Simplicial Convolutional Filters

TL;DR

This paper introduces simplicial convolutional filters for signals on topological spaces like simplicial complexes, extending graph filters to higher-dimensional structures, with applications in denoising and network analysis.

Contribution

It develops a new class of filters based on Hodge Laplacians, analyzing their properties, frequency responses, and design procedures for processing signals on simplicial complexes.

Findings

Filters are linear, shift-invariant, and equivariant.

Frequency responses can distinguish gradient, curl, and harmonic components.

Applications include signal denoising and network analysis.

Abstract

We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted as generalizations of graphs that account for nodes, edges, triangular faces etc. To process such signals, we develop simplicial convolutional filters defined as matrix polynomials of the lower and upper Hodge Laplacians. First, we study the properties of these filters and show that they are linear and shift-invariant, as well as permutation and orientation equivariant. These filters can also be implemented in a distributed fashion with a low computational complexity, as they involve only (multiple rounds of) simplicial shifting between upper and lower adjacent simplices. Second, focusing on edge-flows, we study the frequency responses of these filters and examine how we can use the Hodge-decomposition to delineate gradient, curl and harmonic frequencies. We discuss how these frequencies correspond to the lower- and the upper-adjacent couplings and the kernel of the Hodge Laplacian, respectively, and can be tuned independently by our filter designs. Third, we study different procedures for designing simplicial convolutional filters and discuss their relative advantages. Finally, we corroborate our simplicial filters in several applications: to extract different frequency components of a simplicial signal, to denoise edge flows, and to analyze financial markets and traffic networks.