Geometric and information-theoretic aspects of high-dimensional phenomena

该计划的重点是研究几何，分析和信息论方面的高维现象的边界概率，凸几何和分析。该项目的一部分涉及熵中心极限定理的收敛速度问题，并致力于获得新的渐近展开的相对熵的增长方面的尺寸。在另一部分中，我们提出了一个系统的研究的维数行为的熵和信息的不同类别的概率分布，满足凸性条件。特别是，新的浓度属性的信息内容将被认为是相关的高维数据。计划介绍和探索概率测度的特殊位置，负责独立被加数和的正确行为（当熵幂不等式可以被逆转时）。另一部分解决了Kac和McKean提出的稳定性问题，在Cramer对正规律的表征的熵变式中。该提案的主题是发展高维现象的信息论方法，重点是获得熵和信息的新的渐近界。熵的研究是由纯数学内外的各种应用所决定的。熵在统计物理中起着关键作用（为了捕捉系统中的混乱程度），在统计学中（测量统计估计器的性能），在工程和通信数学理论中。拟议的研究还旨在提供概率，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越来越重要的作用。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.