Optimal Transport, Wasserstein Spaces and Applications to Machine Learning

This thesis explores the theory and applications of optimal transport (OT), with a particular emphasis on its role in analyzing and quantifying the generalization behavior of deep neural networks (DNNs).We present a dual approach: a rigorous theoretical development of optimal transport grounded in foundational mathematics (metric spaces, measure theory, and functional analysis), and a comprehensive empirical investigation of Wasserstein distances in machine learning contexts.The theoretical component covers the Monge and Kantorovich formulations, duality theory, and the geometry of the Wasserstein space.On the computational side, we study the convergence of empirical Wasserstein distances under various sampling schemes and cost functions, validating theoretical rates and examining their dependence on dimension.We explore the use of pushforward Wasserstein distances by Loulakis and Makridakis to study generalization error in deep learning.We demonstrate that these distances, induced by learned feature representations, offer a tighter quantification of generalization error than traditional bounds.Interestingly, our results reveal that the convergence behavior of pushforward distances often deviates from classical OT theory, hinting at a deeper interaction between data geometry and neural network learning dynamics.Our findings highlight the potential of optimal transport as a principled tool in modern machine learning and suggest promising directions for future theoretical work on the convergence of learned transport maps.