UC Berkeley

This dissertation consists of several papers. First, we start off introducing domain adaptation theory and briefly introduce optimal transport. Such an introduction allows the reader to understand why studying problems in optimal transport theory is so valuable. Our first key result establishes bounds between regularized and unregularized optimal transport. Instead of using an entropic regularization, which is used in the Sinkhorn divergence, we regularize using dual potentials in a reproducing kernel Hilbert space. After this, we derive sample complexity bounds for the regularized optimal transport problem, and we show this is a substantial improvement over unregularized optimal transport. With these two results, one can approximate the theoretical optimal transport distance. Next, we prove the first and second moments of the source and target distributions are enough to determine explicitly the optimal transport map and also that this is a linear mapping. Furthermore, we propose an alternative regularization for the transport map between two distributions. After this, we briefly diverge from optimal transport theory and introduce work on prior elicitation. In particular, we extend a result from \cite{Pinelis2017} on non-asymptotic bounds for maximum likelihood estimators to that for M-estimators. Crucially, we show sufficient assumptions for these to hold and use these to theoretically justify our prior elicitation objective. Last, we return to optimal transport and introduce a variant to compare multiple probability measures, which we call sliced multi-marginal optimal transport. There, we propose a paradigm based on random one-dimensional projections.