Revisiting convolutional neural network on graphs with polynomial approximations of Laplace-Beltrami spectral filtering

TL;DR

This paper explores spectral graph convolutional neural networks using polynomial approximations of the Laplace-Beltrami operator, comparing their effectiveness and computational costs in brain image classification tasks.

Contribution

It introduces the LB-CNN framework with polynomial approximations and evaluates their performance on Alzheimer's brain imaging data.

Findings

No significant accuracy improvement over spectral graph-CNN

Polynomials had similar computational costs and classification accuracy

Deep learning can learn spectral filters independent of polynomial type

Abstract

This paper revisits spectral graph convolutional neural networks (graph-CNNs) given in Defferrard (2016) and develops the Laplace-Beltrami CNN (LB-CNN) by replacing the graph Laplacian with the LB operator. We then define spectral filters via the LB operator on a graph. We explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters and define an update of the LB operator for pooling in the LBCNN. We employ the brain image data from Alzheimer's Disease Neuroimaging Initiative (ADNI) and demonstrate the use of the proposed LB-CNN. Based on the cortical thickness of the ADNI dataset, we showed that the LB-CNN didn't improve classification accuracy compared to the spectral graph-CNN. The three polynomials had a similar computational cost and showed comparable classification accuracy in the LB-CNN or spectral graph-CNN. Our findings suggest that even though the shapes of the three polynomials are different, deep learning architecture allows us to learn spectral filters such that the classification performance is not dependent on the type of the polynomials or the operators (graph Laplacian and LB operator).