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3.5

$$\cfrac{\partial E_{(w, b)}}{\partial w}=2\left(w \sum_{i=1}^{m} x_{i}^{2}-\sum_{i=1}^{m}\left(y_{i}-b\right) x_{i}\right)$$ [推导]：已知$E_{(w, b)}=\sum\limits_{i=1}^{m}\left(y_{i}-w x_{i}-b\right)^{2}$，所以 $$\begin{aligned} \cfrac{\partial E_{(w, b)}}{\partial w}&=\cfrac{\partial}{\partial w} \left[\sum_{i=1}^{m}\left(y_{i}-w x_{i}-b\right)^{2}\right] \\ &= \sum_{i=1}^{m}\cfrac{\partial}{\partial w} \left[\left(y_{i}-w x_{i}-b\right)^{2}\right] \\ &= \sum_{i=1}^{m}\left[2\cdot\left(y_{i}-w x_{i}-b\right)\cdot (-x_i)\right] \\ &= \sum_{i=1}^{m}\left[2\cdot\left(w x_{i}^2-y_i x_i +bx_i\right)\right] \\ &= 2\cdot\left(w\sum_{i=1}^{m} x_{i}^2-\sum_{i=1}^{m}y_i x_i +b\sum_{i=1}^{m}x_i\right) \\ &=2\left(w \sum_{i=1}^{m} x_{i}^{2}-\sum_{i=1}^{m}\left(y_{i}-b\right) x_{i}\right) \end{aligned}$$

3.6

$$\cfrac{\partial E_{(w, b)}}{\partial b}=2\left(m b-\sum_{i=1}^{m}\left(y_{i}-w x_{i}\right)\right)$$ [推导]：已知$E_{(w, b)}=\sum\limits_{i=1}^{m}\left(y_{i}-w x_{i}-b\right)^{2}$，所以 $$\begin{aligned} \cfrac{\partial E_{(w, b)}}{\partial b}&=\cfrac{\partial}{\partial b} \left[\sum_{i=1}^{m}\left(y_{i}-w x_{i}-b\right)^{2}\right] \\ &=\sum_{i=1}^{m}\cfrac{\partial}{\partial b} \left[\left(y_{i}-w x_{i}-b\right)^{2}\right] \\ &=\sum_{i=1}^{m}\left[2\cdot\left(y_{i}-w x_{i}-b\right)\cdot (-1)\right] \\ &=\sum_{i=1}^{m}\left[2\cdot\left(b-y_{i}+w x_{i}\right)\right] \\ &=2\cdot\left[\sum_{i=1}^{m}b-\sum_{i=1}^{m}y_{i}+\sum_{i=1}^{m}w x_{i}\right] \\ &=2\left(m b-\sum_{i=1}^{m}\left(y_{i}-w x_{i}\right)\right) \end{aligned}$$

