Topological Signal Processing Over Simplicial Complexes

• Published in: IEEE transactions on signal processing 2020, Vol.68, p.2992-3007
• Main Authors: Barbarossa, Sergio, Sardellitti, Stefania
• Format: Article
• Language: English
• Subjects: Algebraic topology; Apexes; Data mining; Extraterrestrial measurements; Face; Fields (mathematics); graph signal processing; Graph theory; Metric space; Network topology; Signal processing; Topology; topology inference; Wireless networks
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2020.2981920

The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a special case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. Even though the theory can be applied to signals of any order, we focus on signals defined over the edges of a graph and show how building a simplicial complex of order two, i.e. including triangles, yields benefits in the analysis of edge signals. After reviewing the basic principles of algebraic topology, we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. Then we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to traffic analysis over wireless networks and to the processing of discrete vector fields to illustrate the benefits of the proposed methodologies.