Stability to Deformations of Manifold Filters and Manifold Neural Networks

TL;DR

This paper introduces manifold convolutional filters and neural networks, analyzing their stability to smooth deformations, and highlights their limitations in discriminating high-frequency components, with implications for large-scale graph analysis.

Contribution

It generalizes the stability analysis of graph and continuous-time filters to manifold filters, providing new insights into their behavior under deformations.

Findings

Manifold filters are stable to smooth deformations of the manifold.

High frequency discrimination is limited in manifold, graph, and continuous filters.

Using neural networks can improve discrimination of high frequency components.

Abstract

The paper defines and studies manifold (M) convolutional filters and neural networks (NNs). \emph{Manifold} filters and MNNs are defined in terms of the Laplace-Beltrami operator exponential and are such that \emph{graph} (G) filters and neural networks (NNs) are recovered as discrete approximations when the manifold is sampled. These filters admit a spectral representation which is a generalization of both the spectral representation of graph filters and the frequency response of standard convolutional filters in continuous time. The main technical contribution of the paper is to analyze the stability of manifold filters and MNNs to smooth deformations of the manifold. This analysis generalizes known stability properties of graph filters and GNNs and it is also a generalization of known stability properties of standard convolutional filters and neural networks in continuous time. The most important observation that follows from this analysis is that manifold filters, same as graph filters and standard continuous time filters, have difficulty discriminating high frequency components in the presence of deformations. This is a challenge that can be ameliorated with the use of manifold, graph, or continuous time neural networks. The most important practical consequence of this analysis is to shed light on the behavior of graph filters and GNNs in large-scale graphs.