求解鲁棒主成分分析的新交替下降方向法

A New Alternating Direction Method for Robust Principal Component Analysis

鲁棒主成分分析作为统计与数据科学领域的基本工具已被广泛研究,其核心原理是把观测数据分解成低秩部分和稀疏部分.本文基于鲁棒主成分分析的非凸模型,提出了一种新的基于梯度方法和非单调搜索技术的高斯型交替下降方向法.在新算法中,交替更新低秩部分和稀疏部分相关的变量,其中低秩部分的变量是利用一步带有精确步长的梯度下降法进行更新,稀疏部分的变量是采用非单调搜索技术进行更新.本文在一定的条件下建立了新算法的全局收敛理论.最后的数值试验结果表明了新算法的有效性.

Robust principal component analysis,where a given observation data is separated into a low-rank part and a sparse part,has been widely studied since it is a basic tool in the field of statistics and data science.This paper focuses on a non-convex model for robust principal component analysis problems,where the low-rank matrix is factorized as a product of two small-size matrices such that the low-rank requirement is automatically fulfilled.Based on the non-convex model,we develop an alternating direction method equipped with a non-monotone search technique for solving robust principal component analysis problems.In the new algorithm,the variables are alternately updated,the variables of the low rank part are updated by a gradient descent method with an exact step size,and the variables of the sparse part are updated by a non-monotonic search technique.We established the convergence theory of the new algorithm under certain conditions.The final numerical results show the effectiveness of the proposed algorithm.