ITR: Estimation, Approximation and Computation in Learning Theory

学习理论作为一个快速发展的多学科研究领域，近年来引起了数学界的广泛关注。现在有许多紧迫的问题来自统计、工程和计算机科学社区，这些社区在学习理论方面取得了重大进展，为数学家提供了独特的机会和巨大的需求，以发展理论概念和计算工具来协助这一领域的研究。我们建议研究几个对学习理论持续快速发展至关重要的基本理论数学和计算问题。其中包括对F. Cucker和S. Smale学习理论的进一步研究和改进，V. Vapnik的支持向量机（SVM）， T. Poggio的回归理论，C. a . michelli的不确定条件下最优估计的确定性方法以及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中学习函数，学习向量值函数，学习学习函数的最优信息，以及使用函数类的非线性宽度的概念估计近似误差，这对于获得确定性估计很有用，从而导致学习的统计估计。我们将研究在Cucker和Smale近似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没有得到太多强调，并使用对偶理论作为桥梁来比较它们。我们还将研究最近由T. Poggio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。我们也对学习算法的稳定性问题感兴趣，并寻求在复杂空间上构造适合应用的核。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许多实际问题。因此，在高度关注国家安全反恐的时代，这项研究将为处理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问题的机会。

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.