Convolutional Filtering in Simplicial Complexes

TL;DR

This paper introduces convolutional filters for data on simplicial complexes, enabling higher-order network analysis with properties like permutation equivariance and efficient computation.

Contribution

It develops a novel convolutional filtering framework on simplicial complexes using Hodge-Laplacians and incidence matrices, with theoretical properties and practical implementation.

Findings

Filters are permutation and orientation equivariant.

Computational complexity is linear in SC dimension.

Numerical experiments demonstrate effectiveness.

Abstract

This paper proposes convolutional filtering for data whose structure can be modeled by a simplicial complex (SC). SCs are mathematical tools that not only capture pairwise relationships as graphs but account also for higher-order network structures. These filters are built by following the shift-and-sum principle of the convolution operation and rely on the Hodge-Laplacians to shift the signal within the simplex. But since in SCs we have also inter-simplex coupling, we use the incidence matrices to transfer the signal in adjacent simplices and build a filter bank to jointly filter signals from different levels. We prove some interesting properties for the proposed filter bank, including permutation and orientation equivariance, a computational complexity that is linear in the SC dimension, and a spectral interpretation using the simplicial Fourier transform. We illustrate the proposed approach with numerical experiments.