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Stability to Deformations of Manifold Filters and Manifold Neural Networks

Zhiyang Wang, Luana Ruiz, Alejandro Ribeiro

TL;DR

Manifold filters, same as graph filters and standard continuous time filters, have difficulty discriminating high frequency components in the presence of deformations and this is a challenge that can be ameliorated with the use of manifold, graph, or continuous time neural networks.

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.

Stability to Deformations of Manifold Filters and Manifold Neural Networks

TL;DR

Manifold filters, same as graph filters and standard continuous time filters, have difficulty discriminating high frequency components in the presence of deformations and this is a challenge that can be ameliorated with the use of manifold, graph, or continuous time neural networks.

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.

Paper Structure

This paper contains 26 sections, 126 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Manifold Convolutional Filters
  3. Laplace-Beltrami Operator
  4. Manifold Filters
  5. Frequency Representation of Manifold Filters
  6. Stability of Manifold Filters with respect to Manifold Deformations
  7. Stability of Manifold Filters to Absolute Perturbations
  8. Stability of Manifold Filters to Relative Perturbations
  9. Stability of Manifold Neural Networks
  10. Stability of MNNs to LB Operator Perturbations
  11. Stability of MNNs to Manifold Deformations
  12. Discussions
  13. From Manifold Neural Networks to Graph Neural Networks
  14. Discretization in the Space Domain
  15. Discretization in the Time Domain
  16. ...and 11 more sections

Figures (6)

  • Figure 1: Illustration of an $\alpha$-FDT filter. The $x$-axis stands for the spectrum with each sample representing an eigenvalue. The gray shaded areas show the grouping of the eigenvalues according to Definition \ref{['def:alpha-spectrum']}. The red lines show a set of $\alpha$-FDT filters that can discriminate each eigenvalue group.
  • Figure 2: Illustration of a $\gamma$-FRT filter. The $x$-axis stands for the spectrum with each sample representing an eigenvalue. The gray shaded area shows the grouping of the eigenvalues according to Definition \ref{['def:frt-spectrum']}. The red lines show a set of $\alpha$-FDT filters that can discriminate each eigenvalue group.
  • Figure 3: Point cloud models with 300 sampling points in each model. Our goal is to identify chair models from other models such as toilet and table.
  • Figure 4: Difference between error rates on the original test dataset and the deformed one.
  • Figure 5: Difference between error rates on the original test dataset and the deformed one.
  • ...and 1 more figures

Theorems & Definitions (10)

  • proof
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  • proof : Proof of Lemma \ref{['lem:eigenvalue_absolute']}
  • proof : Proof of Lemma \ref{['lem:davis-kahan']}
  • proof : Proof of Lemma \ref{['lem:eigenvalue_relative']}