Application of Greedy Approximations in Numerical Integration and Learning Theory

TemlyakovDMS-0906260 该项目的目标是设计和研究在数值积分和学习理论中实际可实现的贪婪型近似方法。 当代应用数学面临的最大挑战之一是高维问题。 在金融、量子化学、生物学、医学和其他领域，自然会出现数百甚至数千维的高维问题。 这类典型的问题是数值积分和统计估计。 基本问题是如何构造好的数值积分（容积公式）和统计估计方法。 最近的研究表明，贪婪型逼近方法适用于不同的高维问题，包括数值积分和学习理论的问题。 研究者和他的同事研究贪婪近似在数值积分和学习理论中的应用。 初步研究表明，贪婪型近似方法在高维中工作得很好，可以被认为是一个建设性的确定性替代一些强大的概率方法。 贪婪近似有可能成为数值积分和学习理论的一个变革性概念。 理解智能以及它如何学习和吸收信息是本世纪最大的科学挑战之一。 它是设计系统以有效分析数据和提取基本信息的关键。 研究这方面智力的科学学科被称为学习理论。 它在国防和民用领域都有无数现有和潜在的应用。 例如，管理诸如通过监视获得的安全数据库之类的大型数据库需要对数据集进行分类，以便加速重要特征或特定信息的提取。 学习理论发现规则，允许从已经分类的过去数据中分类新数据。 一个典型的应用是搜索大型数据库（例如电子邮件），以确定其中哪些可能与恐怖活动有关。 该项目的目标是利用近似和统计中的基本概念来明确定义和量化学习挑战，并设计新的，更有效的技术（贪婪算法）。

TemlyakovDMS-0906260 The goal of the project is to design and study greedy type approximation methods that are practically implementable in numerical integration and learning theory. One of the biggest challenges of contemporary applied mathematics is high-dimensional problems. High-dimensional problems with dimensions of hundreds, and even thousands, arise naturally in finance, quantum chemistry, biology, medicine, and other areas. Typical problems of this kind are numerical integration and statistical estimation. The fundamental question is how to construct good methods of numerical integration (cubature formulas) and statistical estimation. Recent investigations show that greedy type approximation methods are good for different high-dimensional problems, including problems from numerical integration and learning theory. The investigator and his colleagues study application of greedy approximations in numerical integration and learning theory. Preliminary investigations show that greedy type approximation methods work well in high dimensions and can be considered as a constructive deterministic alternative to some powerful probabilistic methods. The greedy approximation has potential to become a transformative concept of numerical integration andlearning theory. Understanding intelligence and how it learns and assimilates information is one of the great scientific challenges of this decade. It is key to designing systems to efficiently analyze data and extract essential information. The scientific discipline that studies this aspect of intelligence is called learning theory. It has a myriad of existing and potential applications in both the defense and civilian sectors. For instance, managing large data bases such as security data bases obtained through surveillance requires classification of the data sets in order to speed up extraction of significant features or specific information. Learning theory discovers rules that allow the classification of new data from past data that have already been classified. A prototypical application is the search through a large data base (for example emails) to determine which of these have possible links to terrorist activities. It is the goal of this project to utilize fundamental concepts in approximation and statistics to clearly define and quantify the learning challenge and design new, more efficient techniques (greedy algorithms).