Probabilistic Approach in Geometric Functional Analysis

几何泛函分析中的概率方法

• 批准号: 0245380
• 负责人: Mark Rudelson
• 金额: $ 10.02万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245380&HistoricalAwards=false
• 关键词: Probabilistic; Approach; Geometric; Functional; Analysis

The PI plans to study convex bodies using the methods of probability.For many problems of convex geometry, such as finding sections of a convexbody with certain nice properties or approximating a convex body by anotherbody having a better structure, explicit constructions are unknown.In these cases random constructions turn out to be very effective.It is often possible to define a random section or approximation andto show that it has the desired property with high probability.This approach combined with advanced probabilistic tools, such as measureconcentration, have led to major discoveries in convex geometry and functionalanalysis.The PI also plans to study the combinatorial dimensionof a set of functions. This is a new notionwhich arises from questions in probability and computer science.The PI plans to investigate the connection between the entropy and thecombinatorial dimension and apply the results to study properties ofconvex bodies.The proposed research will provide new connections betweenfunctional analysis, convex geometry and probability.The study of combinatorial dimension is likely to have concretepractical applications in machine learning. The non-symmetric convex sets, which are one of the main objectsof the proposed research, arise naturally in a broad classof optimization problems. So, better understanding of the structureof such bodies will result in constructing more effective optimizationalgorithms.

PI计划用概率方法研究凸体。对于凸几何的许多问题，例如寻找具有某些良好性质的凸体的部分或用具有更好结构的另一个凸体近似一个凸体，显式构造是未知的。在这些情况下，随机结构是非常有效的。通常可以定义一个随机截面或近似，并以高概率表明它具有期望的性质。这种方法与先进的概率工具（如测量集中）相结合，导致了凸几何和功能分析的重大发现。PI还计划研究一组函数的组合维数。这是一个从概率论和计算机科学问题中产生的新概念。PI计划研究熵和组合维之间的联系，并将结果应用于研究凸体的性质。提出的研究将提供功能分析，凸几何和概率之间的新联系。组合维的研究可能在机器学习中有具体的实际应用。非对称凸集是本文研究的主要对象之一，它自然地出现在各种各样的优化问题中。因此，更好地理解这些物体的结构将导致构建更有效的优化算法。