ITR: Estimation, Approximation and Computation in Learning Theory

ITR：学习理论中的估计、近似和计算

• 批准号: 0312113
• 负责人: Yuesheng Xu
• 金额: $ 22.5万
• 依托单位: West Virginia University Research Corporation
• 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: 2003-09-01 至 2004-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0312113&HistoricalAwards=false
• 关键词: ITR; Estimation; Approximation; Computation; Learning

ITR: Estimation, Approximation and Computation in Learning Theory Learning theory, a rapidly growing area of multidisciplinary research has recently attracted much attention from the mathematical community. There are now numerous pressing issues coming from the statistical, engineering and computer science communities resulting from their significant progress in learning theory that provide a unique opportunity and vast need for mathematicians to develop both theoretical concepts and computational tools to assist in this area of research. We propose to study several fundamental theoretical mathematical and computational problems crucial for the continued rapid development of learning theory. They include a further study and improvements of the F. Cucker and S. Smale theory of learning, the support vector machine (SVM) of V. Vapnik, the regression theory of T. Poggio, the deterministic approach of C. A. Micchelli for optimal estimation under uncertainty and the relationship between these important ideas. Among other things, we will be concerned with learning a function from other than function values, learning vector valued functions, learning the optimal information for learning a function and estimating the approximation error using notions of nonlinear widths of function classes which is useful for obtaining deterministic estimates that lead to statistical estimates for learning. We shall study efficient numerical solutions of second kind integral equations in high dimensions which come up in the study of the approximation error of Cucker and Smale. We will also focus upon the minimal norm interpolation approach to regression and SVM which is not emphasized much in the learning theory literature and use duality theory as a bridge to compare all of them. We shall also study the kernel density problem whose importance in learning theory has been recently described by T. Poggio, investigate how to choose a kernel from the data and consider probability density estimation problems which are useful in pattern recognition and speech recognition. We are also interested in the question of stability of learning algorithms and seek to construct kernels on complex spaces suitable for applications.Our proposed research addresses a multitude of practical problems arising from the handling of massive amounts of data in high dimensional spaces. Therefore, in a time of heightened concern for national security against terrorism, this research will provide a new tool for dealing with the technological challenges that have recently emerged and an opportunity for applied mathematicians to assist in their solution.

ITR：学习理论中的估计、近似和计算学习理论是近年来数学界的一个研究热点。现在有许多紧迫的问题，来自统计，工程和计算机科学界的学习理论，提供了一个独特的机会和巨大的需要，数学家发展的理论概念和计算工具，以协助在这一领域的研究取得重大进展。我们建议研究几个基本的理论数学和计算问题的学习理论的持续快速发展至关重要。其中包括对F. Cucker和S. Smale的学习理论、V.Vapnik的支持向量机（SVM）、T. Poggio，C. A. Micchelli关于不确定性下最优估计与这些重要思想之间的关系。除此之外，我们将关注从函数值以外的函数学习函数，学习向量值函数，学习函数学习的最佳信息以及使用函数类的非线性宽度的概念来估计近似误差，这对于获得确定性估计是有用的，从而导致学习的统计估计。我们将研究在Cucker和Smale的逼近误差研究中出现的高维第二类积分方程的有效数值解。我们还将专注于回归和SVM的最小范数插值方法，这在学习理论文献中并没有太多的强调，并使用对偶理论作为桥梁来比较它们。我们还将研究核密度问题，它在学习理论中的重要性最近已由T。Poggio，研究如何从数据中选择一个核，并考虑概率密度估计问题，这在模式识别和语音识别中很有用。我们也有兴趣在学习算法的稳定性问题，并寻求构建适合的应用程序的复杂空间上的内核。我们提出的研究解决了大量的实际问题所产生的处理大量的数据在高维空间。因此，在高度关注国家安全反恐的时候，这项研究将为应对最近出现的技术挑战提供一个新的工具，并为应用数学家提供一个帮助解决这些问题的机会。