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 non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a very simple case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. We are interested in the most general case, where the signals are associated with vertices, edges and higher order complexes. After reviewing the basic principles of algebraic topology, we show how to build unitary bases to represent signals defined over sets of increasing order, giving rise to a spectral simplicial complex theory. Then we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. After having established the analysis tools, we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals to illustrate the benefits of the proposed methodologies.