Topological Signal Processing

In many applications, from sensor to social networks, transportation systems, gene regulatory networks, the observations can be represented as a signal defined over the vertices of a graph. The analysis of such signals requires the extension of standard signal processing tools. Graph Signal Processing (GSP) has emerged in the last years as a generalization of signal processing to deal with signals defined over the vertices of a graph. 

We contributed to GSP by deriving a fundamental link between uncertainty principle and sampling theory [1], proposing a novel graph topology inference technique [2], showing how to deal with directed graphs [3] and with topology uncertainties [4]. 

Lately, we extended GSP to Topological Signal Processing (TSP) [5, 6], as a very general framework to deal with signal defined over a topological space, i.e. a set of points along with a set of neighborhood relations. We focused on simplicial complexes, as a generalization of graphs, to incorporate multiway relations of any order. This includes, as a simple example, flow signals defined over the edges of a graph. To illustrate the benefits of the proposed methodologies, we considered applications to data traffic over wireless networks and processing vector fields [5, 6]. 

The current research activity focuses on merging TSP tools with machine learning, giving rise to Topological Deep Learning, where deep neural networks are trained by taking into account the propertied of the topological space where the signals of different layers reside