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|
||
<span class="md-ellipsis">
|
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|
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函数逼近
|
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|
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|
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</span>
|
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|
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</a>
|
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优化
|
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|
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|
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</span>
|
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|
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|
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</a>
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抽样
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推断
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概率论
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计数
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</a>
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概率概念
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分布
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</a>
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贝叶斯
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</a>
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信息论
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机器学习
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机器学习
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深度学习
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</span>
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</a>
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强化学习
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分布式深度学习
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</span>
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计算语言学
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语言学基础
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嵌入与序列模型
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Transformer 与语言模型
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</a>
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高级文本生成
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</span>
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计算机视觉
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图像基础
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目标检测与分割
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ViT 与生成模型
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视频与 3D 视觉
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音频与语音
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数字信号处理
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</a>
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<span class="md-ellipsis">
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自动语音识别
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</span>
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</a>
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</li>
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<a href="../../chapter%2009%3A%20audio%20and%20speech/03.%20text%20to%20speech%20and%20voice/" class="md-nav__link">
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<span class="md-ellipsis">
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语音合成
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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说话人与音频分析
|
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|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<span class="md-ellipsis">
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源分离与降噪
|
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|
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</span>
|
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|
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|
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|
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</a>
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</ul>
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</nav>
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<span class="md-ellipsis">
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多模态学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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多模态学习
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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多模态表征
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</span>
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</a>
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视觉语言模型
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</span>
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</a>
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<span class="md-ellipsis">
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图像与视频 Token 化