3.7

$$ w=\cfrac{\sum_{i=1}^{m}y_i(x_i-\bar{x})}{\sum_{i=1}^{m}x_i^2-\cfrac{1}{m}(\sum_{i=1}^{m}x_i)^2} $$ [推导]：令公式(3.5)等于0 $$ 0 = w\sum_{i=1}^{m}x_i^2-\sum_{i=1}^{m}(y_i-b)x_i $$ $$ w\sum_{i=1}^{m}x_i^2 = \sum_{i=1}^{m}y_ix_i-\sum_{i=1}^{m}bx_i $$ 由于令公式(3.6)等于0可得$b=\cfrac{1}{m}\sum_{i=1}^{m}(y_i-wx_i)$，又因为$\cfrac{1}{m}\sum_{i=1}^{m}y_i=\bar{y}$，$\cfrac{1}{m}\sum_{i=1}^{m}x_i=\bar{x}$，则$b=\bar{y}-w\bar{x}$，代入上式可得 $$\begin{aligned} w\sum_{i=1}^{m}x_i^2 & = \sum_{i=1}^{m}y_ix_i-\sum_{i=1}^{m}(\bar{y}-w\bar{x})x_i \\ w\sum_{i=1}^{m}x_i^2 & = \sum_{i=1}^{m}y_ix_i-\bar{y}\sum_{i=1}^{m}x_i+w\bar{x}\sum_{i=1}^{m}x_i \\ w(\sum_{i=1}^{m}x_i^2-\bar{x}\sum_{i=1}^{m}x_i) & = \sum_{i=1}^{m}y_ix_i-\bar{y}\sum_{i=1}^{m}x_i \\ w & = \cfrac{\sum_{i=1}^{m}y_ix_i-\bar{y}\sum_{i=1}^{m}x_i}{\sum_{i=1}^{m}x_i^2-\bar{x}\sum_{i=1}^{m}x_i} \end{aligned}$$ 由于$\bar{y}\sum_{i=1}^{m}x_i=\cfrac{1}{m}\sum_{i=1}^{m}y_i\sum_{i=1}^{m}x_i=\bar{x}\sum_{i=1}^{m}y_i$，$\bar{x}\sum_{i=1}^{m}x_i=\cfrac{1}{m}\sum_{i=1}^{m}x_i\sum_{i=1}^{m}x_i=\cfrac{1}{m}(\sum_{i=1}^{m}x_i)^2$，代入上式即可得公式(3.7) $$ w=\cfrac{\sum_{i=1}^{m}y_i(x_i-\bar{x})}{\sum_{i=1}^{m}x_i^2-\cfrac{1}{m}(\sum_{i=1}^{m}x_i)^2} $$ 如果要想用Python来实现上式的话，上式中的求和运算只能用循环来实现，但是如果我们能将上式给向量化，也就是转换成矩阵（向量）运算的话，那么我们就可以利用诸如NumPy这种专门加速矩阵运算的类库来进行编写。下面我们就尝试将上式进行向量化，将$ \cfrac{1}{m}(\sum_{i=1}^{m}x_i)^2=\bar{x}\sum_{i=1}^{m}x_i $代入分母可得 $$\begin{aligned} w & = \cfrac{\sum_{i=1}^{m}y_i(x_i-\bar{x})}{\sum_{i=1}^{m}x_i^2-\bar{x}\sum_{i=1}^{m}x_i} \\ & = \cfrac{\sum_{i=1}^{m}(y_ix_i-y_i\bar{x})}{\sum_{i=1}^{m}(x_i^2-x_i\bar{x})} \end{aligned}$$ 又因为$ \bar{y}\sum_{i=1}^{m}x_i=\bar{x}\sum_{i=1}^{m}y_i=\sum_{i=1}^{m}\bar{y}x_i=\sum_{i=1}^{m}\bar{x}y_i=m\bar{x}\bar{y}=\sum_{i=1}^{m}\bar{x}\bar{y} $，$\sum_{i=1}^{m}x_i\bar{x}=\bar{x}\sum_{i=1}^{m}x_i=\bar{x}\cdot m \cdot\frac{1}{m}\cdot\sum_{i=1}^{m}x_i=m\bar{x}^2=\sum_{i=1}^{m}\bar{x}^2$，则上式可化为 $$\begin{aligned} w & = \cfrac{\sum_{i=1}^{m}(y_ix_i-y_i\bar{x}-x_i\bar{y}+\bar{x}\bar{y})}{\sum_{i=1}^{m}(x_i^2-x_i\bar{x}-x_i\bar{x}+\bar{x}^2)} \\ & = \cfrac{\sum_{i=1}^{m}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{m}(x_i-\bar{x})^2} \end{aligned}$$ 若令$\boldsymbol{x}=(x_1,x_2,…,x_m)^T$，$\boldsymbol{x}_{d}=(x_1-\bar{x},x_2-\bar{x},…,x_m-\bar{x})^T$为去均值后的$\boldsymbol{x}$，$\boldsymbol{y}=(y_1,y_2,…,y_m)^T$，$\boldsymbol{y}_{d}=(y_1-\bar{y},y_2-\bar{y},…,y_m-\bar{y})^T$为去均值后的$\boldsymbol{y}$，其中$\boldsymbol{x}$、$\boldsymbol{x}_{d}$、$\boldsymbol{y}$、$\boldsymbol{y}_{d}$均为m行1列的列向量，代入上式可得 $$w=\cfrac{\boldsymbol{x}_{d}^T\boldsymbol{y}_{d}}{\boldsymbol{x}_d^T\boldsymbol{x}_{d}}$$

