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: 0407476
• 负责人: Yuesheng Xu
• 金额: $ 22.5万
• 依托单位: Syracuse University
• 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: 2003-09-23 至 2006-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0407476&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：学习理论学习理论中的估计，近似和计算，多学科研究的快速增长领域最近引起了数学社区的广泛关注。现在，由于其在学习理论方面的重大进展，统计，工程和计算机科学社区带来的统计，工程和计算机科学社区带来了许多紧迫的问题，这些问题为数学家提供了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。我们建议研究一些基本理论数学和计算问题对于持续的学习理论的持续发展至关重要。它们包括对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，T。Poggio的回归理论，C。A。Micchelli对不确定的最佳估计的确定性方法以及这些重要思想之间的关系。除其他事项外，我们将关注从功能值以外的其他功能，学习矢量有价值函数，学习学习功能的最佳信息，并使用功能类的非线性宽度概念来估算近似误差，这对于获得导致学习统计估计的确定性估计值有用。我们将研究高维度中第二类积分方程的有效数值解决方案，这些方程在研究Cucker和Smale的近似误差中出现。我们还将专注于最小的规范插值方法回归和SVM，这在学习理论文献中并不强调，并将二元性理论用作比较所有这些的桥梁。我们还将研究内核密度问题，这些问题在学习理论中的重要性最近由T. Poggio描述，研究如何从数据中选择核并考虑概率密度估计问题，这些问题在模式识别和语音识别方面有用。我们还对学习算法的稳定性问题感兴趣，并试图在适合应用程序的复杂空间上构建内核。我们的拟议研究解决了由于在高维空间中处理大量数据而引起的多种实际问题。因此，在人们对国家安全反对恐怖主义的关注时，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了协助解决方案的机会。