}(\\mathbf{a})-\\vec{y})\\tag{10}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E我们希望找到 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cmathbf%7Ba%7D_%7Bn%2B1%7D\" alt=\"\\mathbf{a}_{n+1}\" eeimg=\"1\"\u002F\u003E 使得 (10) 为 0 。\u003C\u002Fp\u003E\u003Cp\u003E根据多元函数的一阶泰勒展开： \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cvec%7Bf%7D%28%5Cmathbf%7Ba%7D_%7Bn%2B1%7D%29+%5Capprox+%5Cvec%7Bf%7D%28%5Cmathbf%7Ba%7D_%7Bn%7D%29%2BJ%28%5Cmathbf%7Ba%7D_n%29%5Ccdot%28%5Cmathbf%7Ba%7D_%7Bn%2B1%7D-%5Cmathbf%7Ba%7D_n%29\" alt=\"\\vec{f}(\\mathbf{a}_{n+1}) \\approx \\vec{f}(\\mathbf{a}_{n})+J(\\mathbf{a}_n)\\cdot(\\mathbf{a}_{n+1}-\\mathbf{a}_n)\" eeimg=\"1\"\u002F\u003E ，(10) 可以变为（注意 (15) 的第一行括号外的 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=J%5ET\" alt=\"J^T\" eeimg=\"1\"\u002F\u003E 仍然用 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=J%28%5Cmathbf%7Ba%7D_n%29%5ET\" alt=\"J(\\mathbf{a}_n)^T\" eeimg=\"1\"\u002F\u003E 近似）：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cbegin%7Balign%2A%7D+%5Cnabla%5Cvarepsilon%28%5Cmathbf%7Ba%7D_%7Bn%2B1%7D%29%26%3D2J%5ET%28%5Cvec%7Bf%7D%28%5Cmathbf%7Ba%7D_%7Bn%2B1%7D%29-%5Cvec%7By%7D%29%5C%5C+%26+%5Capprox+2J%5ET%28%5Cvec%7Bf%7D%28%5Cmathbf%7Ba%7D_n%29%2BJ%5Ccdot%28%5Cmathbf%7Ba%7D_%7Bn%2B1%7D-%5Cmathbf%7Ba%7D_n%29-%5Cvec%7By%7D%29%3D0+%5Cend%7Balign%2A%7D%5Ctag%7B15%7D\" alt=\"\\begin{align*} \\nabla\\varepsilon(\\mathbf{a}_{n+1})&=2J^T(\\vec{f}(\\mathbf{a}_{n+1})-\\vec{y})\\\\ & \\approx 2J^T(\\vec{f}(\\mathbf{a}_n)+J\\cdot(\\mathbf{a}_{n+1}-\\mathbf{a}_n)-\\vec{y})=0 \\end{align*}\\tag{15}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E即：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cbegin%7Balign%7D+J%5ETJ%28%5Cmathbf%7Ba%7D_%7Bn%2B1%7D-%5Cmathbf%7Ba%7D_n%29%26%3D-J%5ET%28%5Cvec%7Bf%7D%28%5Cmathbf%7Ba%7D_n%29-%5Cvec%7By%7D%29%5C%5C+%5Cmathbf%7Ba%7D_%7Bn%2B1%7D%26%3D%5Cmathbf%7Ba%7D_n-%28J%5ETJ%29%5E%7B-1%7DJ%5ET%5Ccdot+%5Cmathbf%7Br%7D+%5Cend%7Balign%7D%5Ctag%7B16%7D\" alt=\"\\begin{align} J^TJ(\\mathbf{a}_{n+1}-\\mathbf{a}_n)&=-J^T(\\vec{f}(\\mathbf{a}_n)-\\vec{y})\\\\ \\mathbf{a}_{n+1}&=\\mathbf{a}_n-(J^TJ)^{-1}J^T\\cdot \\mathbf{r} \\end{align}\\tag{16}\" eeimg=\"1\"\u002F\u003E \u003C\u002Fp\u003E\u003Cp\u003E(16) 与 (14) 相同。