数据缺损矩阵低秩分解的正则化方法

数据缺损下矩阵低秩逼近问题出现在许多数据处理分析与应用领域. 由于极高的元素缺损率,数据缺损下的矩阵低秩逼近呈现很大的不适定性, 因而寻求有效的数值算法是一个具有挑战性的课题. 本文系统完整地综述了作者近期在这方面的一些研究进展, 给出了基本模型问题的不适定性理论分析, 提出了两种新颖的正则化方法: 元素约束正则化和引导正则化, 分别适用于中等程度的数据缺损和高度元素缺损的矩阵低秩逼近. 本文同时也介绍了相应快速有效的数值算法. 在一些实际的大规模数值例子中, 这些新的正则化算法均表现出比现有其他方法都好的数值特性.

Regularization methods for low-rank factorization of matrices with missing data

The problem of low-rank factorization of matrices with missing entries appears in many application fields for data analysis. Due to the high missing of entries in the matrices, this problem is highly ill-posed. It is a challenging task to give an efficient algorithm for obtaining a physically meaning full solution to the ill-posed problem. This paper surveys our recent develops on this topic, including theoretical analysis on solutions to the ill-posed problem and the regularization. Two novel regularization methods, entry-constraint regularization and inducible regularization, are given for the problem with moderate dada missing and high-level data missing, respectively. Corresponding algorithms for solving the regularized problems are also discussed and illustrated on some simulation data sets and real-world examples in large scale from applications.