The discrete Schrödinger bridge, and the ensuing chaos

Abstract:
Schrödinger studied in the 1930s a thought experiment about hot gas in which a cloud of particles evolves in time from an initial distribution to another one, possibly quite different from the initial one. He posed the problem of determining the most likely evolution among the many possible ones, a problem now known as the Schrödinger bridge problem. H. Föllmer later in the 1980s framed the problem as an entropy regularized variational problem. The Schrödinger problem underlies a number of methods in data science and machine learning, while its mathematical and statistical foundations are still being understood. After introducing the problem, several variations, and their connections to regularized optimal transport, we will study the asymptotics of the discrete Schrödinger bridge towards a continuum counterpart for a large number of particles. This will lead to a central limit theorem as well as a second order Gaussian chaos limit. A novel chaos decomposition of the discrete Schrödinger bridge by polynomial functions of the pair of empirical distributions as a first and second order Taylor approximations in the space of measures is key to the asymptotic analysis. We will conclude with a brief overview of recent developments and interesting venues for future research.

This is joint work with Lang Liu and Soumik Pal <https://arxiv.org/abs/2011.08963>.

Biography:
Zaid Harchaoui is a Professor in the Department of Statistics with an adjunct appointment in the Paul G. Allen School of Computer Science and Engineering, and a Senior Data Science Fellow in the eScience Institute, at the University of Washington at Seattle. He is a co-PI of IFDS, the NSF-TRIPODS institute on the foundations of data science, and of IFML, the NSF-AI Institute on the Foundations of Machine Learning. He received the doctoral degree from Telecom Paris – Institut Polytechnique de Paris. He previously held appointments at the Courant Institute of Mathematical Sciences at New York University, and at INRIA – the French National Institute for Research in Digital Science and Technology. He is currently an action editor at the Journal of Machine Learning Research, and an associate editor at the Journal of the Royal Statistical Society and the IEEE Transactions on Pattern Analysis and Machine Intelligence.