Revealing the Simplicity of High-Dimensional Objects via Pathwise Analysis

One of the main reasons behind the success of high-dimensional statistics and modern machine learning in taming the curse of dimensionality is that many classes of high-dimensional distributions are surprisingly well-behaved and, when viewed correctly, exhibit a simple structure. This emergent simplicity is in the center of the theory of “high-dimensional phenomena”, and is manifested in principles such as “Gaussian-like behavior” (objects of interest often inherit the properties of the Gaussian measure), “dimension-free behavior” (expressed in inequalities which do not depend on the dimension) and “mean-field behavior” (where the behavior of a system having many degrees of freedom can be compared to a “limit object” having a small number thereof).

In this talk, we present a new analytic approach that helps reveal phenomena of this nature. The approach is based on pathwise analysis: We construct a sampling procedure associated with the high-dimensional object, which uses the randomness coming from a Brownian motion. This gives rise to a stochastic process which allows us to make the object tractable by the analysis of the process, via Ito calculus (the theory of diffusing particles) and relate quantities of interest of the object with the behavior of the process, for example, through differentiation with respect to time.
I will try to explain how this approach works and will briefly discuss several results, of relevance to high dimensional statistics and machine learning, that stem from it. These results include concentration inequalities, central limit theorems, and “mean-field” structure theorems.