A Journey Through Reciprocal Space

: from Deep Spectral Learning to Topological Signals

Abstract

This thesis investigates the intersection of theoretical physics, machine learning, and network dynamics, with a focus on Neural Network Interpretability and Signal Dynamics in Simplicial Complexes. It explores the crucial role of spectral properties of the adjacency matrix in understanding and controlling network systems, demonstrating their application in both neural networks and complex systems. The study introduces "Spectral Parameterization," a novel approach to Feed-Forward Neural Network weight analysis, using spectral graph theory to redefine the connections within a network. This method facilitates several advancements, including a new formalism for understanding feature extraction, a reduction in trainable parameters through eigenvalue adjustments, a Structural Pruning Algorithm for optimizing network efficiency, and a Spectral Regularization Technique for network compression without performance loss. Additionally, the dissertation delves into the dynamics of topological signals in Simplicial Complexes, emphasizing the spectral analysis of the Dirac operator to understand pattern formation and synchronization phenomena. The research links the spectral properties of neural networks and Simplicial complexes, culminating in the development of the Recurrent Spectral Network, a learning dynamical system. This work contributes to a deeper comprehension of spectral methods in neural networks and Simplicial complexes, offering both theoretical insights and practical tools for future research and applications in these fields.

Date of Award	21 Feb 2024
Original language	English
Awarding Institution	University of Namur
Sponsors	Department of Mathematics and Computer Science, University of Florence
Supervisor	Teo Carletti (Supervisor), Duccio Fanelli (Co-Supervisor), Anne-Sophie Libert (President), Mattia Frasca (Jury), Michael SCHAUB (Jury) & Cecilia Clementi (Jury)