Signal processing on simplicial complexes

TL;DR

This paper extends graph signal processing to simplicial complexes, enabling analysis of complex high-dimensional relations beyond traditional graphs, with a general framework and numerical demonstrations.

Contribution

It develops a novel signal processing framework on simplicial complexes that generalizes traditional GSP to higher-dimensional structures.

Findings

Framework recovers traditional GSP on graphs

Enables processing of signals on high-dimensional complexes

Numerical examples demonstrate applicability

Abstract

Theoretical development and applications of graph signal processing (GSP) have attracted much attention. In classical GSP, the underlying structures are restricted in terms of dimensionality. A graph is a combinatorial object that models binary relations, and it does not directly model complex n-ary relations. One possible high dimensional generalization of graphs are simplicial complexes. They are a step between the constrained case of graphs and the general case of hypergraphs. In this paper, we develop a signal processing framework on simplicial complexes, such that we recover the traditional GSP theory when restricted to signals on graphs. It is worth mentioning that the framework works much more generally, though the focus of the paper is on simplicial complexes. We demonstrate how to perform signal processing with the framework using numerical examples.