Topological signal processing and learning under uncertainties

In this thesis, we explore the frontiers of topological signal processing and deep learning, extending traditional methods from graphs to simplicial complexes and to more general cell complexes. This work develops a comprehensive framework that blends algebraic topology with signal processing, enabling the analysis of signals defined over topological domains that capture higher-order interactions and structural features of data. We introduce novel methodologies to process signals on cell complexes, which—unlike simplicial complexes—allow for the representation of cycles of any order without the restrictive inclusion property. Real-world data is often subject to perturbations, therefore we further extend our analysis by developing a theoretical framework for assessing the stability of topological filters under uncertainties. By establishing new stability bounds for both graph and simplicial complex filters, and by introducing robust filter design strategies, we provide crucial insights into the resilience of topological methods. These stability results also inform our derivation of novel bounds for graph convolutional networks, linking traditional signal processing stability with the performance of modern deep learning architectures. We also propose a new class of neural network models, termed Generalized Simplicial Attention Neural Networks (GSANs), which leverage masked self-attention mechanisms to process data on simplicial complexes. Finally, the practical impact of our theoretical and algorithmic contributions is showcased in the realm of biological data analysis. In order to handle as much as possible the uncertainty given by the collection of real world data we propose a reconstruction pipeline for Drug Repurposing Knowledge Graph with enriched node and edge features and higher-order interactions, designing a novel deep learning architecture tailored for drug repurposing. Overall, this thesis presents a unified approach that bridges topological signal processing and deep learning, offering new insights and tools for analyzing complex data across a variety of domains.