Topological Signal Processing Over Simplicial Complexes

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...

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Bibliographic Details
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
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Summary: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.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2020.2981920