向量值与泛函值数据处理的再生核Hilbert空间方法

项目名称： 向量值与泛函值数据处理的再生核Hilbert空间方法

项目编号： No.11301208

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 王蕊

作者单位： 吉林大学

项目金额： 23万元

中文摘要： 随着数据信息的与日俱增，有效的数据处理方法已成为科学技术发展的迫切需求。基于再生核Hilbert空间（RKHS）的数据处理方法在标量值数据处理中已经得到了广泛而深入的研究，并取得了极大的成功。面对实际应用中对向量值及线性泛函值数据处理的大量需求，发展处理该类型数据的RKHS方法将具有重要的理论意义和广阔的应用前景。本项目将在RKHS的框架下，研究基于向量值数据和线性泛函值数据的机器学习算法和采样理论。一方面，研究常用向量值RKHS中函数的性质和空间的结构，各种空间之间的包含关系，进而将结果应用于机器学习中，包括改进学习算法的误差估计、建立正则化方法中再生核的更新及最优选取算法。同时，在向量值RKHS的框架下建立向量值函数的采样理论，如最优重构算法、完全重构公式、最优采样点的选取。另一方面，建立用于再生泛函的新型RKHS的理论，并在此基础上发展线性泛函值数据的机器学习算法和采样理论。

中文关键词： 非点值泛函数据；泛函再生核Hilbert空间；向量值再生核Hilbert空间；学习算法；采样理论

英文摘要： With the growth of data, effective data processing methods have become an urgent demand for the develepment of science and technology. For processing scalar-valued data, reproducing kernel Hilbert space based methods have been widely investigated and achieved great success. Because of the large demand of processing vector-valued and linear functional-valued data in practical applications, developing the RKHS based methods for these kinds of data will have theoretical significance and application prospect. This project aims at investigating machine learning algorithms and sampling theory for vector-valued and linear functional-valued data in the framework of RKHS. On one hand, we study the structure of commonly used vector-valued RKHSs, the properties of functions, and the inclusion relation between all kinds of vector-valued RKHSs. Based upon these results, we plan to improve the error estimation of multi-task learning algorithms and study how to update and select the operator-valued reproducing kernels in regularization. Moreover, we will also establish the sampling theory for vector-valued functions, including optimal reconstruction algorithm, perfect reconstruction formulas, and optimal sampling points. On the other hand, we will build the theory of a new type of RKHS, in which the general linear functionals

英文关键词： non-point-evaluation functional data；functional reproducing kernel Hilbert space；vector-valued reproducing kernel Hilbert space；learning algorithm；sampling theory