\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E参考资料：\u003C\u002Fp\u003E\u003Cp\u003E[1] \u003Ca href=\"https:\u002F\u002Flink.zhihu.com\u002F?target=http%3A\u002F\u002Ffourier.eng.hmc.edu\u002Fe176\u002Flectures\u002FNM\u002Fnode36.html\" class=\" external\" target=\"_blank\" rel=\"nofollow noreferrer\"\u003E\u003Cspan class=\"invisible\"\u003Ehttp:\u002F\u002F\u003C\u002Fspan\u003E\u003Cspan class=\"visible\"\u003Efourier.eng.hmc.edu\u002Fe17\u003C\u002Fspan\u003E\u003Cspan class=\"invisible\"\u003E6\u002Flectures\u002FNM\u002Fnode36.html\u003C\u002Fspan\u003E\u003Cspan class=\"ellipsis\"\u003E\u003C\u002Fspan\u003E\u003C\u002Fa\u003E\u003C\u002Fp\u003E\u003Cp\u003E[2] \u003Ca href=\"https:\u002F\u002Flink.zhihu.com\u002F?target=https%3A\u002F\u002Fwww.cs.ccu.edu.tw\u002F~wtchu\u002Fcourses\u002F2014s_OPT\u002FLectures\u002FChapter%25209%2520Newton%2527s%2520Method.pdf\" class=\" external\" target=\"_blank\" rel=\"nofollow noreferrer\"\u003E\u003Cspan class=\"invisible\"\u003Ehttps:\u002F\u002Fwww.\u003C\u002Fspan\u003E\u003Cspan class=\"visible\"\u003Ecs.ccu.edu.tw\u002F~wtchu\u002Fco\u003C\u002Fspan\u003E\u003Cspan class=\"invisible\"\u003Eurses\u002F2014s_OPT\u002FLectures\u002FChapter%209%20Newton%27s%20Method.pdf\u003C\u002Fspan\u003E\u003Cspan class=\"ellipsis\"\u003E\u003C\u002Fspan\u003E\u003C\u002Fa\u003E\u003C\u002Fp\u003E\u003Cp\u003E[3] \u003Ca href=\"https:\u002F\u002Flink.zhihu.com\u002F?target=https%3A\u002F\u002Fgithub.com\u002FMarekKowalski\u002FFaceSwap\u002Ftree\u002Fmaster\u002FFaceSwap\" class=\" external\" target=\"_blank\" rel=\"nofollow noreferrer\"\u003E\u003Cspan class=\"invisible\"\u003Ehttps:\u002F\u002F\u003C\u002Fspan\u003E\u003Cspan class=\"visible\"\u003Egithub.com\u002FMarekKowalsk\u003C\u002Fspan\u003E\u003Cspan class=\"invisible\"\u003Ei\u002FFaceSwap\u002Ftree\u002Fmaster\u002FFaceSwap\u003C\u002Fspan\u003E\u003Cspan class=\"ellipsis\"\u003E\u003C\u002Fspan\u003E\u003C\u002Fa\u003E\u003C\u002Fp\u003E\u003Cp\u003E\u003C\u002Fp\u003E","adminClosedComment":false,"topics":[{"url":"https:\u002F\u002Fwww.zhihu.com\u002Fapi\u002Fv4\u002Ftopics\u002F19602355","type":"topic","id":"19602355","name":"凸优化"},{"url":"https:\u002F\u002Fwww.zhihu.com\u002Fapi\u002Fv4\u002Ftopics\u002F19616819","type":"topic","id":"19616819","name":"最优化"}],"voteupCount":69,"voting":0,"commentCount":29,"contributions":[],"isTitleImageFullScreen":false,"upvotedFollowees":[],"commercialInfo":{"isCommercial":false,"plugin":{}},"suggestEdit":{"status":false,"reason":"","tip":"","url":"","title":""},"reason":"","annotationAction":[],"canTip":false,"canOpenTipjar":false,"tipjarorsCount":0,"isLabeled":false,"hasPublishingDraft":false,"isFavorited":false,"favlistsCount":67,"isNormal":true,"status":0,"activityToppingInfo":{"state":"untopped"},"shareText":"牛顿法和高斯-牛顿法 - 来自知乎专栏，作者: 杨指北 https:\u002F\u002Fzhuanlan.zhihu.com\u002Fp\u002F103724149 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