New High Dimensional Phenomena and Applications

该项目的研究处于数学中几个主题的交叉点，重点是研究测量和其他高维现象的概率，几何和信息理论方面。在复杂系统中，随机性规则被很好地理解，并且有足够多的潜在事件彼此独立，大规模的聚合行为倾向于偏离中值行为很少。这种现象，被称为测量的集中，一直是令人兴奋的发展的主题，因为集中工具允许人们分析相当一般的系统的许多基本属性。由于受到挑战性的纯数学问题的强烈激励，该研究领域在信息论，统计学，计算机科学和机器学习等学科中也有广泛的应用。这个研究项目的目的是特别是在增长的维度的作用的正确理解，特别是在高维作为一个统一的力量的问题。例如，高维模型在实践中是有用的，可以理解一个现象在时间上的整个演化，而不仅仅是控制局部规则，这可能会导致一个人的低维直觉误入歧途。该项目将对纠正数学中的这些误解产生重要影响，并将在这些想法应用于统计和机器学习等领域时产生进一步的影响。该项目还将对数学科学的本科生和研究生教育产生更广泛的影响。该项目的主题涉及长期存在的未决问题或与近期发展有关的挑战性问题。更具体地说，研究人员计划开发新的浓度工具，从球形浓度现象及其扩展到格拉斯曼和Stiefel流形。有了新的工具，调查的目标之一将是与Kannan，Lovasz和Simonovits的K-L-S猜想有关的问题的循环。该项目将探索对数凹和更一般的凸测度的高维投影的精细浓度特性;特别是，这项工作将研究欧几里得空间上凸测度的新积分几何特征，这些特征负责谱隙和Cheeger等周常数。该项目的一部分是致力于渐近展开的中心极限定理的相对熵和费舍尔信息，包括贝里-埃辛界，以及它们的应用，以最佳运输的总和独立的随机向量。该项目还涉及信息理论的不平等和经验分布的运输问题。

The project's research lies at the crossroads of several themes in mathematics and is focused on the study of probabilistic, geometric, and information-theoretic aspects of concentration of measure and other high-dimensional phenomena. In complex systems where rules of randomness are well understood and sufficiently many underlying events are independent of each other, aggregate behavior at large scales tends to deviate very little from the median behavior. This phenomenon, known as concentration of measure, has been the subject of exciting developments, since concentration tools allow one to analyze many essential properties of rather general systems. Being strongly motivated by challenging purely mathematical questions, this research area also has a wide range of applications in disciplines such as information theory, statistics, computer science, and machine learning, among others. This research project is aimed in particular at a correct understanding of the role of the growing dimension, especially in the problems where high dimension serves as a unifying force. High-dimensional models are useful in practice for instance to understand the entire evolution of a phenomenon in time, not just the governing local rules, which may lead one's low-dimensional intuition astray. The project will have an important impact correcting such misconceptions in mathematics, and will have further impact when these ideas are applied to areas such as statistics and machine learning. The project will also have broader impact on educating undergraduate and graduate students in the mathematical sciences. The project's themes refer either to long-standing open problems or to challenging questions related to recent developments. More specifically, the investigator plans to develop new concentration tools starting from the spherical concentration phenomenon and its extensions to Grassman and Stiefel manifolds. With new tools, one of the targets of investigation will be the circle of problems related to the K-L-S conjecture of Kannan, Lovasz, and Simonovits. The project will explore refined concentration properties of high dimensional projections of log-concave and more general convex measures; in particular, the work will investigate new integral geometric characteristics of convex measures on Euclidean spaces that are responsible for spectral gap and Cheeger isoperimetric constants. Part of the project is devoted to asymptotic expansions in the central limit theorem for the relative entropy and Fisher information, including Berry-Esseen bounds, and their applications to optimal transport for sums of independent random vectors. The project also deals with information-theoretic inequalities and transport problems about empirical distributions.