Measure Transportation And Notions Of Dimensionality In High Dimensional Probability

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

High dimensional probability is a mathematical theory which aims to explain the behavior of systems with very large number of degrees of freedom. Such systems are of utmost importance in the physical sciences and in a world where social media and communication networks constantly generate huge data sets. This project aims to develop theoretical tools in high dimensional probability via two methods. The first method will exploit the intuition that in many practical situations, although the system contains numerous degrees of freedom, the relevant information is contained in a much smaller set of variables. The project will aim to quantify this intuition by developing the notion of intrinsic dimensionality. The second method aims to represent a given complex system as the transformation of a much simpler system. The project also includes educational efforts, including mentoring undergraduate and graduate students, and the dissemination of the work in professional meetings and to the public. The long-term goal is to improve understanding of high-dimensional probability measures via two approaches. The first approach is to develop an original concept of intrinsic dimensionality in the context of functional inequalities. The notion of dimension plays a crucial role in functional inequalities via the curvature-dimension condition, but it ignores the fact that the measures of interest can live on lower-dimensional spaces. This omission can lead to inefficient functional inequalities; this project hopes to bridge this gap. The second approach is to develop new measure transportation methods based on stochastic processes and probabilistic flows. For example, Lipschitz transport maps provide a powerful way to transfer functional inequalities from simple source measures to complicated target measures. Only a few such Lipschitz transport maps are known to exist, which limits the applicability of the transport method. The project will utilize the tools of stochastic analysis and renormalization group methods to build transport maps with desirable regularity. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.