Dottorato in TECNOLOGIE DELL'INFORMAZIONE E DELLE COMUNICAZIONI (ICT)

Titolo della tesi: Signal Processing and Learning over Topological Spaces

The aim of this thesis is to introduce a variety of signal processing methodologies specifically designed to model, interpret, and learn from data structured within topological spaces. These spaces are loosely characterized as a collection of points together with a neighborhood notion among points. The methodologies and tools discussed herein hold particular relevance and utility when applied to signals defined over combinatorial topological spaces, such as cell complexes, or within metric spaces that exhibit non-trivial properties, such as Riemann manifolds with non-flat metrics. One of the primary motivations behind this research is to address and surmount the constraints encountered with traditional graph-based representations when they are employed to depict intricate systems. This thesis emphasizes the necessity to account for sophisticated, multiway, and geometry-sensitive interactions that are not adequately captured by conventional graph models. The contributions of this work include but are not limited to the development of methodologies for sparse signal representations over simplicial/cell complexes, attentional deep neural networks for data defined over simplicial/cell complexes, and the establishment of an inaugural signal processing framework specifically devised for signals that are defined over the tangent bundles of Riemann manifolds, that we formally connect with cellular sheaves over undirected graphs. The implications of these developments are potentially profound for the signal processing community, which only recently has started to synergize its historical founding concepts with tools from algebraic topology and differential geometry. Moreover, we put particular attention on making the presentation on this work, on the one hand, accessible to a broader SP audience and, on the other hand, complementary to related results from the (pure) machine learning community.