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@@ -34,7 +34,7 @@ b_{ij}&=-\frac{1}{2}(dist^2_{ij}-b_{ii}-b_{jj})\\
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## 10.14
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$$\begin{aligned}
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-\sum^m_{i=1}\left\| \sum^{d'}_{j=1}z_{ij}\boldsymbol{w}-\boldsymbol x_i \right\|^2_2&=\sum^m_{i=1}\boldsymbol z^{\mathrm{T}}_i\boldsymbol z_i-2\sum^m_{i=1}\boldsymbol z^{\mathrm{T}}_i\mathbf{W}^{\mathrm{T}}\boldsymbol x_i +\text { const }\\
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+\sum^m_{i=1}\left\| \sum^{d'}_{j=1}z_{ij}\boldsymbol{w}_j-\boldsymbol x_i \right\|^2_2&=\sum^m_{i=1}\boldsymbol z^{\mathrm{T}}_i\boldsymbol z_i-2\sum^m_{i=1}\boldsymbol z^{\mathrm{T}}_i\mathbf{W}^{\mathrm{T}}\boldsymbol x_i +\text { const }\\
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&\propto -\operatorname{tr}(\mathbf{W}^{\mathrm{T}}(\sum^m_{i=1}\boldsymbol x_i\boldsymbol x^{\mathrm{T}}_i)\mathbf{W})
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\end{aligned}$$
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[推导]：已知$\mathbf{W}^{\mathrm{T}} \mathbf{W}=\mathbf{I},\boldsymbol z_i=\mathbf{W}^{\mathrm{T}} \boldsymbol x_i$，则
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Sm1les