3.9

$$\hat{\boldsymbol{w}}^{*}=\underset{\hat{\boldsymbol{w}}}{\arg \min }(\boldsymbol{y}-\mathbf{X} \hat{\boldsymbol{w}})^{\mathrm{T}}(\boldsymbol{y}-\mathbf{X} \hat{\boldsymbol{w}})$$ [推导]：公式(3.4)是最小二乘法运用在一元线性回归上的情形，那么对于多元线性回归来说，我们可以类似得到 $$\begin{aligned} \left(\boldsymbol{w}^{*}, b^{*}\right)&=\underset{(\boldsymbol{w}, b)}{\arg \min } \sum_{i=1}^{m}\left(f\left(\boldsymbol{x}_{i}\right)-y_{i}\right)^{2} \\ &=\underset{(\boldsymbol{w}, b)}{\arg \min } \sum_{i=1}^{m}\left(y_{i}-f\left(\boldsymbol{x}_{i}\right)\right)^{2}\\ &=\underset{(\boldsymbol{w}, b)}{\arg \min } \sum_{i=1}^{m}\left(y_{i}-\left(\boldsymbol{w}^\mathrm{T}\boldsymbol{x}_{i}+b\right)\right)^{2} \end{aligned}$$ 为便于讨论，我们令$\hat{\boldsymbol{w}}=(\boldsymbol{w};b)=(w_1;…;w_d;b)\in\mathbb{R}^{(d+1)\times 1},\hat{\boldsymbol{x}}_i=(x_{i1};…;x_{id};1)\in\mathbb{R}^{(d+1)\times 1}$，那么上式可以简化为 $$\begin{aligned} \hat{\boldsymbol{w}}^{*}&=\underset{\hat{\boldsymbol{w}}}{\arg \min } \sum_{i=1}^{m}\left(y_{i}-\hat{\boldsymbol{w}}^\mathrm{T}\hat{\boldsymbol{x}}_{i}\right)^{2} \\ &=\underset{\hat{\boldsymbol{w}}}{\arg \min } \sum_{i=1}^{m}\left(y_{i}-\hat{\boldsymbol{x}}_{i}^\mathrm{T}\hat{\boldsymbol{w}}\right)^{2} \\ \end{aligned}$$ 根据向量内积的定义可知，上式可以写成如下向量内积的形式 $$\begin{aligned} \hat{\boldsymbol{w}}^{*}&=\underset{\hat{\boldsymbol{w}}}{\arg \min } \begin{bmatrix} y_{1}-\hat{\boldsymbol{x}}_{1}^\mathrm{T}\hat{\boldsymbol{w}} & \cdots & y_{m}-\hat{\boldsymbol{x}}_{m}^\mathrm{T}\hat{\boldsymbol{w}} \\ \end{bmatrix} \begin{bmatrix} y_{1}-\hat{\boldsymbol{x}}_{1}^\mathrm{T}\hat{\boldsymbol{w}} \\ \vdots \\ y_{m}-\hat{\boldsymbol{x}}_{m}^\mathrm{T}\hat{\boldsymbol{w}} \end{bmatrix} \\ \end{aligned}$$ 其中 $$ \begin{aligned} \begin{bmatrix} y_{1}-\hat{\boldsymbol{x}}_{1}^\mathrm{T}\hat{\boldsymbol{w}} \\ \vdots \\ y_{m}-\hat{\boldsymbol{x}}_{m}^\mathrm{T}\hat{\boldsymbol{w}} \end{bmatrix}&=\begin{bmatrix} y_{1} \\ \vdots \\ y_{m} \end{bmatrix}-\begin{bmatrix} \hat{\boldsymbol{x}}_{1}^\mathrm{T}\hat{\boldsymbol{w}} \\ \vdots \\ \hat{\boldsymbol{x}}_{m}^\mathrm{T}\hat{\boldsymbol{w}} \end{bmatrix}\\ &=\boldsymbol{y}-\begin{bmatrix} \hat{\boldsymbol{x}}_{1}^\mathrm{T} \\ \vdots \\ \hat{\boldsymbol{x}}_{m}^\mathrm{T} \end{bmatrix}\cdot\hat{\boldsymbol{w}}\\ &=\boldsymbol{y}-\mathbf{X}\hat{\boldsymbol{w}} \end{aligned} $$ 所以 $$\hat{\boldsymbol{w}}^{*}=\underset{\hat{\boldsymbol{w}}}{\arg \min }(\boldsymbol{y}-\mathbf{X} \hat{\boldsymbol{w}})^{\mathrm{T}}(\boldsymbol{y}-\mathbf{X} \hat{\boldsymbol{w}})$$

