稀疏逼近及其应用

项目名称： 稀疏逼近及其应用

项目编号： No.10871015

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 数理科学和化学

项目作者： 陈迪荣

作者单位： 北京航空航天大学

项目金额： 25万元

中文摘要： 本项目发展了稀疏逼近理论和方法，针对函数、算子和学习问题，建立了稀疏逼近新理论和方法，并揭示其内在联系。建立数据获取的自适应方法，研究取样点的分布流形，给出其分布稀疏性的定性和定量刻画；研究基于小波的稀疏逼近理论和方法，分别研究Lp-误差和点态意义下的小波m-项逼近、基于数据的小波m-项逼近；分析现有学习算法的稀疏性， 研究函数型数据(functional data)的学习问题，将传统的一些算法（包括SVM, Boosting等）算法推广到函数型数据情形、研究其相容性以及收敛速度估计。对正则化谱聚类算法, 建立了一致误差界估计和收敛速度。这是该类算法中迄今为止唯一的量化结果。研究Hilbert变换H小波分解的点态收敛问题以及小波收缩估计的收敛性。我们去除样本集合的独立性假设,建立针对指数强混合型样本的最小平方正则化回归算法的学习率.采取的研究方法有，综合利用函数论、泛函分析、概率统计和图像处理方法。

中文关键词： 稀疏逼近；m-项逼近；小波收缩算法；学习理论

英文摘要： The project focuses on the problem of sparse approximation theory and methods. We established the novel sparse approximation theory for functions learning theory and the operator theory and reveal the internal relationship between them. In details, we established adaptive data acquisition methods and explore stochastic distribution of sampling points on the manifold, and characterize the properties of sparse approximation both qualitatively and quantitatively. Based on wavelet theory, we studied Lp-errors and m-terms wavelet approximation. By analyzing the sparse of learning algorithm we will further study the functional data. We generalize the finite dimension case to functional case for some learning algorithms (including SVM, Boosting and so on) and study the consistency and rates of convergence. The methods of function theory, functional analysis, probability statistics and image processing will be used in our study. The following key problems were solved: to establish the corresponding wavelet transforms based on non-uniform sampling data; to establish Littlewood-Paley theory starting with the function approximation from sparse data; to apply wavelet m-term approximation and wavelet shrinkage algorithm to functional data learning.

英文关键词： sparse approximation; m-term approximation; wavelet shrinkage; learning theory