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@@ -48,7 +48,7 @@ $$
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## 10.4
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## 10.4
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$$
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$$
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-\sum^m_{i=1}dist^2_{ij}=tr(\boldsymbol B)+mb_{jj}
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+\sum^m_{i=1}dist^2_{ij}=\operatorname{tr}(\boldsymbol B)+mb_{jj}
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$$
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$$
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[解析]：首先根据式10.3有
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[解析]：首先根据式10.3有
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@@ -119,7 +119,7 @@ $$
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由公式（10.6）和（10.9）可得
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由公式（10.6）和（10.9）可得
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-tr(\boldsymbol B)&=\frac{1}{2m}\sum^m_{i=1}\sum^m_{j=1}dist^2_{ij}\\
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+\operatorname{tr}(\boldsymbol B)&=\frac{1}{2m}\sum^m_{i=1}\sum^m_{j=1}dist^2_{ij}\\
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&=\frac{m}{2}dist^2_{\cdot}
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&=\frac{m}{2}dist^2_{\cdot}
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\end{aligned}
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\end{aligned}
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$$
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$$
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@@ -133,7 +133,7 @@ $$
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由公式（10.5）和（10.7）可得
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由公式（10.5）和（10.7）可得
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-b_{ii}&=\frac{1}{m}\sum^m_{j=1}dist^2_{ij}-\frac{1}{m}tr(\boldsymbol B)\\
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+b_{ii}&=\frac{1}{m}\sum^m_{j=1}dist^2_{ij}-\frac{1}{m}\operatorname{tr}(\boldsymbol B)\\
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&=dist^2_{i\cdot}-\frac{1}{2}dist^2_{\cdot}
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&=dist^2_{i\cdot}-\frac{1}{2}dist^2_{\cdot}
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\end{aligned}
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\end{aligned}
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$$
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$$
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@@ -379,7 +379,7 @@ $$
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## 10.31
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## 10.31
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$$
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$$
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\begin{aligned}
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\begin{aligned}
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-&\min\limits_{\boldsymbol Z}tr(\boldsymbol Z \boldsymbol M \boldsymbol Z^T)\\
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+&\min\limits_{\boldsymbol Z}\operatorname{tr}(\boldsymbol Z \boldsymbol M \boldsymbol Z^T)\\
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&s.t. \boldsymbol Z^T\boldsymbol Z=\boldsymbol I.
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&s.t. \boldsymbol Z^T\boldsymbol Z=\boldsymbol I.
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\end{aligned}
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\end{aligned}
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$$
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$$
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@@ -393,13 +393,12 @@ $$
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&=\sum^m_{i=1}\|\boldsymbol Z(\boldsymbol I_i-\boldsymbol W_i)\|^2_2\\
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&=\sum^m_{i=1}\|\boldsymbol Z(\boldsymbol I_i-\boldsymbol W_i)\|^2_2\\
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&=\sum^m_{i=1}(\boldsymbol Z(\boldsymbol I_i-\boldsymbol W_i))^T\boldsymbol Z(\boldsymbol I_i-\boldsymbol W_i)\\
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&=\sum^m_{i=1}(\boldsymbol Z(\boldsymbol I_i-\boldsymbol W_i))^T\boldsymbol Z(\boldsymbol I_i-\boldsymbol W_i)\\
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&=\sum^m_{i=1}(\boldsymbol I_i-\boldsymbol W_i)^T\boldsymbol Z^T\boldsymbol Z(\boldsymbol I_i-\boldsymbol W_i)\\
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&=\sum^m_{i=1}(\boldsymbol I_i-\boldsymbol W_i)^T\boldsymbol Z^T\boldsymbol Z(\boldsymbol I_i-\boldsymbol W_i)\\
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-&=tr((\boldsymbol I-\boldsymbol W)^T\boldsymbol Z^T\boldsymbol Z(\boldsymbol I-\boldsymbol W))\\
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-&=tr(\boldsymbol Z(\boldsymbol I-\boldsymbol W)(\boldsymbol I-\boldsymbol W)^T\boldsymbol Z^T)\\
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-&=tr(\boldsymbol Z\boldsymbol M\boldsymbol Z^T)
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+&=\operatorname{tr}((\boldsymbol I-\boldsymbol W)^T\boldsymbol Z^T\boldsymbol Z(\boldsymbol I-\boldsymbol W))\\
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+&=\operatorname{tr}(\boldsymbol Z(\boldsymbol I-\boldsymbol W)(\boldsymbol I-\boldsymbol W)^T\boldsymbol Z^T)\\
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+&=\operatorname{tr}(\boldsymbol Z\boldsymbol M\boldsymbol Z^T)
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\end{aligned}
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\end{aligned}
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$$
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$$
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-
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其中，$\boldsymbol M=(\boldsymbol I-\boldsymbol W)(\boldsymbol I-\boldsymbol W)^T$。
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其中，$\boldsymbol M=(\boldsymbol I-\boldsymbol W)(\boldsymbol I-\boldsymbol W)^T$。
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[解析]：约束条件$\boldsymbol Z^T\boldsymbol Z=\boldsymbol I$是为了得到标准化（标准正交空间）的低维数据。
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[解析]：约束条件$\boldsymbol Z^T\boldsymbol Z=\boldsymbol I$是为了得到标准化（标准正交空间）的低维数据。
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