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</span>
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|
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|
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|
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</a>
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</li>
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跨模态生成
|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2010%3A%20multimodal%20learning/05.%20unified%20multimodal%20architectures/" class="md-nav__link">
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<span class="md-ellipsis">
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统一多模态架构
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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自主系统
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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自主系统
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感知
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</span>
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</a>
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机器人学习
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</span>
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</a>
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<span class="md-ellipsis">
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视觉-语言-动作模型
|
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</span>
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|
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|
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|
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</a>
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<a href="../../chapter%2011%3A%20autonomous%20systems/04.%20self-driving/" class="md-nav__link">
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自动驾驶
|
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</span>
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
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<span class="md-ellipsis">
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太空与极端机器人
|
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|
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|
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|
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</span>
|
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|
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|
||
|
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</a>
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图神经网络
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</span>
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<span class="md-nav__icon md-icon"></span>
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图神经网络
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</label>
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几何深度学习
|
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<span class="md-ellipsis">
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|
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硬件基础
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</a>
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ARM 与 NEON
|
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x86 与 AVX
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Triton、TPU 与 Pallas
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AI 推理
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量化
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边缘推理
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扩缩与部署
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ML 系统设计
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应用 AI
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前沿 AI
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<h1 id="_1">矩阵分解<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>矩阵分解将复杂矩阵拆分为更简单的因子，用于求解方程组、计算逆矩阵和数据压缩。本文涵盖高斯消元、LU、QR、Cholesky、特征分解和SVD——这些算法是PCA、推荐系统和机器学习数值稳定性的基石。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>矩阵分解（或因子分解）将一个矩阵拆分成更容易处理的更简单的部分。可以把它类比为因数分解：<span class="arithmatex">\(12 = 3 \times 4\)</span> 比单独的12更容易理解。</p>
|
||
</li>
|
||
<li>
|
||
<p>我们分解矩阵是为了更快地求解方程组、稳定地计算逆矩阵、寻找特征值、压缩数据以及理解变换的几何结构。</p>
|
||
</li>
|
||
<li>
|
||
<p>最基本的技术是<strong>高斯消元</strong>（行化简）。思路很简单：给定方程组 <span class="arithmatex">\(A\mathbf{x} = \mathbf{b}\)</span>，使用三种允许的操作简化 <span class="arithmatex">\(A\)</span>，直到答案显而易见。</p>
|
||
</li>
|
||
<li>
|
||
<p>这些操作是：交换两行、将一行乘以非零标量、或将一行的倍数加到另一行上。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，要消除主元下方的第一列，从下面的行中减去第1行的倍数：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[
|
||
\begin{bmatrix} 2 & 1 & 5 \\ 4 & 3 & 7 \\ 6 & 5 & 9 \end{bmatrix} \xrightarrow{R_2 - 2R_1} \begin{bmatrix} 2 & 1 & 5 \\ 0 & 1 & -3 \\ 6 & 5 & 9 \end{bmatrix} \xrightarrow{R_3 - 3R_1} \begin{bmatrix} 2 & 1 & 5 \\ 0 & 1 & -3 \\ 0 & 2 & -6 \end{bmatrix}
|
||
\]</div>
|
||
<ul>
|
||
<li>目标是<strong>行阶梯形（REF）</strong>：每个主元（每行第一个非零条目）下方全为零，且每个主元在其上方主元的右侧。矩阵呈现阶梯形状。</li>
