笔记

1.GCN

图上的图信号：
$$x=[1,2,3,4,4]$$

先滤波，再做非线性变换
$$x^{\prime }=g_{\theta }\ast x_k$$$$x_{k+1}=\sigma (w{x}^{\prime })$$

使用图卷积滤波：
$$g_{\theta }\ast x=U{g}_{\theta }{U}^{T}x$$
其中：$L=I_N-D^{-1/2}A{D}^{-1/2}=U\Lambda U^T$, $g_{\theta }=diag(\theta )$

简化滤波过程：

1.1 切比雪夫近似

由Hammond et al. (2011）得，有切比雪夫近似：
$$g_{{\theta }^{\prime }}(\Lambda )\approx \sum _{k=0}^K{\theta }_k^{\prime }{T}_k(\tilde{\Lambda })$$

其中：$\tilde{\Lambda }=\frac{2}{{\lambda }_{max}}\Lambda -I_N$, $T_k=2x{T}_{k-1}(x)-T_{k-2}(x)$, $T_0(x)=1$, $T_1(x)=x$

因此有：

$$\begin{gathered} g_{{\theta }^{\prime }}\ast x=U{g}_{\theta }(\Lambda )U^{T}x \\ \approx U\sum _{k=0}^K{\theta }_k^{\prime }{T}_k(\tilde{\Lambda })U^{T}x \\ =\sum _{k=0}^K{\theta }_k^{\prime }U{T}_k(\tilde{\Lambda })U^{T}x \\ =\sum _{k=0}^K{\theta }_k^{\prime }{T}_k(U\tilde{\Lambda }{U}^T)x \\ =\sum _{k=0}^K{\theta }_k^{\prime }{T}_k(\tilde{L})x \end{gathered}$$

其中：$\tilde{L}=\frac{2}{{\lambda }_{max}}L-I_N$

1.2 限制阶数K=1

令$K=1$, 因为$T_0(x)=1,T_1(x)=x$，则有

$$\begin{gathered} g_{{\theta }^{\prime }}\ast x=\sum _{k=0}^K{\theta }_k^{\prime }{T}_k(\tilde{L})x \\ \approx ({\theta }_0{T}_0(\tilde{L})+{\theta }_1{T}_1(\tilde{L}))x \\ =({\theta }_0+{\theta }_1\tilde{L})x \end{gathered}$$

其中：$\tilde{L}=\frac{2}{{\lambda }_{max}}L-I_N$

1.3 假设${\lambda }_{max}=2$

假设${\lambda }_{max}=2$，则有$\tilde{L}=L-I_N$，有：
$$\begin{gathered} g_{{\theta }^{\prime }}\ast x=({\theta }_0+{\theta }_1\tilde{L})x \\ =({\theta }_0+{\theta }_1(L-I_N))x \end{gathered}$$

1.4 设${\theta }_0,{\theta }_1$

设${\theta }_0=-{\theta }_1$，有
$$\begin{gathered} g_{{\theta }^{\prime }}\ast x=({\theta }_0+{\theta }_1(L-I_N))x \\ =\theta (I_N-L+I_N)x \\ =\theta (I_N-(I_N-D^{-1/2}A{D}^{-1/2})+I_N)x \\ =\theta (I_N+D^{-1/2}A{D}^{-1/2})x \end{gathered}$$

1.5 renormalization trick

$$\begin{gathered} g_{{\theta }^{\prime }}\ast x=\theta (I_N+D^{-1/2}A{D}^{-1/2})x \\ \approx \theta ({\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2})x \end{gathered}$$

1.6 总结

综上：
$$x^{\prime }=g_{{\theta }^{\prime }}\ast x\approx \theta ({\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2})x$$

代入非线性方程，有：
$$\begin{gathered} x_{(k+1)}=\sigma (wx) \\ \approx \sigma (w({\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2})x_k) \end{gathered}$$

对于特征矩阵$X$：
$$X_{(k+1)}\approx \sigma (({\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2})\Theta X_k)$$