3.10

$$\cfrac{\partial E_{\hat{\boldsymbol w}}}{\partial \hat{\boldsymbol w}}=2\mathbf{X}^{\mathrm{T}}(\mathbf{X}\hat{\boldsymbol w}-\boldsymbol{y})$$ [推导]：将$E_{\hat{\boldsymbol w}}=(\boldsymbol{y}-\mathbf{X}\hat{\boldsymbol w})^{\mathrm{T}}(\boldsymbol{y}-\mathbf{X}\hat{\boldsymbol w})$展开可得 $$E_{\hat{\boldsymbol w}}= \boldsymbol{y}^{\mathrm{T}}\boldsymbol{y}-\boldsymbol{y}^{\mathrm{T}}\mathbf{X}\hat{\boldsymbol w}-\hat{\boldsymbol w}^{\mathrm{T}}\mathbf{X}^{\mathrm{T}}\boldsymbol{y}+\hat{\boldsymbol w}^{\mathrm{T}}\mathbf{X}^{\mathrm{T}}\mathbf{X}\hat{\boldsymbol w}$$ 对$\hat{\boldsymbol w}$求导可得 $$\cfrac{\partial E_{\hat{\boldsymbol w}}}{\partial \hat{\boldsymbol w}}= \cfrac{\partial \boldsymbol{y}^{\mathrm{T}}\boldsymbol{y}}{\partial \hat{\boldsymbol w}}-\cfrac{\partial \boldsymbol{y}^{\mathrm{T}}\mathbf{X}\hat{\boldsymbol w}}{\partial \hat{\boldsymbol w}}-\cfrac{\partial \hat{\boldsymbol w}^{\mathrm{T}}\mathbf{X}^{\mathrm{T}}\boldsymbol{y}}{\partial \hat{\boldsymbol w}}+\cfrac{\partial \hat{\boldsymbol w}^{\mathrm{T}}\mathbf{X}^{\mathrm{T}}\mathbf{X}\hat{\boldsymbol w}}{\partial \hat{\boldsymbol w}}$$ 由矩阵微分公式$\cfrac{\partial\boldsymbol{a}^{\mathrm{T}}\boldsymbol{x}}{\partial\boldsymbol{x}}=\cfrac{\partial\boldsymbol{x}^{\mathrm{T}}\boldsymbol{a}}{\partial\boldsymbol{x}}=\boldsymbol{a},\cfrac{\partial\boldsymbol{x}^{\mathrm{T}}\mathbf{A}\boldsymbol{x}}{\partial\boldsymbol{x}}=(\mathbf{A}+\mathbf{A}^{\mathrm{T}})\boldsymbol{x}$可得 $$\cfrac{\partial E_{\hat{\boldsymbol w}}}{\partial \hat{\boldsymbol w}}= 0-\mathbf{X}^{\mathrm{T}}\boldsymbol{y}-\mathbf{X}^{\mathrm{T}}\boldsymbol{y}+(\mathbf{X}^{\mathrm{T}}\mathbf{X}+\mathbf{X}^{\mathrm{T}}\mathbf{X})\hat{\boldsymbol w}$$ $$\cfrac{\partial E_{\hat{\boldsymbol w}}}{\partial \hat{\boldsymbol w}}=2\mathbf{X}^{\mathrm{T}}(\mathbf{X}\hat{\boldsymbol w}-\boldsymbol{y})$$

