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A Convergence Rate for Manifold Neural Networks

Joyce ChewDeanna NeedellMichael Perlmutter
Dec 2022
摘要
High-dimensional data arises in numerous applications, and the rapidlydeveloping field of geometric deep learning seeks to develop neural networkarchitectures to analyze such data in non-Euclidean domains, such as graphs andmanifolds. Recent work by Z. Wang, L. Ruiz, and A. Ribeiro has introduced amethod for constructing manifold neural networks using the spectraldecomposition of the Laplace Beltrami operator. Moreover, in this work, theauthors provide a numerical scheme for implementing such neural networks whenthe manifold is unknown and one only has access to finitely many sample points.The authors show that this scheme, which relies upon building a data-drivengraph, converges to the continuum limit as the number of sample points tends toinfinity. Here, we build upon this result by establishing a rate of convergencethat depends on the intrinsic dimension of the manifold but is independent ofthe ambient dimension. We also discuss how the rate of convergence depends onthe depth of the network and the number of filters used in each layer.
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