Transport of probability distributions across different Euclidean spaces

In this thesis, we study three problems related to the transport of measures lying on different Euclidean spaces, the first two being in the context of optimal transport and the last one being in the context of generative modeling. In the optimal transport part, we first study the behavior of the common generalizations of optimal transport, including the so-called Gromov-Wasserstein distance, between Gaussian distributions in incomparable spaces. Secondly, we design a computationally efficient and scalable OT distance between Gaussian mixtures possibly living in different Euclidean spaces. Finally, inthe generative modeling part, we study the expressivity of generative models relativelyto the Lipschitz constant of their push-forward mapping.