3.27

$$ \ell(\boldsymbol{\beta})=\sum_{i=1}^{m}(-y_i\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i+\ln(1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i})) $$ [推导]：将公式(3.26)代入公式(3.25)可得 $$ \ell(\boldsymbol{\beta})=\sum_{i=1}^{m}\ln\left(y_ip_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta})+(1-y_i)p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})\right) $$ 其中$ p_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta})=\cfrac{e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i}}{1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i}},p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})=\cfrac{1}{1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i}} $，代入上式可得 $$\begin{aligned} \ell(\boldsymbol{\beta})&=\sum_{i=1}^{m}\ln\left(\cfrac{y_ie^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i}+1-y_i}{1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i}}\right) \\ &=\sum_{i=1}^{m}\left(\ln(y_ie^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i}+1-y_i)-\ln(1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i})\right) \end{aligned}$$ 由于$ y_i $=0或1，则 $$ \ell(\boldsymbol{\beta}) = \begin{cases} \sum_{i=1}^{m}(-\ln(1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i})), & y_i=0 \\ \sum_{i=1}^{m}(\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i-\ln(1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i})), & y_i=1 \end{cases} $$ 两式综合可得 $$ \ell(\boldsymbol{\beta})=\sum_{i=1}^{m}\left(y_i\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i-\ln(1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i})\right) $$ 由于此式仍为极大似然估计的似然函数，所以最大化似然函数等价于最小化似然函数的相反数，也即在似然函数前添加负号即可得公式(3.27)。值得一提的是，若将公式(3.26)这个似然项改写为$p(y_i|\boldsymbol x_i;\boldsymbol w,b)=[p_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta})]^{y_i}[p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})]^{1-y_i}$，再将其代入公式(3.25)可得 $$\begin{aligned} \ell(\boldsymbol{\beta})&=\sum_{i=1}^{m}\ln\left([p_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta})]^{y_i}[p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})]^{1-y_i}\right) \\ &=\sum_{i=1}^{m}\left[y_i\ln\left(p_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta})\right)+(1-y_i)\ln\left(p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})\right)\right] \\ &=\sum_{i=1}^{m} \left \{ y_i\left[\ln\left(p_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta})\right)-\ln\left(p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})\right)\right]+\ln\left(p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})\right)\right\} \\ &=\sum_{i=1}^{m}\left[y_i\ln\left(\cfrac{p_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta})}{p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})}\right)+\ln\left(p_0(\hat{\boldsymbol x}_i;\boldsymbol{\beta})\right)\right] \\ &=\sum_{i=1}^{m}\left[y_i\ln\left(e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i}\right)+\ln\left(\cfrac{1}{1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i}}\right)\right] \\ &=\sum_{i=1}^{m}\left(y_i\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i-\ln(1+e^{\boldsymbol{\beta}^{\mathrm{T}}\hat{\boldsymbol x}_i})\right) \end{aligned}$$ 显然，此种方式更易推导出公式(3.27)

3.30

$$\frac{\partial \ell(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}=-\sum_{i=1}^{m}\hat{\boldsymbol x}_i(y_i-p_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta}))$$ [解析]：此式可以进行向量化，令$p_1(\hat{\boldsymbol x}_i;\boldsymbol{\beta})=\hat{y}_i$，代入上式得 $$\begin{aligned} \frac{\partial \ell(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} &= -\sum_{i=1}^{m}\hat{\boldsymbol x}_i(y_i-\hat{y}_i) \\ & =\sum_{i=1}^{m}\hat{\boldsymbol x}_i(\hat{y}_i-y_i) \\ & ={\mathbf{X}^{\mathrm{T}}}(\hat{\boldsymbol y}-\boldsymbol{y}) \\ & ={\mathbf{X}^{\mathrm{T}}}(p_1(\mathbf{X};\boldsymbol{\beta})-\boldsymbol{y}) \\ \end{aligned}$$

3.32

$$J=\cfrac{\boldsymbol w^{\mathrm{T}}(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})^{\mathrm{T}}\boldsymbol w}{\boldsymbol w^{\mathrm{T}}(\boldsymbol{\Sigma}_{0}+\boldsymbol{\Sigma}_{1})\boldsymbol w}$$ [推导]： $$\begin{aligned} J &= \cfrac{\|\boldsymbol w^{\mathrm{T}}\boldsymbol{\mu}_{0}-\boldsymbol w^{\mathrm{T}}\boldsymbol{\mu}_{1}\|_2^2}{\boldsymbol w^{\mathrm{T}}(\boldsymbol{\Sigma}_{0}+\boldsymbol{\Sigma}_{1})\boldsymbol w} \\ &= \cfrac{\|(\boldsymbol w^{\mathrm{T}}\boldsymbol{\mu}_{0}-\boldsymbol w^{\mathrm{T}}\boldsymbol{\mu}_{1})^{\mathrm{T}}\|_2^2}{\boldsymbol w^{\mathrm{T}}(\boldsymbol{\Sigma}_{0}+\boldsymbol{\Sigma}_{1})\boldsymbol w} \\ &= \cfrac{\|(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})^{\mathrm{T}}\boldsymbol w\|_2^2}{\boldsymbol w^{\mathrm{T}}(\boldsymbol{\Sigma}_{0}+\boldsymbol{\Sigma}_{1})\boldsymbol w} \\ &= \cfrac{\left[(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})^{\mathrm{T}}\boldsymbol w\right]^{\mathrm{T}}(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})^{\mathrm{T}}\boldsymbol w}{\boldsymbol w^{\mathrm{T}}(\boldsymbol{\Sigma}_{0}+\boldsymbol{\Sigma}_{1})\boldsymbol w} \\ &= \cfrac{\boldsymbol w^{\mathrm{T}}(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})^{\mathrm{T}}\boldsymbol w}{\boldsymbol w^{\mathrm{T}}(\boldsymbol{\Sigma}_{0}+\boldsymbol{\Sigma}_{1})\boldsymbol w} \end{aligned}$$

