Computational Optimal Transport for Learning from Geometrically Structured Data

Abstract

The increasing prominence of geometrically structured data in machine learning has driven the need for principled methods to compare and analyze such data effectively. From molecular graphs in drug discovery to brain imaging data in neuroscience, many domains require techniques that respect the underlying geometric relationships in data. Optimal transport (OT) provides a powerful mathematical framework for comparing probability distributions while preserving geometric structures, making it a natural tool for learning from structured data. However, classical OT methods often face computational and theoretical challenges when applied to complex data modalities, particularly when distributions lie on non-Euclidean spaces or require partial alignment. This thesis explores computational advancements in OT for structured data analysis, focusing on three key directions: (1) the development of efficient transport distances tailored for probability measures on spherical domains—the simplest type of manifold, (2) the extension of Gromov-Wasserstein distances to handle partial correspondences between distributions lying in different metric spaces, and (3) the linearization of this extension to improve computational scalability while preserving its ability to model partial alignments. These contributions provide new metrics, algorithms, and theoretical insights that broaden the applicability of OT-based techniques in machine learning. Numerical experiments demonstrate the effectiveness of these methods across a range of real-world machine learning applications, including self-supervised learning, representation learning, variational inference, and shape retrieval and interpolation. By advancing computational OT, this thesis contributes to the broader goal of geometric machine learning, enabling more structured and interpretable methods for learning from complex data.