Convergence of Manifold Filter-Combine Networks

TL;DR

This paper introduces Manifold Filter-Combine Networks (MFCNs), a framework for manifold neural networks inspired by graph neural networks, with a method for implementation on point clouds that converges as data increases.

Contribution

It proposes a new MFCN framework, draws parallels with GNNs, and provides a convergence proof for the implementation method on high-dimensional point clouds.

Findings

The MFCN framework generalizes GNNs to manifolds.

The proposed implementation method is consistent and converges with increasing data.

The approach effectively approximates the manifold using sparse graphs.

Abstract

In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). The 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 the manifold analog of various popular GNNs. We then propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating the manifold by a sparse graph. We 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.