3.37

$$\mathbf{S}_b\boldsymbol w=\lambda\mathbf{S}_w\boldsymbol w$$ [推导]：由公式(3.36)可得拉格朗日函数为 $$L(\boldsymbol w,\lambda)=-\boldsymbol w^{\mathrm{T}}\mathbf{S}_b\boldsymbol w+\lambda(\boldsymbol w^{\mathrm{T}}\mathbf{S}_w\boldsymbol w-1)$$ 对$\boldsymbol w$求偏导可得 $$\begin{aligned} \cfrac{\partial L(\boldsymbol w,\lambda)}{\partial \boldsymbol w} &= -\cfrac{\partial(\boldsymbol w^{\mathrm{T}}\mathbf{S}_b\boldsymbol w)}{\partial \boldsymbol w}+\lambda \cfrac{\partial(\boldsymbol w^{\mathrm{T}}\mathbf{S}_w\boldsymbol w-1)}{\partial \boldsymbol w} \\ &= -(\mathbf{S}_b+\mathbf{S}_b^{\mathrm{T}})\boldsymbol w+\lambda(\mathbf{S}_w+\mathbf{S}_w^{\mathrm{T}})\boldsymbol w \end{aligned}$$ 由于$\mathbf{S}_b=\mathbf{S}_b^{\mathrm{T}},\mathbf{S}_w=\mathbf{S}_w^{\mathrm{T}}$，所以 $$\cfrac{\partial L(\boldsymbol w,\lambda)}{\partial \boldsymbol w} = -2\mathbf{S}_b\boldsymbol w+2\lambda\mathbf{S}_w\boldsymbol w$$ 令上式等于0即可得 $$-2\mathbf{S}_b\boldsymbol w+2\lambda\mathbf{S}_w\boldsymbol w=0$$ $$\mathbf{S}_b\boldsymbol w=\lambda\mathbf{S}_w\boldsymbol w$$ $$(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})^{\mathrm{T}}\boldsymbol{w}=\lambda\mathbf{S}_w\boldsymbol w$$ 若令$(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})^{\mathrm{T}}\boldsymbol{w}=\gamma$，则 $$\gamma(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})=\lambda\mathbf{S}_w\boldsymbol w$$ $$\boldsymbol{w}=\frac{\gamma}{\lambda}\mathbf{S}_{w}^{-1}(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})$$ 由于最终要求解的$\boldsymbol{w}$不关心其大小，只关心其方向，所以其大小可以任意取值。由于$\boldsymbol{\mu}_{0}$和$\boldsymbol{\mu}_{1}$的大小是固定的，所以$\gamma$这个标量的大小只受$\boldsymbol{w}$的大小影响，因此可以调整$\boldsymbol{w}$的大小使得$\gamma=\lambda$，西瓜书中所说的“不妨令$\mathbf{S}_b\boldsymbol w=\lambda(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})$”也等价于令$\gamma=\lambda$，因此，此时$\frac{\gamma}{\lambda}=1$，求解出的$\boldsymbol{w}$即为公式(3.39)

3.38

$$\mathbf{S}_b\boldsymbol{w}=\lambda(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})$$ [推导]：参见公式(3.37)

3.39

$$\boldsymbol{w}=\mathbf{S}_{w}^{-1}(\boldsymbol{\mu}_{0}-\boldsymbol{\mu}_{1})$$ [推导]：参见公式(3.37)

