The Brownian transport map

Abstract:

The existence of a transport map from the standard Gaussian leads to succinct​representations for, potentially complicated, measures.​ Inspired by result from optimal transport, we introduce the Brownian transport map that pushes forward the Wiener measure to a target measure in a finite-dimensional Euclidean space.

Using tools from Ito’s and Malliavin’s calculus, we show that the map is Lipschitz in several cases of interest. Specifically, our results apply when the target measure satisfies one of the following:

– More log-concave than the Gaussian, recovering a result of Caffarelli.

– Bounded convex support with a semi log-concave density, providing an affirmative answer to a question first posed for the Brenier map.

– A mixture of Gaussians, explaining recent results about dimension-free functional inequalities for such measures.

– log-concave and isotropic. In this case, we establish a direct connection between the Poincare constant and the (averaged) Lipschitz constant of the Brownian transport map. Since the Poincare constant is the object of the famous KLS conjecture, we essentially show that the conjecture is equivalent to the existence of a suitable transportation map.

Joint work with Yair Shenfeld​

Bio:

Dan Mikulincer is an instructor (postdoc) at MIT Mathematics.  Previously, he finished his Ph.D. at the Weizmann Institute, Faculty of Mathematics, where he was advised by Ronen Eldan.  He completed his B.Sc. in Mathematics and Computer Science at Ben-Gurion University (BGU), where he also studied Cognitive Neuroscience. He spent the summer of 2019 at Microsoft Research AI, hosted by Sébastien Bubeck, and is also an Azrieli fellow. His research interests broadly lie at the union of high-dimensional geometry, probability, statistics, information theory, and their relation to data science and learning theory.  He is particularly interested in normal approximations and dimension-free phenomena.