Entropic and Displacement interpolation of probability distributions
Tryphon Georgiou
University of California, Irvine 

We will discuss two problems with a long history and a timely presence. Optimal mass transport (OMT) was posed as a problem in 1781 by Gaspar Monge. It provides a natural geometry for interpolating distributions (displacement interpolation) and for modeling flows. As such it has been the cornerstone of many recent developments in physics, probability theory, and image processing. The Schrödinger bridge problem (SBP) was posed by Erwin Schrödinger in 1931, in an attempt to provide a classical interpretation of quantum mechanics. It is rooted in statistical mechanics and large deviations theory, and provides an alternative model for flows of the distribution of particles (entropic interpolation -Schrödinger bridge). We will discuss the relation between the two problems, their practical relevance in the control of particles, ensembles, thermal noise, time-series analysis, images interpolation, etc., and a computational approach based on the Hilbert metric. We will finally discuss bridges over discrete Markov chains and present an approach to robust transport over networks based on the bridge problem. The talk is based on joint works with Yongxin Chen (MSKCC), Michele Pavon (University of Padova), and Allen Tannenbaum (Stony Brook).