Geometric and information-theoretic aspects of high-dimensional phenomena

本建议的重点是研究高维现象的几何，分析和信息论方面的边界概率，凸几何和分析。项目的一部分涉及熵中心极限定理的收敛速度问题，并致力于获得相对熵关于增长维数的新的渐近展开式。在另一部分，提出了对满足凸性条件的不同类别的概率分布的熵和信息的维行为进行系统的研究。特别是，信息内容的新集中特性将被考虑到依赖的高维数据。计划引入和探索概率测度的特殊位置，负责独立和的和的正确行为（当熵幂不等式可以逆转时）。另一部分讨论了稳定性问题，由Kac和McKean提出，在克莱默正常律的熵变特征中。该建议的主要主题是发展高维现象的信息论方法，重点是在熵和信息上获得新的渐近界。熵的研究是由纯数学内外的各种应用决定的。熵在统计物理学（为了捕获系统中的无序量）、统计学（衡量统计估计器的性能）、工程和通信数学理论中起着关键作用。本研究还旨在提供概率论、几何泛函分析和信息论之间的新联系，并证明熵界在纯数学领域中日益重要的作用。该项目的一个组成部分是研究生和本科生的参与和培训。

This proposal focuses on the study of geometric, analytic and information-theoretic aspects of high dimensional phenomenaon the border of probability, convex geometry and analysis. One part of the project concerns the problem of rates ofconvergence 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'scharacterization 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 capturethe 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 entropybounds in purely mathematical fields.An integral component of the project is the involvement and training of the graduate and undergraduate students.