$\begin{array}{l}\left(g * x\right)&=U\left(U^{T} x \odot U^{T} g\right)\\&=U\left(U^{T} g \odot U^{T} x\right)\\&=g_{\theta}\left(U \Lambda U^{T}\right) x\\&=U g_{\theta}(\Lambda) U^{T} x\end{array}$
ps:后面两部推导参考 Courant-Fischer min-max theorem ：$\underset{\operatorname{dim}(U)=k}{min} \;\;\underset{x \in U,\|x\|=1}{max} x^{H}Ax= \lambda_{k} $。
Where
Symmetric normalized Laplacian ：$L^{\text {sym }}=D^{-\frac{1}{2}} L D^{-\frac{1}{2}}=D^{-\frac{1}{2}}(D-A) D^{-\frac{1}{2}}=I_{n}-D^{-\frac{1}{2}} A D^{-\frac{1}{2}}=U \Lambda U^{T}$
$U$ is the matrix of eigenvectors of the Symmetric normalized Laplacian.
$Λ$ a diagonal matrix of its eigenvalues of the Symmetric normalized Laplacian.
$U^{\top} x$ being the graph Fourier transform of $x$.
We can understand $g_{\theta }$ as a function of the eigenvalues of L, i.e. $g_{\theta }(Λ)$.
接下来将介绍的图上频域卷积的工作，都是在 $g_{\theta}(\Lambda)$ 的基础上做文章，参数 $\theta$ 即为模型需要学习的卷积核参数。
$g_{\theta}(\Lambda)=\left[\begin{array}{cccc}\hat{g}_{\theta}\left(\lambda_{0}\right) & 0 & \cdots & 0 \\0 & \hat{g}_{\theta}\left(\lambda_{1}\right) & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & \hat{g}_{\theta}\left(\lambda_{n-1}\right)\end{array}\right]$
2.1.2 Improvement 1 :Polynomial parametrization for localized filters 在二代 GCN 中采用:
$g_{\theta}(\Lambda)=\sum \limits_{k=0}^{K-1} \theta_{k} \Lambda^{k}$
which is
$\hat{g}_{\theta}\left(\lambda_{i}\right)=\sum \limits _{k=0}^{K-1} \theta_{k} \lambda_{i}{ }^{k}$