协方差结构分析中的特征值问题的理论和算法

对协方差矩阵的结构进行分析，是多元统计分析的重点和难点．这一方向的两类重要模型是主成分分析和典型相关分析，二者归根结底都可以转化为线性的代数特征值问题，而这正是数值代数领域的研究核心之一．这一方向有着明确的应用背景和广泛的应用前景，是目前国内外数据科学领域的研究热点．但截至目前，现有研究手段大多来自于统计学和数据科学而鲜见计算数学尤其是数值代数方向的工具．本项目拟从数值代数的角度对问题进行深入研究，主要从以下三个方面开展工作：完善求解主成分分析模型的经典算法的理论分析和Oja类型的流算法的收敛性分析；建立求解典型相关分析模型的经典算法的理论分析；发展求解典型相关分析模型的经典算法和Oja类型的流算法，并对两类算法进行收敛性分析．本项目将最终形成一套从数值代数角度针对主成分分析模型和典型相关分析模型的分析框架和一套求解这两类模型的程序集或软件包．

Analyzing the structure of covariance matrices, is an important and difficult problem in multivariate statistical analysis. Principal component analysis(PCA) and canonical correlation analysis(CCA) are two important models in this area. Either of them can be finally transformed to a linear algebraic eigenvalue problem, which is one of the central tasks in the research on numerical linear algebra. This topic has substantial background and extensive prospect in practical applications, and hence recently it becomes a hotspot in data science both domestically and internationally. However, till now the current research tools almost come from statistics and data science and hardly originate in computational mathematics, especially numerical linear algebra. In this project, we propose to make research on this topic in a numerical linear algebra view, and work on these three threads: 1) consummate the theoretical analysis of the classical method for PCA and the convergence analysis of the Oja-type streaming algorithm for PCA; 2) build the theoretical analysis of the classical method for CCA; and 3) develop the classical and Oja-type streaming algorithms for CCA and complete their convergence analysis. Finally the project would produce a numerical linear algebra framework to analyze PCA and CCA, and a package to solve the two models.