Probabilistic Approach in Geometric Functional Analysis

首席研究员（PI）计划使用概率方法研究分析，凸几何和计算机科学中出现的问题。对于许多几何和分析问题，例如找到具有某些良好性质的凸体的截面或从大集合中提取具有某种结构的小样本，显式结构是未知的。在这些情况下，随机结构被证明是非常有效的。通常可以定义一个随机部分或样本，并以高概率显示它具有所需的属性。这种方法结合了先进的概率工具，如测量浓度，导致了凸几何和泛函分析的重大发现。PI计划研究一组函数的组合维数。这是一种新的复杂性度量，它源于概率和计算机科学中的问题。PI还将研究随机矩阵的谱特性，并将研究结果应用于有效纠错码的构建。这项研究将提供泛函分析，凸几何和概率之间的新的连接。这个项目的一个重要部分是概率方法在计算机科学具体问题中的应用。组合维数的研究在机器学习中有着重要的实际应用价值。几何方法的信号恢复和纠错码将导致更有效和更稳定的算法的建设。

The Principal Investigator (PI) plans to study problems arising in analysis, convex geometry and computer science using the methods of probability. For many problems of geometry and analysis, such as finding sections of a convex body with certain nice properties or extracting a small sample with a certain structure from a large set, the 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 sample and show that it has the desired property with high probability. This approach combined with advanced probabilistic tools, like measure concentration, led to major discoveries in convex geometry and functional analysis. The PI plans to study the combinatorial dimension of a set of functions. This is a new measure of complexity, which arises from questions in probability and computer science. The PI will also investigate the spectral properties of random matrices and apply the results of this research to construction of effective error-correcting codes. This research will provide new connections between functional analysis, convex geometry and probability. An important part of this project is the application of probabilistic methods to concrete problems of computer science. The study of combinatorial dimension is likely to have practical applications in machine learning. The geometric approach to the signal recovery and error-correcting codes will lead to the construction of more efficient and stable algorithms.