Convolutional Learning on Simplicial Complexes

TL;DR

This paper introduces a novel simplicial complex convolutional neural network (SCCNN) that generalizes existing methods by capturing multi-hop adjacencies and inter-simplicial couplings, with theoretical analysis and empirical validation.

Contribution

It presents a new SCCNN architecture that incorporates symmetries, spectral analysis, and stability considerations for learning on simplicial complexes, advancing the state-of-the-art.

Findings

Higher-order convolutions improve prediction accuracy.

Inter-simplicial couplings enhance data representation.

SCCNNs are permutation and orientation equivariant.

Abstract

We propose a simplicial complex convolutional neural network (SCCNN) to learn data representations on simplicial complexes. It performs convolutions based on the multi-hop simplicial adjacencies via common faces and cofaces independently and captures the inter-simplicial couplings, generalizing state-of-the-art. Upon studying symmetries of the simplicial domain and the data space, it is shown to be permutation and orientation equivariant, thus, incorporating such inductive biases. Based on the Hodge theory, we perform a spectral analysis to understand how SCCNNs regulate data in different frequencies, showing that the convolutions via faces and cofaces operate in two orthogonal data spaces. Lastly, we study the stability of SCCNNs to domain deformations and examine the effects of various factors. Empirical results show the benefits of higher-order convolutions and inter-simplicial couplings in simplex prediction and trajectory prediction.