Topological Signal Processing: Making Sense of Data Building on Multiway Relations

Abstract

Uncovering hidden relations in complex data sets is a key step to making sense of the data, which is a hot topic in our era of data deluge. Graph-based representations are examples of tools able to encode relations in a mathematical structure enabling the uncovering of patterns like clusters and paths. However, graphs only capture pairwise relations encoded in the presence of edges, but there are many forms of interaction that cannot be reduced to pairwise relations. To overcome the limitations of graph-based representations, it is necessary to incorporate multiway relations. In this article, we exploit tools from algebraic topology to handle multiway relations. Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study a topological space, that is, a set of points, along with a set of neighborhoods. More specifically, we illustrate topological signal processing (TSP), a framework encompassing a class of methods for analyzing signals defined over a topological space. Given its generality, TSP incorporates graph signal processing (GSP) as a particular case. After motivating the use of topological and geometrical methods for detecting patterns in the data, we present the signal processing tools based on algebraic topology and then illustrate their advantages with respect to graph-based methodology.