ETD

• machine learning
• optimal transport
• optimization

Unbalanced optimal transport (UOT) has emerged in the field of machine learning as a flexible method to match or compare point clouds and more generally nonnegative and finite Radon measures. In Chapter 1 we provide a review of the main theory of UOT and we define one of the central objects of our work, the unbalanced Monge map.
In Chapter 2, we study the entropy-regularized UOT problem. We present the entropic duality theorem which leads to the Sinkhorn algorithm for the approximation of the entropic optimal transport plan. We provide results on the convergence of the entropy-regularized UOT to UOT and we define the second central object of the thesis, the entropic map, which we prove convergences to the unbalanced Monge map in the Euclidean case.
In Chapter 3 we develop the framework of cost-regularized unbalanced optimal transport (CR-UOT) and we study its entropy-regularized formulation, proving its convergence to CR-UOT. Then, we focus on a Gromov-Wasserstein-related case of CR-UOT using a family of inner product costs parametrized by linear transformations that admit unbalanced Monge maps, under suitable hypotheses. Finally, we show that such Monge maps can be approximated using entropic maps, which can therefore be used in matching applications.
In Chapter 4 we describe a block coordinate descent algorithm to solve the entropy-regularized CR-UOT problem with inner product costs and we apply the associated entropic map in single-cell multi-omics alignment tasks.