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 define spectral filters via the LB operator on a graph and explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters. We then update the LB operator for pooling in the LB-CNN. We employ the brain image data from Alzheimer's Disease Neuroimaging Initiative (ADNI) and Open Access Series of Imaging Studies (OASIS) to demonstrate the use of the proposed LB-CNN. Based on the cortical thickness of two datasets, we showed that the LB-CNN slightly improves 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. The LB-CNN trained via the ADNI dataset can achieve reasonable classification accuracy for the OASIS dataset. 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).
Keywords: Chebyshev polynomial; Graph convolutional neural network; Hermite polynomial; Laguerre polynomial; Laplace–Beltrami operator; Signals on surfaces.