|
||
</ul>
|
||
<p><img alt="高斯消元：行操作产生三角形形式，然后从下往上求解" src="../../images/gaussian_elimination.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>进一步得到<strong>简化行阶梯形（RREF）</strong>，使每个主元为1且是该列中唯一的非零条目。每个矩阵有唯一的RREF。</p>
|
||
</li>
|
||
<li>
|
||
<p>一旦转换为三角形形式，我们通过<strong>回代</strong>求解：最下面一行直接给出最后一个变量，然后向上求解。</p>
|
||
</li>
|
||
<li>
|
||
<p>这是所有其他分解方法所建立的基础，分解的目标就是将矩阵简化为三角形形式，从而可以通过回代求解变量。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>LU分解</strong>将高斯消元形式化，将方阵分解为 <span class="arithmatex">\(A = LU\)</span>（或通过行交换得到 <span class="arithmatex">\(A = PLU\)</span>），其中 <span class="arithmatex">\(L\)</span> 是下三角矩阵，<span class="arithmatex">\(U\)</span> 是上三角矩阵。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="LU分解：将一个困难的矩阵拆分为两个简单的三角矩阵" src="../../images/lu_decomposition.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>求解 <span class="arithmatex">\(A\mathbf{x} = \mathbf{b}\)</span>：先通过前向代入（从上到下）求解 <span class="arithmatex">\(L\mathbf{y} = \mathbf{b}\)</span>，然后通过回代（从下到上）求解 <span class="arithmatex">\(U\mathbf{x} = \mathbf{y}\)</span>。两次简单的三角求解代替了一次困难的一般求解。</p>
|
||
</li>
|
||
<li>
|
||
<p>相比原始高斯消元的优势在于可复用。一旦得到 <span class="arithmatex">\(L\)</span> 和 <span class="arithmatex">\(U\)</span>，就可以对许多不同的 <span class="arithmatex">\(\mathbf{b}\)</span> 向量求解，而无需重新进行分解。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果你需要用1000个不同的右端项求解同一个方程组（这在模拟中很常见），只需分解一次然后重复使用。</p>
|
||
</li>
|
||
<li>
|
||
<p>当矩阵是对称正定矩阵时（如协方差矩阵），我们可以做得更好。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>Cholesky分解</strong>将其分解为 <span class="arithmatex">\(A = LL^T\)</span>，其中 <span class="arithmatex">\(L\)</span> 是下三角矩阵。例如：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[
|
||
\begin{bmatrix} 4 & 2 \\ 2 & 5 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}
|
||
\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这大约比LU快两倍，并且保证数值稳定。可以将其视为矩阵的"平方根"。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果分解失败（平方根下出现负值），则该矩阵不是正定的。因此Cholesky分解也可以作为正定性的检验方法。</p>
|
||
</li>
|
||
<li>
|
||
<p>方阵 <span class="arithmatex">\(A\)</span> 的<strong>特征向量</strong>是特殊方向，该变换在这些方向上只进行拉伸或压缩，而不旋转。<strong>特征值</strong>是缩放因子：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[A\mathbf{x} = \lambda\mathbf{x}\]</div>
|
||
<p><img alt="特征向量保持在同一直线上（仅被缩放），普通向量被旋转" src="../../images/eigenvector.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>大多数向量在乘以矩阵时方向会改变。但特征向量是特殊的：输出方向与输入方向相同，仅被 <span class="arithmatex">\(\lambda\)</span> 缩放。如果 <span class="arithmatex">\(\lambda = 2\)</span>，特征向量长度加倍。如果 <span class="arithmatex">\(\lambda = -1\)</span>，它翻转方向。如果 <span class="arithmatex">\(\lambda = 0\)</span>，它被压缩为零。</p>
|
||
</li>
|
||
<li>
|
||
<p>例如，对于：</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[
|
||
A = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}
|
||
\]</div>
|
||
<p>向量 <span class="arithmatex">\([1, 0]^T\)</span> 是特征向量，<span class="arithmatex">\(\lambda = 3\)</span>，因为 <span class="arithmatex">\(A[1, 0]^T = [3, 0]^T = 3[1, 0]^T\)</span>。</p>
|
||
<ul>
|
||
<li>
|
||
<p>求特征值需要解<strong>特征多项式</strong> <span class="arithmatex">\(\det(A - \lambda I) = 0\)</span>。根即为特征值。然后将每个 <span class="arithmatex">\(\lambda\)</span> 代回 <span class="arithmatex">\((A - \lambda I)\mathbf{x} = \mathbf{0}\)</span> 中，求出对应的特征向量。</p>
|
||
</li>
|
||
<li>
|
||
<p>关键性质：</p>
|
||
<ul>
|
||
<li><span class="arithmatex">\(A\)</span> 的迹等于其特征值之和。</li>
|
||
<li><span class="arithmatex">\(A\)</span> 的行列式等于其特征值之积。</li>
|
||
<li>对称矩阵的特征向量互相垂直，特征值为实数。</li>
|
||
<li>正定矩阵的所有特征值为正。</li>
|
||
<li>协方差矩阵（我们将在统计学中遇到）总是半正定的。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>通过特征多项式计算特征值对于大型矩阵来说是不切实际的。相反，使用迭代方法：</p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>幂迭代</strong>：反复乘以 <span class="arithmatex">\(A\)</span> 并归一化。收敛到主特征向量（最大特征值）。简单但只能找到一个特征对。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>QR算法</strong>：最常用的方法。使用QR分解反复分解和重组矩阵，直到矩阵收敛到三角形形式，对角线上的元素即为所有特征值。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>反迭代</strong>：寻找最接近给定目标值的特征向量。当你大致知道想要哪个特征值时很有用。</p>
|
||
</li>
|
||
<li>
|
||
<p>对于大型稀疏矩阵，<strong>Arnoldi</strong>和<strong>Lanczos</strong>迭代利用稀疏性提高效率。</p>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>如果方阵有一组完整的线性无关的特征向量，它可以被<strong>对角化</strong>：<span class="arithmatex">\(A = PDP^{-1}\)</span>，其中 <span class="arithmatex">\(D\)</span> 是以特征值为对角元的对角矩阵，<span class="arithmatex">\(P\)</span> 的列是特征向量。</p>
|
||
</li>
|
||
<li>
|
||
<p>这有什么用？对角矩阵非常容易处理。需要计算 <span class="arithmatex">\(A^{100}\)</span>？不用将 <span class="arithmatex">\(A\)</span> 自乘100次，计算 <span class="arithmatex">\(PD^{100}P^{-1}\)</span> 即可——而对角矩阵的幂只需独立地对每个对角元求幂。这将一个昂贵的运算变成了廉价运算。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>特征基</strong>是完全由特征向量构成的基。在这个基下，矩阵变成对角矩阵，变换仅仅是沿每个特征向量方向的独立缩放。这就像是找到了变换的自然坐标系。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>QR分解</strong>将任意矩阵 <span class="arithmatex">\(A\)</span> 分解为 <span class="arithmatex">\(A = QR\)</span>，其中 <span class="arithmatex">\(Q\)</span> 是正交矩阵（其列是标准正交的），<span class="arithmatex">\(R\)</span> 是上三角矩阵。可以理解为将"方向"信息（<span class="arithmatex">\(Q\)</span>）与"缩放和混合"信息（<span class="arithmatex">\(R\)</span>）分开。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>Gram-Schmidt过程</strong>逐列构建 <span class="arithmatex">\(Q\)</span>。取 <span class="arithmatex">\(A\)</span> 的第一列并归一化。取第二列，减去其在第一列上的投影（使其垂直），再归一化。对每一列重复此过程。结果是一组标准正交向量。</p>
