New High Dimensional Phenomena and Applications

该项目的研究处于数学中几个主题的十字路口，专注于研究度量集中的概率、几何和信息理论方面以及其他高维现象。在复杂系统中，随机性规则被很好地理解，并且足够多的潜在事件彼此独立，大尺度上的聚集行为倾向于非常小地偏离中值行为。这一现象被称为测量集中，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性质。出于对纯粹数学问题的强烈挑战，这一研究领域在信息论、统计学、计算机科学和机器学习等学科中也有广泛的应用。这项研究的目的特别是为了正确理解增长维度的作用，特别是在高维度作为一种统一力量的问题上。例如，高维模型在实践中是有用的，可以及时理解现象的整个演变，而不仅仅是可能导致一个人的低维直觉误入歧途的主导局部规则。该项目将对纠正数学中的这些误解产生重要影响，当这些想法应用于统计学和机器学习等领域时，将产生进一步的影响。该项目还将对培养数学科学本科生和研究生产生更广泛的影响。该项目的主题要么是长期悬而未决的问题，要么是与最近事态发展有关的具有挑战性的问题。更具体地说，研究人员计划从球形集中现象开始开发新的集中工具，并将其扩展到Grassman和Stiefel流形。有了新的工具，调查对象之一将是与Kannan、Lovasz和Simonovits的K-L-S猜想有关的一圈问题。该项目将探索对数凹和更一般的凸度量的高维投影的精细集中性质；特别是，该工作将研究欧氏空间上的凸度量的新的积分几何特征，这些特征决定了谱间隙和Cheeger等周常数。该项目的一部分致力于中心极限定理的渐近展开，包括相对熵和Fisher信息，包括Berry-Esseen界，以及它们在独立随机向量和的最优传输中的应用。该项目还涉及关于经验分布的信息论不平等和运输问题。

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.