Mathematics 797TT - ST-InfoTheory&OptimalTransport

This is a class on the mathematical foundations and the applications of distances and divergences between probability measures. Among the topics to be treated: Entropies, Kullback-Leibler and general f-divergences; Optimal transport and Wasserstein metrics; Maximum mean discrepancy and reproducing kernel Hilbert spaces, Stein discrepancies, and integral probability metrics. We will provide the analytical and probabilistic foundations for these objects with an emphasis on variational representations, We will discuss inequalities in information theory, and also present algorithms for the computations or statistical estimation of divergences or metrics, including generative adversarial networks and gradient flows. We illustrate the theory with examples from applied mathematics and data science, for example mutual information, generative adversarial networks, and so on. Prerequisite: A good working knowledge of Analysis and Probability Theory and an adventurous spirit.