Topological Signal Processing and Deep Learning

In many applications, from sensor to social networks, vehicular networks, big data or biological networks, the signals of interest are defined over the vertices of a graph. Over the last few years, there was significant advancement in the development of processing tools for the analysis of signals defined over a graph, or graph signals for short. Graph signal processing (GSP) extends classical discrete-time signal processing to signals defined over a discrete domain having a very general structure, represented by a graph, that subsumes discrete-time as a very simple case. Since the signal domain is not defined once for all, but it comes to depend on the graph topology, one of the most interesting and intriguing aspects of GSP is that the analysis tools come to depend on the graph topology as well. This paves the way to a plethora of methods, each emphasizing different aspects of the problem. An important feature of graph signals is that the signal domain is not a metric space, as for example with biological networks, where the vertices may be genes, proteins, enzymes, etc, and the presence of an edge between two molecules means that those molecules undergo a chemical reaction. This marks a fundamental difference with respect to time signals where the time domain is inherently a metric space. In this research area, we have provided fundamental contributions in terms of: i) uncertainty principle; ii) sampling and recovery; iii) adaptive and distributed graph signal processing; iv) graph topology inference; v) GSP tools for directed graphs. 

However, in complex interconnected systems, the interactions often cannot be reduced to simple pairwise relationships, and graph representations might result incomplete and inefficient. For instance, in biological networks, multi-way interactions among complex substances (such as genes, proteins, or metabolites) cannot be evoked using simply pairwise relationships; also, in the brain, groups of neurons typically activate at the same time. These applications have sparked a renewed interest in extending GSP tools to incorporate multi-way relationships among data, thus leading to the emergent field of topological signal processing (TSP). In this context, the seminal works done by our group illustrated the benefits obtained by processing signals defined over simplicial and cell complexes, which are specific examples of hyper-graphs with a rich algebraic description that can easily encode multi-way data. Then, we have designed a novel class of signals defined over topological spaces that are maximally concentrated on the topological domain (e.g., over a set of nodes, edges, triangles, etc.) and perfectly localized on the dual domain (e.g., a set of frequencies), with the aim of learning sparse representations of data defined over topological domains. Finally, we have developed attentional neural network architectures that exploit and/or learn the topological structure (i.e., simplical and/or cell complex) of the data, reaching state of the art performance in several recognized benchmark datasets.   

Selected recent papers:

• C. Battiloro, I. Spinelli, L. Telyatnikov, M. Bronstein, S. Scardapane, P. Di Lorenzo, From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Module, submitted to ICLR 2024.  

• L. Giusti, C. Battiloro, L.Testa, P. Di Lorenzo, S. Sardellitti, S. Barbarossa, Cell Attention Networks, Proc. of IJCNN 2023, preprint arxiv:2209.08179.

• C. Battiloro, P.  Di Lorenzo, A. Ribeiro, Parametric Dictionary Learning for Topological Signal Representation, proc. of EUSIPCO 2023, Helsinki, Finland. 

• C. Battiloro, L. Testa, L. Giusti, S. Sardellitti, P. Di Lorenzo, and S. Barbarossa, Generalized Simplicial Attention Neural Networks, submitted to IEEE Transactions on Signal and Information Processing over Networks, 2023.

• D. M. Cinque, C. Battiloro, P. Di Lorenzo, Pooling Strategies for Simplicial Convolutional Networks, Proc. of ICASSP 2023, Rhodes, Greece.

• C. Battiloro, P. Di Lorenzo, and S. Barbarossa, Topological Slepians: Maximally Localized Representations of Signals over Simplicial Complexes, Proc. of ICASSP 2023, Rhodes, Greece. 

• S. Scardapane, I. Spinelli, and P. Di Lorenzo, Distributed Training of Graph Convolutional Networks, IEEE Transactions on Signal and Information Processing over Networks, vol. 7, pp. 87-100, 2021.