3.43

$$\begin{aligned} \mathbf{S}_b &= \mathbf{S}_t - \mathbf{S}_w \\ &= \sum_{i=1}^N m_i(\boldsymbol\mu_i-\boldsymbol\mu)(\boldsymbol\mu_i-\boldsymbol\mu)^{\mathrm{T}} \end{aligned}$$ [推导]：由公式(3.40)、公式(3.41)、公式(3.42)可得： $$\begin{aligned} \mathbf{S}_b &= \mathbf{S}_t - \mathbf{S}_w \\ &= \sum_{i=1}^m(\boldsymbol x_i-\boldsymbol\mu)(\boldsymbol x_i-\boldsymbol\mu)^{\mathrm{T}}-\sum_{i=1}^N\sum_{\boldsymbol x\in X_i}(\boldsymbol x-\boldsymbol\mu_i)(\boldsymbol x-\boldsymbol\mu_i)^{\mathrm{T}} \\ &= \sum_{i=1}^N\left(\sum_{\boldsymbol x\in X_i}\left((\boldsymbol x-\boldsymbol\mu)(\boldsymbol x-\boldsymbol\mu)^{\mathrm{T}}-(\boldsymbol x-\boldsymbol\mu_i)(\boldsymbol x-\boldsymbol\mu_i)^{\mathrm{T}}\right)\right) \\ &= \sum_{i=1}^N\left(\sum_{\boldsymbol x\in X_i}\left((\boldsymbol x-\boldsymbol\mu)(\boldsymbol x^{\mathrm{T}}-\boldsymbol\mu^{\mathrm{T}})-(\boldsymbol x-\boldsymbol\mu_i)(\boldsymbol x^{\mathrm{T}}-\boldsymbol\mu_i^{\mathrm{T}})\right)\right) \\ &= \sum_{i=1}^N\left(\sum_{\boldsymbol x\in X_i}\left(\boldsymbol x\boldsymbol x^{\mathrm{T}} - \boldsymbol x\boldsymbol\mu^{\mathrm{T}}-\boldsymbol\mu\boldsymbol x^{\mathrm{T}}+\boldsymbol\mu\boldsymbol\mu^{\mathrm{T}}-\boldsymbol x\boldsymbol x^{\mathrm{T}}+\boldsymbol x\boldsymbol\mu_i^{\mathrm{T}}+\boldsymbol\mu_i\boldsymbol x^{\mathrm{T}}-\boldsymbol\mu_i\boldsymbol\mu_i^{\mathrm{T}}\right)\right) \\ &= \sum_{i=1}^N\left(\sum_{\boldsymbol x\in X_i}\left(- \boldsymbol x\boldsymbol\mu^{\mathrm{T}}-\boldsymbol\mu\boldsymbol x^{\mathrm{T}}+\boldsymbol\mu\boldsymbol\mu^{\mathrm{T}}+\boldsymbol x\boldsymbol\mu_i^{\mathrm{T}}+\boldsymbol\mu_i\boldsymbol x^{\mathrm{T}}-\boldsymbol\mu_i\boldsymbol\mu_i^{\mathrm{T}}\right)\right) \\ &= \sum_{i=1}^N\left(-\sum_{\boldsymbol x\in X_i}\boldsymbol x\boldsymbol\mu^{\mathrm{T}}-\sum_{\boldsymbol x\in X_i}\boldsymbol\mu\boldsymbol x^{\mathrm{T}}+\sum_{\boldsymbol x\in X_i}\boldsymbol\mu\boldsymbol\mu^{\mathrm{T}}+\sum_{\boldsymbol x\in X_i}\boldsymbol x\boldsymbol\mu_i^{\mathrm{T}}+\sum_{\boldsymbol x\in X_i}\boldsymbol\mu_i\boldsymbol x^{\mathrm{T}}-\sum_{\boldsymbol x\in X_i}\boldsymbol\mu_i\boldsymbol\mu_i^{\mathrm{T}}\right) \\ &= \sum_{i=1}^N\left(-m_i\boldsymbol\mu_i\boldsymbol\mu^{\mathrm{T}}-m_i\boldsymbol\mu\boldsymbol\mu_i^{\mathrm{T}}+m_i\boldsymbol\mu\boldsymbol\mu^{\mathrm{T}}+m_i\boldsymbol\mu_i\boldsymbol\mu_i^{\mathrm{T}}+m_i\boldsymbol\mu_i\boldsymbol\mu_i^{\mathrm{T}}-m_i\boldsymbol\mu_i\boldsymbol\mu_i^{\mathrm{T}}\right) \\ &= \sum_{i=1}^N\left(-m_i\boldsymbol\mu_i\boldsymbol\mu^{\mathrm{T}}-m_i\boldsymbol\mu\boldsymbol\mu_i^{\mathrm{T}}+m_i\boldsymbol\mu\boldsymbol\mu^{\mathrm{T}}+m_i\boldsymbol\mu_i\boldsymbol\mu_i^{\mathrm{T}}\right) \\ &= \sum_{i=1}^Nm_i\left(-\boldsymbol\mu_i\boldsymbol\mu^{\mathrm{T}}-\boldsymbol\mu\boldsymbol\mu_i^{\mathrm{T}}+\boldsymbol\mu\boldsymbol\mu^{\mathrm{T}}+\boldsymbol\mu_i\boldsymbol\mu_i^{\mathrm{T}}\right) \\ &= \sum_{i=1}^N m_i(\boldsymbol\mu_i-\boldsymbol\mu)(\boldsymbol\mu_i-\boldsymbol\mu)^{\mathrm{T}} \end{aligned}$$

