Spectral domain graph convolution

Background knowledge of graph convolution

Classic convolutional neural networks have achieved success in many fields

• Image classification
• Visual semantic question answering
• Image segmentation
• Target Detection
• Talk by looking at pictures
• video processing

But the limitations of classic convolutional networks: they cannot handle graph-structured data.

The limitations of classic convolutional networks: they cannot handle graph-structured data.

• Limitations of classic convolutional networks in processing graph-structured data
  • Can only process data with fixed input dimensions
  • Local input data must be in order
  • 
    
    Voice, image, and video (regular structure) meet the above two requirements, but they are not suitable for spectral domain graph convolution of graph-structured data (non-Euclidean space data)
  • According to the graph theory and the convolution theorem, the data is converted from the spatial domain to the spectral domain for processing.
  • Have a relatively solid theoretical foundation

1.1 Implementation ideas of spectral domain graph convolution

What is convolution

According to the convolution theorem, the Fourier transform of the convolution of two signals in the spatial domain (or time domain) is equal to the product of the Fourier transform of the two signals in the frequency domain.
The meaning of convolution operation:

• Convert spatial domain signal to frequency domain and then multiply
• Convert the multiplied result to the empty domain

How to define the Fourier transform on a graph

Based on graph theory, graph Fourier transform

1.2 Laplacian matrix

Definition of Laplacian Matrix – Symbol Settings

The definition of Laplacian matrix: degree matrix minus adjacency matrix

The Laplace matrix is ​​a symmetric positive semi-definite matrix
. As a symmetric positive semi-definite matrix, the Laplace matrix has the following properties:

1. A symmetric matrix of order n must have n linearly independent eigenvectors
2. The eigenvectors corresponding to different eigenvalues ​​of the symmetric matrix are orthogonal to each other. The matrix composed of these orthogonal eigenvectors is an orthogonal matrix.
3. The eigenvectors of a real symmetric matrix must be real vectors
4. The eigenvalues ​​of a positive semidefinite matrix must be nonnegative

Spectral Decomposition of Laplacian Matrix

Eigen decomposition, also known as spectral decomposition, is a method of decomposing a matrix into the product of matrices represented by its eigenvalues ​​and eigenvectors. An

n-order symmetric matrix must have n linearly independent eigenvectors ( symmetric matrix properties ). n linearly independent vectors in an n-dimensional linear space can constitute a set of basis. ( Matrix theory knowledge )
The n eigenvectors of the Laplacian matrix are all linearly independent. They are a set of basis in n-dimensional space.
The eigenvectors corresponding to different eigenvalues ​​of the symmetric matrix are orthogonal to each other. The matrix composed of these orthogonal eigenvectors is an orthogonal matrix (property of the symmetric matrix). The n
eigenvectors of the Laplacian matrix are one in the n-dimensional space. set of orthonormal basis

Conclusion: The Laplacian matrix is ​​a kind of Laplacian operator on the graph.

In the Euclidean space, the two-dimensional Laplacian operator can be understood as the difference between the central node and the surrounding nodes, and then the sum.
A similar Laplacian operator on a graph can be defined as follows.


Conclusion: The Laplacian matrix is ​​a kind of Laplacian operator on the graph.

1.3 Graph Fourier Transform

The signal on the graph is generally expressed as a vector. Assuming there are n nodes, the nodes on the graph are recorded as:

each node has a signal value, similar to the pixel value on the image. The value on node i is x(i) = xi

Fourier transform formula: one is continuous and the other is discrete.

The cosine functions of different frequencies in Fourier transform can be regarded as basis functions, and their Fourier coefficients represent the amplitude of the basis.
The essence of the inverse Fourier transform is to express any function as a linear combination of several orthogonal basis functions. The essence of the forward
Fourier transform is to find the coefficients of the linear combination. The specific method is to use the original function and the basis function. Find the inner product of conjugate.

Fourier transform actually uses the eigenvector of the Laplacian matrix as the basis function of the graph Fourier transform. The signal on any graph can be expressed as:

Why use Laplacian eigenvectors as basis

The classical Fourier transform has the following rules: the basis function of the Fourier transform is the eigenfunction of the Laplacian operator
and because the Laplacian matrix is ​​the Laplacian operator on the graph, so, similarly, The basis function of the graph Fourier transform is the eigenvector of the graph Laplacian matrix

1.4 Convolution theorem

Considering the two signals as input signals and convolution kernels respectively, the convolution operation can be defined as:

1. Convert spatial domain signal to frequency domain and then multiply
2. Convert the multiplied result to the empty domain

Two or three classic graph convolution models

SCNN

ChebNet

GCN