【图论】Graph Fourier Transform

Orignal Ariticle: link

本期，根据维基百科、知乎整理出图上的傅里叶变换。

Graph Fourier Transform

In mathematics, graph Fourier transform is a mathematical transform which eigendecomposes the Laplacian matrix of a graph into eigenvalues and eigenvectors. Analogously to classical Fourier Transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis.

Graph Fourier Transform (GFT) is defined as:
$$\mathcal{G}\mathcal{F}[f]({\lambda }_l)=\hat{f}({\lambda }_l)=<f\text{f}f,{u\text{u}u}_l>=\sum _{i=1}^{N}f(i)u_l^T(i).$$
where

• $G=(V,E)$: undirected weighted graph
• $V$: the set of nodes with $|V|=N$
• $E$: the set of edges
• $f:V\to \mathbb{R}$: a graph signal that is a function defined on the vertices of the graph $G$, which maps every vertex $\{v_i{\}}_{i=1,\cdots ,N}$ to a real number $f(i)$. Any graph signal can be projected on the eigenvectors $(0={\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_N)$ of the Laplacian matrix $\mathbf{L}$
• ${\lambda }_l$: the $l$th eigenvalue of the Laplacian matrix $\mathbf{L}$
• ${u\text{u}u}_l$: the $l$th eigenvector of the Laplacian matrix $\mathbf{L}$

It can also be represented as
$$\begin{array}{r}\left[\begin{array}{c}\hat{f}({\lambda }_1)\\ \hat{f}(({\lambda }_2)\\ ⋮\\ \hat{f}(({\lambda }_N)\end{array}\right]={\left[\begin{array}{cccc}{u}_1(1)& u_2(1)& \cdots & u_N(1)\\ u_1(2)& u_2(2)& \cdots & u_N(2)\\ ⋮& ⋮& \ddots & ⋮\\ u_1(N)& u_2(N)& \cdots & u_N(N)\end{array}\right]}^T\left[\begin{array}{c}f(1)\\ f(2)\\ ⋮\\ f(N)\end{array}\right]\end{array},$$
which is equal to
$$\hat{f}\text{f^}\hat{f}={\mathbf{U}}^{T}f\text{f}f.$$
Since Laplacian matrix $\mathbf{L}$ is a real symmetric matrix, its eigenvectors ${u\text{u}u}_1,{u\text{u}u}_2,\cdots ,{u\text{u}u}_N$ form an orthogonal basis. Hence an inverse graph Fourier transform (IGFT) exists, and it is written as

$$\mathcal{I}\mathcal{G}\mathcal{F}[\hat{f}](i)=f(i)=<\hat{f}\text{f^}\hat{f},{u\text{u}u}_l>=\sum _{i=1}^n\hat{f}({\lambda }_l)u_l(i).$$

It can be also represented as

$$\begin{array}{r}\left[\begin{array}{c}f(1)\\ f(2)\\ ⋮\\ f(N)\end{array}\right]=\left[\begin{array}{cccc}{u}_1(1)& u_2(1)& \cdots & u_N(1)\\ u_1(2)& u_2(2)& \cdots & u_N(2)\\ ⋮& ⋮& \ddots & ⋮\\ u_1(N)& u_2(N)& \cdots & u_N(N)\end{array}\right]\left[\begin{array}{c}\hat{f}({\lambda }_1)\\ \hat{f}(({\lambda }_2)\\ ⋮\\ \hat{f}(({\lambda }_N)\end{array}\right]\end{array},$$

which is equal to
$$f\text{f}f=\mathbf{U}\hat{f}\text{f^}\hat{f}.$$

Orthogonal basis for the Fourier transform

The eigenvectors of Laplacian matrix are a set of linearly independent orthogonal bases, thus any graph signal can be expressed as a linear combination of the eigenvectors of Laplacian matrix
$$f\text{f}f={\hat{f}\text{f^}\hat{f}}_1{u\text{u}u}_1+{\hat{f}\text{f^}\hat{f}}_2{u\text{u}u}_2+\cdots +{\hat{f}\text{f^}\hat{f}}_N{u\text{u}u}_N.$$

Convolution Operation on Graph

Generalized convolution in the vertex domain is multiplication in the graph spectral domain:
$$f\text{f}f\ast h\text{h}h=\text{IGFT}(\hat{f}\text{f^}\hat{f}\hat{h}\text{h^}\hat{h})$$
which can be also expressed as
$$f\text{f}f\ast h\text{h}h(i)=\sum _{l=1}^N\hat{f}({\lambda }_l)\hat{h}({\lambda }_l)u_l(i).$$

• $h\text{h}h$: convolution kernel

Steps of Convolution Operation on Graph:

1. $\hat{f}\text{f^}\hat{f}={\mathbf{U}}^{T}f\text{f}f$
2. $\hat{h}\text{h^}\hat{h}={\mathbf{U}}^{T}h\text{h}h$
3. $\hat{h}\text{h^}\hat{h}\hat{f}\text{f^}\hat{f}={\mathbf{U}}^{T}h\text{h}h\odot {\mathbf{U}}^{T}f\text{f}f=\hat{f}\text{f^}\hat{f}\hat{h}\text{h^}\hat{h}$, i,e,

$$\begin{array}{rl}f\text{f}f\ast h\text{h}h& =\mathbf{U}\hat{f}\text{f^}\hat{f}\hat{h}\text{h^}\hat{h}\\ & =\mathbf{U}\left[\begin{array}{ccc}\hat{h}({\lambda }_1)& & \\ & \ddots & \\ & & \hat{h}({\lambda }_N)\end{array}\right]{\mathbf{U}}^{T}f\text{f}f\end{array}$$