Marcel Klatt

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  Empirical (Regularized) Optimal Transport: Statistical Theory and Applications
  Marcel Klatt
  In recent years, the theory of optimal transport (OT) has found its way into data analysis. Especially regularized OT methods have encountered growing interest, as the routine use of OT in applications is still hampered by its computational complexity. Among others, the most prominent proposal is entropy regularization that serves to define an entropy regularized OT plan and a corresponding divergence also known as Sinkhorn divergence. This talk invites to a small trip through distributional limit theory for certain empirical (regularized) OT quantities defined for distributions supported on finite metric spaces. In particular, we will explore the statistical differences between asymptotic distributions for empirical non-regularized OT quantities and their regularized counterparts. Specific focus is set to the empirical regularized OT plan for which we can prove that it asymptotically follows a Gaussian law. As a consequence we discuss applications in colocalization analysis of protein interaction networks based on regularized OT. In the final part of the talk, we consider the non-regularized OT plan that is a solution to a finite dimensional basic linear program. In fact, distributional limit theory for such a quantity is not as straightforward and brings into play the combinatorial nature and the concept of degeneracy inherent in linear programming. This is joint work with Carla Tameling, Axel Munk and Yoav Zemel.