Soutenance publique de thèse de doctorat en Sciences mathématiques - Lorenzo GIAMBAGLI

A Journey Through Reciprocal Space: from Deep Spectral Learning to Topological Signals

Jury

• Prof. Anne-Sophie LIBERT (département de matéhmatiques, UNamur), présidente
• Prof. Timoteo CARLETTI (département de mathématique, UNamur), promoteur et secrétaire
• Prof. Duccio FANELLI (Università di Firenze), co-promoteur
  
• Prof. Michael SCHAUB (RWTH Aachen University)
  
• Prof. Mattia FRASCA (Università degli studi di Catania)
  
• Prof. Cecilia CLEMENTI (Freie Universität Berlin)

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.

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