Finite Impulse Response Filters for Simplicial Complexes

TL;DR

This paper introduces finite impulse response filters for signals on simplicial complexes, leveraging the Hodge Laplacian to enable spectral filtering and subspace-specific control, with applications in denoising and component extraction.

Contribution

It proposes a novel FIR filtering framework based on the Hodge Laplacian for simplicial complex signals, extending graph filtering to higher-dimensional structures.

Findings

Effective in sub-component extraction

Demonstrates denoising capabilities

Enables nuanced control over signal subspaces

Abstract

In this paper, we study linear filters to process signals defined on simplicial complexes, i.e., signals defined on nodes, edges, triangles, etc. of a simplicial complex, thereby generalizing filtering operations for graph signals. We propose a finite impulse response filter based on the Hodge Laplacian, and demonstrate how this filter can be designed to amplify or attenuate certain spectral components of simplicial signals. Specifically, we discuss how, unlike in the case of node signals, the Fourier transform in the context of edge signals can be understood in terms of two orthogonal subspaces corresponding to the gradient-flow signals and curl-flow signals arising from the Hodge decomposition. By assigning different filter coefficients to the associated terms of the Hodge Laplacian, we develop a subspace-varying filter which enables more nuanced control over these signal types. Numerical experiments are conducted to show the potential of simplicial filters for sub-component extraction, denoising and model approximation.