3.44

$$\max\limits_{\mathbf{W}}\cfrac{ \operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_b \mathbf{W})}{\operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_w \mathbf{W})}$$ [解析]：此式是公式(3.35)的推广形式，证明如下： 设$\mathbf{W}=(\boldsymbol w_1,\boldsymbol w_2,…,\boldsymbol w_i,…,\boldsymbol w_{N-1})\in\mathbb{R}^{d\times(N-1)}$，其中$\boldsymbol w_i\in\mathbb{R}^{d\times 1}$为$d$行1列的列向量，则 $$\left\{ \begin{aligned} \operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_b \mathbf{W})&=\sum_{i=1}^{N-1}\boldsymbol w_i^{\mathrm{T}}\mathbf{S}_b \boldsymbol w_i \\ \operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_w \mathbf{W})&=\sum_{i=1}^{N-1}\boldsymbol w_i^{\mathrm{T}}\mathbf{S}_w \boldsymbol w_i \end{aligned} \right.$$ 所以公式(3.44)可变形为 $$\max\limits_{\mathbf{W}}\cfrac{ \sum_{i=1}^{N-1}\boldsymbol w_i^{\mathrm{T}}\mathbf{S}_b \boldsymbol w_i}{\sum_{i=1}^{N-1}\boldsymbol w_i^{\mathrm{T}}\mathbf{S}_w \boldsymbol w_i}$$ 对比公式(3.35)易知上式即公式(3.35)的推广形式

3.45

$$\mathbf{S}_b\mathbf{W}=\lambda\mathbf{S}_w\mathbf{W}$$ [推导]：同公式(3.35)一样，我们在此处也固定公式(3.44)的分母为1，那么公式(3.44)此时等价于如下优化问题 $$\begin{array}{cl}\underset{\boldsymbol{w}}{\min} & -\operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_b \mathbf{W}) \\ \text { s.t. } & \operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_w \mathbf{W})=1\end{array}$$ 根据拉格朗日乘子法可知，上述优化问题的拉格朗日函数为 $$L(\mathbf{W},\lambda)=-\operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_b \mathbf{W})+\lambda(\operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_w \mathbf{W})-1)$$ 根据矩阵微分公式$\cfrac{\partial}{\partial \mathbf{X}} \text { tr }(\mathbf{X}^{\mathrm{T}} \mathbf{B} \mathbf{X})=(\mathbf{B}+\mathbf{B}^{\mathrm{T}})\mathbf{X}$对上式关于$\mathbf{W}$求偏导可得 $$\begin{aligned} \cfrac{\partial L(\mathbf{W},\lambda)}{\partial \mathbf{W}} &= -\cfrac{\partial\left(\operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_b \mathbf{W})\right)}{\partial \mathbf{W}}+\lambda \cfrac{\partial\left(\operatorname{tr}(\mathbf{W}^{\mathrm{T}}\mathbf{S}_w \mathbf{W})-1\right)}{\partial \mathbf{W}} \\ &= -(\mathbf{S}_b+\mathbf{S}_b^{\mathrm{T}})\mathbf{W}+\lambda(\mathbf{S}_w+\mathbf{S}_w^{\mathrm{T}})\mathbf{W} \end{aligned}$$ 由于$\mathbf{S}_b=\mathbf{S}_b^{\mathrm{T}},\mathbf{S}_w=\mathbf{S}_w^{\mathrm{T}}$，所以 $$\cfrac{\partial L(\mathbf{W},\lambda)}{\partial \mathbf{W}} = -2\mathbf{S}_b\mathbf{W}+2\lambda\mathbf{S}_w\mathbf{W}$$ 令上式等于$\mathbf{0}$即可得 $$-2\mathbf{S}_b\mathbf{W}+2\lambda\mathbf{S}_w\mathbf{W}=\mathbf{0}$$ $$\mathbf{S}_b\mathbf{W}=\lambda\mathbf{S}_w\mathbf{W}$$