High-Dimensional Phenomena, Limit Theorems, and Applications

该项目的研究涵盖了数学中的几个主题，重点研究高维现象的概率、几何和信息论方面，包括越来越多的随机变量的各种函数的测度集中和渐近行为。集中工具是许多令人兴奋的发展的主题，因为它们有助于探索一般复杂系统的最基本性质，其中它们的许多小部分的随机性导致稳定的极限行为。由于与具有挑战性的数学问题有关，这一研究领域已被证明对统计学、信息论、计算机科学、机器学习等其他领域的应用非常有用。该项目的目标之一是澄清在高维模型中作为统一来源的增长维度的作用，特别是它对整个时间演变的影响，而不是局部规则。拟议的研究将通过在不同的数学领域之间建立新的联系，并为它们提供强大的跨学科工具，从而产生更广泛的影响。该项目还将对数学科学教育产生重要影响。更具体地说，研究者打算为包括格拉斯曼流形在内的具有足够多对称性的空间开发新的高级集中工具。计划将它们应用于与Kannan、Lovasz和Simonovits的薄壳（方差）问题和K-L-S猜想有关的对数凹和更一般的双曲测度的多维投影的全局性质的研究。另一类应用是处理相关条件下相关数据的随机求和模型。该项目的一部分致力于信息理论距离的极限定理和中心极限定理的渐近展开，如相对熵（Kullback-Leibler距离）和相对Fisher信息，这将伴随着适当的Berry-Esseen边界。本项目还处理了泊松近似问题中的edgeworth型展开和信息界。拟议的主题或涉及长期悬而未决的问题，或涉及与最近事态发展有关的具有挑战性的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project's research covers several topics in mathematics and is focused on the study of probabilistic, geometric, and information-theoretic aspects of high dimensional phenomena, including concentration of measure and asymptotic behavior of various functions of a growing number of random variables. The concentration tools are the subject of many exciting developments, since they help explore most essential properties of general complex systems where randomness of their numerous small parts results in a stable limit behavior. Being connected with challenging mathematical problems, this research area has proved to be very useful for applications in other fields such as statistics, information theory, computer science, machine learning. One of the objectives of the project is to clarify the role of growing dimension as a unifying source in high-dimensional models and, in particular, its influence on the entire evolution in time as opposed to local rules. Proposed research will have a broader impact by creating new connections between different mathematical fields and providing them with powerful interdisciplinary tools. The project will also have an important impact on educating in mathematical sciences. More specifically, the investigator intends to develop new advanced concentration tools for spaces with sufficiently many symmetries including Grassmanian manifolds. It is planned to apply them in the study of global properties of multidimensional projections for log-concave and more general hyperbolic measures, that are related to the thin shell (variance) problem and the K-L-S conjecture of Kannan, Lovasz and Simonovits. Another sort of applications deals with randomized models of summation for dependent data under correlation conditions. Part of the project is devoted to limit theorems and asymptotic expansions in the central limit theorem for information-theoretic distances such as the relative entropy (Kullback-Leibler distance) and relative Fisher information, which will be accompanied by proper Berry-Esseen bounds. The project also deals with Edgeworth-type expansions and informational bounds in the problem of Poisson approximation. The proposed themes refer either to long-standing open problems or to challenging questions related to recent developments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.