High-Dimensional Phenomena, Limit Theorems, and Applications

该项目的研究涵盖了数学中的几个主题，并专注于研究高维现象的概率、几何和信息理论方面，包括测量的集中度和越来越多的随机变量的各种函数的渐近行为。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属性，在这些系统中，其众多小部分的随机性导致稳定的极限行为。这一研究领域与具有挑战性的数学问题相联系，在统计学、信息论、计算机科学、机器学习等领域的应用被证明是非常有用的。该项目的目标之一是澄清不断增长的维度在高维度模型中作为统一来源的作用，特别是它对整个时间演变的影响，而不是当地规则。拟议的研究将产生更广泛的影响，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。该项目还将对数学科学教育产生重要影响。更具体地说，研究者打算为具有足够多对称性的空间(包括Grassmanian流形)开发新的高级集中工具。计划将它们应用于研究与薄壳(方差)问题和Kannan，Lovasz和Simonovits的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。另一类应用涉及相关条件下相关数据的随机化求和模型。该项目的一部分将致力于信息论距离的中心极限定理中的极限定理和渐近展开，例如相对熵(Kullback-Leibler距离)和相对Fisher信息，它们将伴随着适当的Berry-Esseen界。该项目还讨论了泊松近似问题的Edgeworth展开和信息界。建议的主题涉及长期悬而未决的问题或与最近发展相关的挑战性问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.