图半监督学习

$$\begin{array}\\ \displaystyle E(f)&=\frac12\sum\limits_{i=1}^m\sum\limits_{j=1}^m\boldsymbol W_{ij}(f(\boldsymbol x_i)-f(\boldsymbol x_j))^2 \\ &=\frac12\left( \sum\limits_{i=1}^md_if^2(x_i)+\sum\limits_{j=1}^md_jf^2(x_j) -2\sum\limits_{i=1}^m\sum\limits_{j=1}^m\boldsymbol W_{ij}f(\boldsymbol x_i)f(\boldsymbol x_j) \right)\\&=\sum\limits_{i=1}^md_if^2(x_i) -\sum\limits_{i=1}^m\sum\limits_{j=1}^m\boldsymbol W_{ij}f(\boldsymbol x_i)f(\boldsymbol x_j)\\&=f^TDf-f^TWf \\&= f^{T}(D-W)f\end{array}$$

$$\begin{aligned} E(f) &=\left(\boldsymbol{f}_{l}^{\mathrm{T}} \boldsymbol{f}_{u}^{\mathrm{T}}\right)\left(\left[\begin{array}{ll} \mathbf{D}_{l l} & \mathbf{0}_{l u} \\ \mathbf{0}_{u l} & \mathbf{D}_{u u} \end{array}\right]-\left[\begin{array}{ll} \mathbf{W}_{l l} & \mathbf{W}_{l u} \\ \mathbf{W}_{u l} & \mathbf{W}_{u u} \end{array}\right]\right)\left[\begin{array}{l} \boldsymbol{f}_{l} \\ \boldsymbol{f}_{u} \end{array}\right] \\&=\left[\begin{array}{ll} \boldsymbol{f}_{l}^{\mathrm{T}} & \boldsymbol{f}_{u}^{\mathrm{T}} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{D}_{l l}-\boldsymbol{W}_{l l} & -\boldsymbol{W}_{l u} \\ -\boldsymbol{W}_{u l} & \boldsymbol{D}_{u u}-\boldsymbol{W}_{u u} \end{array}\right]\left[\begin{array}{l} \boldsymbol{f}_{l} \\ \boldsymbol{f}_{u} \end{array}\right] \\ &=\left[\begin{array}{ll}\boldsymbol{f}_{l}^{\mathrm{T}}\left(\boldsymbol{D}_{l l}-\boldsymbol{W}_{l l}\right)-\boldsymbol{f}_{u}^{\mathrm{T}} \boldsymbol{W}_{u l} & -\boldsymbol{f}_{l}^{\mathrm{T}} \boldsymbol{W}_{l u}+\boldsymbol{f}_{u}^{\mathrm{T}}\left(\boldsymbol{D}_{u u}-\boldsymbol{W}_{u u}\right)\end{array}\right]\left[\begin{array}{l} \boldsymbol{f}_{l} \\ \boldsymbol{f}_{u} \end{array}\right] \\ &=\left(\boldsymbol{f}_{l}^{\mathrm{T}}\left(\boldsymbol{D}_{l l}-\boldsymbol{W}_{l l}\right)-\boldsymbol{f}_{u}^{\mathrm{T}} \boldsymbol{W}_{u l}\right) \boldsymbol{f}_{l}+\left(-\boldsymbol{f}_{l}^{\mathrm{T}} \boldsymbol{W}_{l u}+\boldsymbol{f}_{u}^{\mathrm{T}}\left(\boldsymbol{D}_{u u}-\boldsymbol{W}_{u u}\right)\right) \boldsymbol{f}_{u} \\ &=\boldsymbol{f}_{l}^{\mathrm{T}}\left(\boldsymbol{D}_{l l}-\boldsymbol{W}_{l l}\right) \boldsymbol{f}_{l}-\boldsymbol{f}_{u}^{\mathrm{T}} \boldsymbol{W}_{u l} \boldsymbol{f}_{l}-\boldsymbol{f}_{l}^{\mathrm{T}} \boldsymbol{W}_{l u} \boldsymbol{f}_{u}+\boldsymbol{f}_{u}^{\mathrm{T}}\left(\boldsymbol{D}_{u u}-\boldsymbol{W}_{u u}\right) \boldsymbol{f}_{u} \\ &=\boldsymbol{f}_{l}^{\mathrm{T}}\left(\boldsymbol{D}_{l l}-\boldsymbol{W}_{l l}\right) \boldsymbol{f}_{l}-2 \boldsymbol{f}_{u}^{\mathrm{T}} \boldsymbol{W}_{u l} \boldsymbol{f}_{l}+\boldsymbol{f}_{u}^{\mathrm{T}}\left(\boldsymbol{D}_{u u}-\boldsymbol{W}_{u u}\right) \boldsymbol{f}_{u} \end{aligned}$$

$$\begin{aligned} \frac{\partial E(f)}{\partial \boldsymbol{f}_{u}} &=\frac{\partial \boldsymbol{f}_{l}^{\mathrm{T}}\left(\boldsymbol{D}_{l l}-\boldsymbol{W}_{l l}\right) \boldsymbol{f}_{l}-2 \boldsymbol{f}_{u}^{\mathrm{T}} \boldsymbol{W}_{u l} \boldsymbol{f}_{l}+\boldsymbol{f}_{u}^{\mathrm{T}}\left(\boldsymbol{D}_{u u}-\boldsymbol{W}_{u u}\right) \boldsymbol{f}_{u}}{\partial \boldsymbol{f}_{u}} \\ &=-2 \boldsymbol{W}_{u l} \boldsymbol{f}_{l}+2\left(\boldsymbol{D}_{u u}-\boldsymbol{W}_{u u}\right) \boldsymbol{f}_{u}=0 \end{aligned}$$

$$F(t+1)=\alpha SF(t)+(1-\alpha)Y$$

$$F^*=\lim_{t\to\infty}F(t)=(1-\alpha)(I-\alpha S)^{-1}Y$$

$$\mathbf{F}(t)=(\alpha \mathbf{S})^{t} \mathbf{Y}+(1-\alpha)\left(\sum_{i=0}^{t-1}(\alpha \mathbf{S})^{i}\right) \mathbf{Y}\\ \mathbf{F}^{*}=\lim _{t \rightarrow \infty}\mathbf{F}(t)=\lim _{t \rightarrow \infty}(\alpha \mathbf{S})^{t} \mathbf{Y}+\lim _{t \rightarrow \infty}(1-\alpha)\left(\sum_{i=0}^{t-1}(\alpha \mathbf{S})^{i}\right) \mathbf{Y}\\ \lim _{t \rightarrow \infty} \sum_{i=0}^{t-1}(\alpha \mathbf{S})^{i}=\frac{\mathbf{I}-\lim _{t \rightarrow \infty}(\alpha \mathbf{S})^{t}}{\mathbf{I}-\alpha \mathbf{S}}=\frac{\mathbf{I}}{\mathbf{I}-\alpha \mathbf{S}}=(\mathbf{I}-\alpha \mathbf{S})^{-1}\\ \mathbf{F}^{*}=\lim _{t \rightarrow \infty} \mathbf{F}(t)=(1-\alpha)(\mathbf{I}-\alpha \mathbf{S})^{-1} \mathbf{Y}$$