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Concentration, Convexity, and Structure

浓度、凸性和结构

基本信息

批准号：
1812240
负责人：
Grigoris Paouris
金额：
$ 15万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-15 至 2021-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812240&HistoricalAwards=false
关键词：
Concentration Convexity Structure

项目摘要

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.

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。已经观察到，这些不同的区域具有共同的特征，即基本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促进（非渐近）高维概率。研究者打算研究与高维概率中最广泛应用的原理有关的几个问题：测度集中现象。这个原则通常是经常观察到的高维系统聚集在典型形式周围的趋势背后的主要原因。为了量化这种现象，需要对高维对象（例如，测度或随机向量）进行精确的不等式，其中可能缺乏独立性。所研究的问题有很强的几何成分。该研究的结果将对高度依赖于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浓度的函数的量或分离其特征（比如关于高斯测度）;特别是，确定控制小波动（方差）和小球概率的量。该项目对这个问题进行了系统的研究，并提出了一些新的方法来计算偏差不等式（特别是在小球区域）。计划在更一般的措施，如对数凹概率措施上测试这些方法。该项目还将研究几何量的极限定理，以补充渐近水平的浓度不等式。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。