  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).
