Optimal transport in high dimension : obtaining regularity and robustness using convexity and projections.

Summary In recent years, optimal transport has gained popularity in machine learning as a way to compare probability measures. Unlike more traditional dissimilarities for probability distributions, such as Kullback-Leibler divergence, optimal transport distances (or Wasserstein distances) allow for the comparison of distributions with disjoint supports by taking into account the geometry of the underlying space. This advantage is however hampered by the fact that these distances are usually computed by solving a linear program, which poses, when the underlying space is high dimensional, well documented statistical challenges commonly referred to as the ``dimensional curse''. Beyond this purely metric aspect, another interest of optimal transport theory is that it provides mathematical tools to study maps that can transform, or transport, one measure into another. Such maps play an increasingly important role in various fields of science (biology, brain imaging) or subfields of machine learning (generative models, domain adaptation), among others. In this thesis, we propose a new estimation framework for computing variants of Wasserstein distances. The goal is to reduce the effects of high dimensionality by taking advantage of the low dimensional structures hidden in the distributions. This can be done by projecting the measures onto a subspace chosen to maximize the Wasserstein distance between their projections. In addition to this new methodology, we show that this framework is more broadly consistent with a link between regularization of Wasserstein distances and robustness.In the following contribution, we start from the same problem of estimating the optimal transport in high dimension, but adopt a different perspective: rather than modifying the cost function, we return to the more fundamental view of Monge and propose to use Brenier's theorem and Caffarelli's regularity theory to define a new procedure for estimating lipschitzian transport maps which are the gradient of a strongly convex function.