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一个快速发展的多学科研究领域，最近引起了数学界的广泛关注。由于统计、工程和计算机科学界在学习理论方面取得的重大进展，现在出现了许多紧迫的问题，这为数学家开发理论概念和计算工具来协助这一研究领域提供了独特的机会和巨大的需求。我们建议研究对学习理论的持续快速发展至关重要的几个基本理论数学和计算问题。其中包括对F. Cucker和S. Smale学习理论、V. Vapnik的支持向量机（SVM）、T. Poggio的回归理论、C. A. Micchelli的最优估计的确定性方法的进一步研究和改进。不确定性以及这些重要想法之间的关系。除其他事项外，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、学习学习函数的最佳信息以及使用函数类非线性宽度的概念估计逼近误差，这对于获得确定性很有用导致学习统计估计的估计。我们将研究在Cucker和Smale的逼近误差研究中提出的高维第二类积分方程的有效数值解。我们还将重点关注回归和支持向量机的最小范数插值方法，这在学习理论文献中并没有得到太多强调，并使用对偶理论作为比较所有这些方法的桥梁。我们还将研究 T. Poggio 最近描述的在学习理论中的重要性的核密度问题，研究如何从数据中选择核，并考虑在模式识别和语音识别中有用的概率密度估计问题。我们还对学习算法的稳定性问题感兴趣，并寻求在适合应用的复杂空间上构建内核。我们提出的研究解决了在高维空间中处理大量数据所产生的众多实际问题。因此，在国家安全反恐日益受到关注的时代，这项研究将为应对最近出现的技术挑战提供新的工具，并为应用数学家提供协助解决这些挑战的机会。