ITR: Estimation, Approximation and Computation in Learning Theory

ITR：学习理论中的估计、近似和计算学习理论是一个迅速发展的多学科研究领域，最近引起了数学界的极大关注。现在，统计、工程和计算机科学界在学习理论方面取得了重大进展，产生了许多紧迫的问题，这为数学家提供了一个独特的机会和巨大的需求，以开发理论概念和计算工具来协助这一研究领域。我们建议研究几个基本的理论、数学和计算问题，这些问题对于学习理论的持续快速发展至关重要。其中包括对F.Cucker和S.Smer的学习理论、V.Vapnik的支持向量机、T.Poggio的回归理论、C.A.Micchelli的不确定性最优估计确定性方法的进一步研究和改进，以及这些重要思想之间的关系。在其他方面，我们将关注从函数值以外的其他方面学习函数，学习向量值函数，学习用于学习函数的最佳信息，以及使用函数类的非线性宽度的概念来估计逼近误差，这对于获得导致用于学习的统计估计的确定性估计是有用的。在研究Cucker和Smel逼近误差时，我们将研究高维第二类积分方程解的有效数值解。我们还将重点介绍学习理论文献中不太重视的回归和支持向量机的最小范数内插方法，并以对偶理论为桥梁对它们进行比较。我们还将研究T.Poggio最近描述的核密度问题，它在学习理论中的重要性，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，并试图在复杂空间上构造适合应用的核。我们提出的研究解决了高维空间中处理海量数据所产生的大量实际问题。因此，在高度关注国家安全打击恐怖主义的时候，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协助解决这些挑战的机会。

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.