Vector valued optimal transport: from dynamic to static formulations

Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (\`a la Benamou-Brenier) to static formulations (\`a la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product space $\mathbb{R}^d \times G$, where $G$ is a weighted graph over a finite set of nodes and the graph geometry strongly influences the associated dynamic and static distances. We obtain sharp inequalities relating four notions of vector valued optimal transport and prove that the distances are mutually bi-H\"older equivalent. We discuss the theoretical and practical advantages of each metric and indicate potential applications in multispecies PDE and data analysis. In particular, one of the static formulations discussed in the paper is amenable to linearization, a technique that has been explored in recent years to accelerate the computation of pairwise optimal transport distances.

It begins with a boundary: A geometric view on probabilistically robust learning

Although deep neural networks have achieved super-human performance on many classification tasks, they often exhibit a worrying lack of robustness towards adversarially generated examples. Thus, considerable effort has been invested into reformulating Empirical Risk Minimization (ERM) into an adversarially robust framework. Recently, attention has shifted towards approaches which interpolate between the robustness offered by adversarial training and the higher clean accuracy and faster training times of ERM. In this paper, we take a fresh and geometric view on one such method -- Probabilistically Robust Learning (PRL) (Robey et al., ICML, 2022). We propose a geometric framework for understanding PRL, which allows us to identify a subtle flaw in its original formulation and to introduce a family of probabilistic nonlocal perimeter functionals to address this. We prove existence of solutions using novel relaxation methods and study properties as well as local limits of the introduced perimeters.