|
||
</li>
|
||
<li>
|
||
<p>QR分解是QR算法求特征值背后的引擎。它也直接用于求解最小二乘问题：当 <span class="arithmatex">\(A\mathbf{x} = \mathbf{b}\)</span> 没有精确解（方程多于未知数）时，QR找到最佳近似解。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>SVD</strong>（奇异值分解）是最通用、也可以说是最重要的分解。每个矩阵（任意形状、任意秩）都有SVD：<span class="arithmatex">\(A = U\Sigma V^T\)</span></p>
|
||
<ul>
|
||
<li><span class="arithmatex">\(V^T\)</span>（<span class="arithmatex">\(n \times n\)</span>，正交）：旋转输入</li>
|
||
<li><span class="arithmatex">\(\Sigma\)</span>（<span class="arithmatex">\(m \times n\)</span>，对角）：沿正交坐标轴缩放（奇异值，非负，递减排列）</li>
|
||
<li><span class="arithmatex">\(U\)</span>（<span class="arithmatex">\(m \times m\)</span>，正交）：旋转输出</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="SVD：任何变换 = 旋转，然后缩放，再旋转" src="../../images/svd.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>几何上，SVD表明每个线性变换，无论多么复杂，都只是一个旋转、一个沿坐标轴的拉伸、再一个旋转的组合。一个圆变成了一个椭圆。</p>
|
||
</li>
|
||
<li>
|
||
<p>奇异值（<span class="arithmatex">\(\sigma_1 \geq \sigma_2 \geq \ldots\)</span>）揭示了每个方向的"重要性"。大的奇异值对应最重要的方向。<span class="arithmatex">\(A\)</span> 的秩等于非零奇异值的个数。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>低秩近似</strong>：只保留最大的 <span class="arithmatex">\(k\)</span> 个奇异值，将其他置零，就得到了 <span class="arithmatex">\(A\)</span> 的最佳秩-<span class="arithmatex">\(k\)</span> 近似。这就是图像压缩的原理：一张 <span class="arithmatex">\(1000 \times 1000\)</span> 的图像可能只需要 <span class="arithmatex">\(k = 50\)</span> 个奇异值就能看起来几乎一模一样，压缩了20倍。</p>
|
||
</li>
|
||
<li>
|
||
<p>SVD也提供了伪逆：<span class="arithmatex">\(A^+ = V\Sigma^+U^T\)</span>，其中 <span class="arithmatex">\(\Sigma^+\)</span> 是对非零奇异值取倒数。</p>
|
||
</li>
|
||
<li>
|
||
<p>特征分解只对方阵有效，而SVD对任意矩阵都有效。这是它的关键优势。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>PCA</strong>（主成分分析）使用特征分解（或SVD）进行降维。</p>
|
||
</li>
|
||
<li>
|
||
<p>想象一个数据集，每个样本有100个特征（堆叠成矩阵的100维向量）。其中许多特征是相关的、冗余的。</p>
|
||
</li>
|
||
<li>
|
||
<p>PCA找到数据实际变化的那些方向，让你只保留重要的部分。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="PCA找到数据中方差最大的方向" src="../../images/pca.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>第一主成分（PC1）是方差最大的方向。</p>
|
||
</li>
|
||
<li>
|
||
<p>第二主成分（PC2）捕获剩余部分的最大方差，且与第一主成分垂直。</p>
|
||
</li>
|
||
<li>
|
||
<p>如果大部分方差只集中在少数几个方向上，你可以将数据投影到这些维度上，丢弃其余部分，损失极小。</p>
|
||
</li>
|
||
<li>
|
||
<p>步骤：</p>
|
||
<ul>
|
||
<li>标准化数据（减去均值，除以标准差），使所有特征贡献平等</li>
|
||
<li>计算协方差矩阵</li>
|
||
<li>求其特征值和特征向量</li>
|
||
<li>选择 <span class="arithmatex">\(k\)</span> 个最大特征值对应的特征向量（即主成分）</li>
|
||
<li>将数据投影到这些主成分上</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>标准化至关重要：如果不做标准化，用公里测量的特征会主导用厘米测量的特征，而不论其实际重要性如何。</p>
|
||
</li>
|
||
<li>
|
||
<p>在实践中，PCA用于可视化（将高维数据投影到2D或3D）、降噪（丢弃主要是噪声的低方差方向），以及通过减少输入特征数量来加速机器学习模型。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>核PCA</strong>将PCA扩展到非线性关系。它通过核函数将数据映射到更高维空间，在那里结构变得线性，然后应用标准PCA并投影回来。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>Schur分解</strong>将方阵分解为 <span class="arithmatex">\(A = QTQ^\ast\)</span>，其中 <span class="arithmatex">\(Q\)</span> 是酉矩阵，<span class="arithmatex">\(T\)</span> 是上三角矩阵。每个方阵都有Schur分解，即使它不能被对角化。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>非负矩阵分解（NMF）</strong> 将一个矩阵分解为两个非负矩阵：<span class="arithmatex">\(A \approx WH\)</span>，其中 <span class="arithmatex">\(W\)</span> 和 <span class="arithmatex">\(H\)</span> 的所有条目都 <span class="arithmatex">\(\geq 0\)</span>。与可能产生负条目的SVD不同，NMF只做加法，从不做减法。这使得各部分可解释：在主题建模中，<span class="arithmatex">\(W\)</span> 给出每个文档的主题权重，<span class="arithmatex">\(H\)</span> 给出每个主题的词权重，全部非负，这与我们对文档"包含多少某个主题"的思考方式相符。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>谱定理</strong>指出，对称（或Hermitian）矩阵总可以用正交（或酉）矩阵对角化。它们的特征值总是实数，特征向量总是正交的。这是PCA的理论基础。</p>
|
||
</li>
|
||
</ul>
|
||
<h2 id="colabjupyter-notebook">编程练习（使用CoLab或Jupyter Notebook）<a class="headerlink" href="#colabjupyter-notebook" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>计算对称矩阵的特征值和特征向量。验证特征向量互相垂直，并从特征分解重建矩阵。</li>
|
||
</ol>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">],</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="p">[</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]])</span>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">eigenvalues</span><span class="p">,</span> <span class="n">eigenvectors</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Eigenvalues: </span><span class="si">{</span><span class="n">eigenvalues</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Eigenvectors orthogonal: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">eigenvectors</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span><span class="w"> </span><span class="n">eigenvectors</span><span class="p">[:,</span><span class="mi">1</span><span class="p">])</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a><span class="c1"># Reconstruct: A = P D P^T</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a><span class="n">D</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">eigenvalues</span><span class="p">)</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a><span class="n">A_reconstructed</span> <span class="o">=</span> <span class="n">eigenvectors</span> <span class="o">@</span> <span class="n">D</span> <span class="o">@</span> <span class="n">eigenvectors</span><span class="o">.</span><span class="n">T</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Reconstruction matches: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="n">A_reconstructed</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div>
