Concentration, Convexity, and Structure

在数学、统计力学、量子信息等各种科学学科中，高维结构扮演着核心角色。据观察，这些不同的区域有一个共同的特征，那就是基本的概率原理支配着潜在的高维行为。在大多数情况下，有效的近似和研究是通过(非渐近的)高维概率来实现的。研究人员打算研究与高维概率中应用最广泛的原理有关的几个问题：测量现象的集中。这一原理通常是高维系统聚集在典型形式周围的主要原因。为了量化这一现象，人们需要高维对象(例如，测量或随机向量)的精确不等式，其中可能缺乏独立性性质。研究中的问题具有很强的几何成分。这项研究的结果将对严重依赖高维对象的学科产生影响，包括渐近几何分析、几何概率、机器学习、稀疏恢复、随机矩阵和随机多项式理论。该项目的主要目标是找到控制其集中度的函数的量或分离出其特征(例如相对于高斯度量)；特别是确定控制小波动(方差)和小球概率的量。该项目对这一问题进行了系统的研究，并提出了一些计算偏差不等式的新方法(特别是在小球制度下)。计划在更一般的度量上测试这些方法，例如对数凹概率度量。该项目还将研究在渐近水平上补充浓度不等的几何量的极限定理。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。

In various scientific disciplines such as mathematics, statistical mechanics, quantum information, and others, high-dimensional structures play a central role. It has been observed that these distinct areas share the common feature that basic probabilistic principles govern the underlying high-dimensional behavior. In most cases, efficient approximation and study is facilitated by (non-asymptotic) high-dimensional probability. The investigator intends to work on several questions related to the most widely applied principle in high-dimensional probability: the concentration of measure phenomenon. This principle is commonly the main reason behind the frequently-observed tendency of high-dimensional systems to congregate around typical forms. To quantify this phenomenon, one needs precise inequalities for high-dimensional objects (for instance, measures or random vectors), where independence properties can be lacking. The questions under study have a strong geometric component. Results of the study will have implications in disciplines that depend vitally on high-dimensional objects, including asymptotic geometric analysis, geometric probability, machine learning, sparse recovery, random matrices, and random polynomial theory.The main goal of the project is to find the quantities or to isolate characteristics of a function that govern its concentration (say with respect to the Gaussian measure); in particular, to determine the quantities that control small fluctuations (variance) and small ball probabilities. The project undertakes a systematic study of this problem and initiates some new methods to compute deviation inequalities (especially in the small ball regime). It is planned to test these methods on more general measures such as log-concave probability measures. The project will also investigate limit theorems for geometric quantities that complement concentration inequalities at the asymptotic level.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.