# Topological Signal Processing over Simplicial Complexes

## TL;DR

This paper develops tools for analyzing signals over topological spaces, especially simplicial complexes, extending graph signal processing to higher-order structures and demonstrating applications to real data.

## Contribution

It introduces a sampling theory for signals on simplicial complexes and proposes a method to infer topology from data, broadening the scope of topological signal processing.

## Key findings

- Building a simplicial complex of order two improves edge signal analysis.
- The proposed topology inference method accurately recovers complex structures.
- Applications demonstrate benefits in analyzing real edge signals and vector fields.

## Abstract

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 real edge signals and to the analysis of discrete vector fields to illustrate the benefits of the proposed methodologies.