Sub-networks and Spectral Anisotropy in Deep Neural Networks

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Examensarbete för masterexamen
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Deep neural networks (DNNs) have achieved remarkable success across diverse domains, yet the fundamental reasons behind their efficacy and ability to generalize remain elusive. This thesis examines how over-parameterized DNNs learn and generalize by investigating two interconnected phenomena: the emergence of sparse, critical sub-networks (aligned with the Lottery Ticket Hypothesis) and the structural symmetry-breaking. Additionally, we explore the geometric structure of the parameter space, with a particular focus on the anisotropy of the Fisher Information Matrix (FIM) spectrum. We demonstrate that different layers in a deep network exhibit varying degrees of symmetry breaking, which we link to the presence of sub-networks that encapsulate the model’s core representational capacity. Using two distinct criteria—magnitudebased and change-based—we identify critical sub-networks and show that, despite the over-parameterization of DNNs, these sparse sub-networks play a central role in achieving high performance. By analyzing the spectrum of the FIM, we reveal that DNNs evolve along a limited number of dominant eigendirections, spanning a subspace where training dynamics converge. This finding highlights an intrinsic anisotropy in the parameter manifold. Furthermore, we investigate how this anisotropy correlates with the emergence of sub-networks and the internal structure of the subspace. Overall, this thesis provides a novel perspective on the roles of implicit regularization, loss landscape geometry, and sparse substructures in modern deep neural networks, offering insights into the geometric nature of DNNs.

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Deep Neural Networks, Information Geometry, Generalization, Spectral Analysis, Lottery Ticket Hypothesis

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