$U^{T}=\left[\begin{array}{c}\overrightarrow{u_{0}} \\\overrightarrow{u_{1}} \\\cdots \\u_{n-1}\end{array}\right]=\left[\begin{array}{cccc}u_{0}^{0} & u_{0}^{1} & \cdots & u_{0}^{n-1} \\u_{1}^{0} & u_{1}^{1} & \cdots & u_{1}^{n-1} \\\cdots & \cdots & \cdots & \cdots \\u_{n-1}^{0} & u_{n-1}^{1} & \cdots & u_{n-1}^{n-1}\end{array}\right]$
其中 $\overrightarrow{u_{0}}$ 为特征值为 $\lambda_{0}$ 对应的特征向量， $\overrightarrow{u_{1}}$ 、 $\overrightarrow{u_{2}}$ 、 $\ldots$ 类似
图上傅里叶变换矩阵形式如下：
$F(\lambda)=\left[\begin{array}{c}\hat{f}\left(\lambda_{0}\right) \\\hat{f}\left(\lambda_{1}\right) \\\cdots \\\hat{f}\left(\lambda_{n-1}\right)\end{array}\right]=\left[\begin{array}{cccc}u_{0}^{0} & u_{0}^{1} & \cdots & u_{0}^{n-1} \\u_{1}^{0} & u_{1}^{1} & \cdots & u_{1}^{n-1} \\\cdots & \cdots & \cdots & \cdots \\u_{n-1}^{0} & u_{n-1}^{1} & \cdots & u_{n-1}^{n-1}\end{array}\right] \cdot\left[\begin{array}{c}f(0) \\f(1) \\\cdots \\f(n-1)\end{array}\right]$
常见的傅里叶变换形式为：（下面推导要用）
$\hat{f}=U^{T} f$
$f$ 在图上的逆傅里叶变换：
$f=U \hat{f}$
下面叙述正文~~~~~
AbstractOur convolutional architecture via a localized first-order approximation of spectral graph convolutions.
1 IntroductionTarget：classfy node in graph.
Our methods：a graph-based semi-supervised method.
Type of loss function: graph-based regularization. Function as following:
$\begin {array}{l}\mathcal{L}=\mathcal{L}_{0}+\lambda \mathcal{L}_{\text {reg }}\\\quad \text { with } \quad \mathcal{L}_{\text {reg }}=\sum \limits _{i, j} A_{i j}\left\|f\left(X_{i}\right)-f\left(X_{j}\right)\right\|^{2}=f(X)^{\top} \Delta f(X)\end{array}$
Where
$\mathcal{L}_{0} $ denotes the supervised loss.It is the filter error on the dataset with label.
$\mathcal{L}_{\text {reg }}$ means smoothness between nodes. If two nodes is similary, their own value $f\left(X_{i}\right)$ and $f\left(X_{j}\right)$ may be more equal.
$f$ is the lable function in the graph;
$X$ is a matrix of node feature vectors $X_i$;
$\bigtriangleup = D − A$ is Laplacian matrix;
Adjacency matrix $A \in \mathbb{R}^{N \times N}$ (binary or weighted. For common sitution is weighted adjacency matrix ).
2 Fast approximate convolutions on graphsThe goal of this model is to learn a function of signals/features on a graph $\mathcal{G}=(\mathcal{V}, \mathcal{E})$.
It takes feature matrix $X$ and adjacency matrix $A$ as input:
A feature description $x_{i}$ for every node $i$ ; summarized in a $N \times D$ feature matrix $X$($N$: number of nodes, $D$: number of input features)
A representative description of the graph structure in matrix form; typically in the form of an adjacency matrix $A$ (or some function thereof)
It produces a node-level output $Z$ (an $N×F$ feature matrix, where $F$ is the number of output features per node). Graph-level outputs can be modeled by introducing some form of pooling operation.
Every neural network layer can then be written as a non-linear function
$H^{(l+1)}=f\left(H^{(l)}, A\right),$
with $H^{(0)}=X$ and $H^{(L)}=Z$ (or $z$ for graph-level outputs), $L$ being the number of layers. The specific models then differ only in how $f(\cdot, \cdot)$ is chosen and parameterized.
Conditional GCN model ：