Geometric and information-theoretic aspects of high-dimensional phenomena

This proposal focuses on the study of geometric, analytic and

information-theoretic aspects of high dimensional phenomena

on the border of probability, convex geometry and analysis.

One part of the project concerns the problem of rates of

convergence in the entropic central limit theorem, and is devoted

to obtaining new asymptotic expansions for the relative entropy

with respect to the growing dimension. In other part,

it is proposed to perform a systematic study of the

dimensional behavior of the entropy and information for

different classes of probability distributions, satisfying

convexity conditions. In particular, new concentration properties

of the information content will be considered for dependent

high-dimensional data. It is planned to introduce and explore

special positions of probability measures, responsible for

correct behaviour of sums of independent summands

(when the entropy power inequality can be reversed).

Another part addresses the stability problem, raised by Kac

and McKean, in the entropic variant of Cramer's

characterization of the normal law.

The main theme of the proposal is the development of the

information-theoretic approach to high dimensional phenomena,

with focus on obtaining new asymptotic bounds on the entropy and

information. The study of entropy is dictated by various

applications within and beyond pure mathematics. Entropy plays

a key role in statistical physics (in order to capture

the amount of disorder in a system), in statistics

(to measure the performance of statistical estimators),

in engineering and mathematical theory of communication.

The proposed research also aims to provide new connections between

probability, geometric functional analysis and information theory,

and to demonstrate an increasing role of entropy

bounds in purely mathematical fields.

An integral component of the project is the involvement and

training of the graduate and undergraduate students.