Manifold Filter-Combine Networks

TL;DR

This paper introduces Manifold Filter-Combine Networks (MFCNs), a framework for manifold neural networks that extends graph neural network concepts to high-dimensional data, with proven convergence and practical demonstrations.

Contribution

It proposes a novel filter-combine framework for MNNs, including a method for implementation on high-dimensional point clouds with theoretical convergence guarantees.

Findings

Method converges to a continuum limit as data points increase

Effective on real-world and synthetic datasets

Provides a manifold analogue of popular GNNs

Abstract

In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). Our filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as manifold analogues of various popular GNNs. We propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating an underlying manifold by a sparse graph. We then prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity, and we numerically demonstrate its effectiveness on real-world and synthetic data sets.