|
||
<ol>
|
||
<li>实现幂迭代求最大特征值，以及反迭代求最小特征值。与 <code>jnp.linalg.eigh</code> 比较。然后尝试自己实现QR算法。</li>
|
||
</ol>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">],</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a> <span class="p">[</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]])</span>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="c1"># Power iteration: finds the LARGEST eigenvalue</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">v</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a><span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">):</span>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a> <span class="n">v</span> <span class="o">=</span> <span class="n">A</span> <span class="o">@</span> <span class="n">v</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a> <span class="n">v</span> <span class="o">=</span> <span class="n">v</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Largest eigenvalue: </span><span class="si">{</span><span class="n">v</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">v</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="c1"># Inverse iteration: multiply by A^{-1} instead of A, finds the SMALLEST eigenvalue</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="n">v</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">])</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">):</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a> <span class="n">v</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a> <span class="n">v</span> <span class="o">=</span> <span class="n">v</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Smallest eigenvalue: </span><span class="si">{</span><span class="mf">1.0</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">v</span><span class="w"> </span><span class="o">@</span><span class="w"> </span><span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="n">v</span><span class="p">))</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a>
|
||
<a id="__codelineno-1-20" name="__codelineno-1-20" href="#__codelineno-1-20"></a><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"jnp.linalg.eigh: </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">A</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div>
|
||
<ol>
|
||
<li>计算矩阵的SVD，然后仅使用前k个奇异值重建矩阵，观察近似质量随k的变化。</li>
|
||
</ol>
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a><span class="n">A</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">],</span>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a> <span class="p">[</span><span class="mf">4.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">,</span> <span class="mf">6.0</span><span class="p">],</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a> <span class="p">[</span><span class="mf">7.0</span><span class="p">,</span> <span class="mf">8.0</span><span class="p">,</span> <span class="mf">9.0</span><span class="p">]])</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">U</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Vt</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]:</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="n">approx</span> <span class="o">=</span> <span class="n">U</span><span class="p">[:,</span> <span class="p">:</span><span class="n">k</span><span class="p">]</span> <span class="o">@</span> <span class="n">jnp</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">S</span><span class="p">[:</span><span class="n">k</span><span class="p">])</span> <span class="o">@</span> <span class="n">Vt</span><span class="p">[:</span><span class="n">k</span><span class="p">,</span> <span class="p">:]</span>
|
||
<a id="__codelineno-2-11" name="__codelineno-2-11" href="#__codelineno-2-11"></a> <span class="n">error</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">A</span> <span class="o">-</span> <span class="n">approx</span><span class="p">)</span>
|
||
<a id="__codelineno-2-12" name="__codelineno-2-12" href="#__codelineno-2-12"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"k=</span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s2">, reconstruction error: </span><span class="si">{</span><span class="n">error</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||
</code></pre></div>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
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|
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|
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|
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