2.SGC

2.1一阶切比雪夫滤波器

在GCN中，经过近似，一阶(K=1)切比雪夫滤波器近似为传播矩阵：
$$S_{1-order}=I_N+D^{-1/2}A{D}^{-1/2}$$
由于$L=I_N-D^{-1/2}A{D}^{-1/2}$, 因此有
$$S_{1-order}=2I_N-L$$

$$\begin{gathered} x^{\prime }=S_{1-order}x \\ =(I_N+D^{-1/2}A{D}^{-1/2})x \\ =(2I_N-L)x \\ =(2I_N-U\Lambda U^T)x \\ =(2U{U}^{-1}-U\Lambda U^T)x \\ =U(2I-\Lambda )U^T \end{gathered}$$
其中，由于L是实对称矩阵，因此有$U^T=U^{-1}$

由此可得：相当于滤波函数为
$$g_{\theta }(\Lambda )=2I-\Lambda$$
也即
$$g_{\theta }(\lambda )=2-\lambda$$
其中，$\lambda$是拉普拉斯矩阵$L$的特征值，表示频率

在经过K次累积后(K层网络)，有
$$g_{\theta }(\lambda {)}^K=(2-\lambda {)}^K$$
其函数图像为

2.2 增强正则化邻接矩阵

当GCN采用renormalization trick策略后，传播矩阵由$S_{1-order}$改为${\tilde{S}}_{adj}$，其中：
$${\tilde{S}}_{adj}={\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2}$$ 其中$\tilde{A}=A+I$, $\tilde{D}=D+I$
相应的，定义增强正则化矩阵$\tilde{L}=I_N-{\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2}$, 其特征值为$\tilde{\lambda }$

相应的，使用${\tilde{S}}_{adj}$做传播矩阵，有

$$\begin{gathered} x^{\prime }={\tilde{S}}_{adj}x \\ =({\tilde{D}}^{-1/2}\tilde{A}{\tilde{D}}^{-1/2})x \\ =(I_N-\tilde{L})x \\ =(I_N-U\tilde{\Lambda }{U}^T)x \\ =(U{U}^{-1}-U\tilde{\Lambda }{U}^T)x \\ =U(I-\tilde{\Lambda })U^T \end{gathered}$$

也即
$$g_{\theta }(\lambda )=1-\tilde{\lambda }$$
其中，$\lambda$是拉普拉斯矩阵$L$的特征值，表示频率

在经过K次累积后(K层网络)，有
$$g_{\theta }(\tilde{\lambda }{)}^K=(1-\tilde{\lambda }{)}^K$$

SGC证明：
$$0={\lambda }_0<{\lambda }_n<\widetilde{{\lambda }_n}<\lambda$$
因此有图像

或句话说，renormalization trick策略使得传播矩阵的最大特征值变小了，在1.6左右，而不是原先的2

2.3 正则化邻接矩阵

为说明其优点，可以先考虑$S_{adj}=D^{-1/2}A{D}^{-1/2}$做传播矩阵，有
$$\begin{gathered} x^{\prime }=S_{adj}x \\ =(D^{-1/2}A{D}^{-1/2})x \\ =(I_N-L)x \\ =U(I-\Lambda )U^T \end{gathered}$$

也即
$$g_{\theta }(\lambda )=1-\lambda$$

在经过K次累积后(K层网络)，有
$$g_{\theta }(\lambda {)}^K=(1-\lambda {)}^K$$

综上，三种传播矩阵$S_{1-order}$, $S_{adj}$, ${\tilde{S}}_{adj}$做传播矩阵，分别有滤波函数为

FAGCN

设计两个传播矩阵：
$$\begin{gathered} {\mathcal{F}}_L=\alpha I+D^{-1/2}A{D}^{-1/2} \\ =(\alpha +1)I-L \end{gathered}$$

$$\begin{gathered} {\mathcal{F}}_L=\alpha I-D^{-1/2}A{D}^{-1/2} \\ =(\alpha -1)I+L \end{gathered}$$

分别相当于滤波函数

$$g1(\lambda )=(1-\lambda +\alpha )$$$$g2(\lambda )=(\lambda -1+\alpha )$$

其图像分别为