The shape of learning: topological and geometric perspectives on neural representations

Abstract

This thesis presents a unified framework for analysing the “expressivity” of Deep Neural Networks by combining factor analysis with topological data analysis (TDA). The central idea is to study the learning in Neural Networks as a geometric and topological object that encodes the representational capacity of the model. Towards the same, the geometric structure of Archetypal Analysis is exploited to study the structural manifold underlying the neural embeddings. The archetypal subspace provides a scalable foundation for applying tools from computational topology, enabling the characterisation of topological transitions that reflect changes in representational structure and network complexity. Building upon this methodological foundation, the framework is applied to the model selection problem in the context of transfer learning, where the objective is to identify an optimal model from a collection of pre-trained architectures. By quantifying the topological and geometric signatures of the embedding space, the approach provides an interpretable measure of expressivity that correlates with the generalization behavior of the Neural Network. Furthermore, the study extends the analysis of expressivity to generative learning, through the topological inference of their generative manifolds forging a relationship between manifold geometry, data diversity, and generative performance. Overall, this research contributes a topology-guided and geometrically interpretable framework for understanding, comparing, and selecting neural architectures. It advances the theoretical and empirical interpretations of deep representation learning and establishes a scalable approach to bridging topological structure, expressivity, and generalisation in neural architecture.