Advancing Optimal Transport Metrics for Data Comparisons, 2025

Supervisors: Jakub W. Both and Enrico Facca

Short description of the project:

This thesis presents two advancements in the application of Optimal Transport (OT) metrics within the DarSIA toolbox. This motivation arises from the growing need for OT metrics that remain both computationally efficient and flexible when applied to real, imperfect data. The well defined Wasserstein distance is used as the underlying metric because it quantifies
the effort required to move one distribution into another, providing a more geometric notion of discrepancy than pointwise norms. The first contribution introduces a Fast Fourier
Transform (FFT) based solver designed to enhance computational efficiency when solving
the Beckmann formulation of the OT problem. The FFT is benchmarked against the pre-
existing solvers in DarSIA. The second contribution incorporates an unbalanced OT model
that allows for variable mass distributions through a penalty parameter, thereby removing
the strict requirement of mass conservation that characterizes the standard formulation.
Both the FFT solver and the unbalanced OT formulation are integrated into DarSIA and
validated on real data, demonstrating their practical value and potential for broader application, while also highlighting areas that warrant further investigation.