We introduce a novel approach for robust principal component analysis (RPCA) for a partially observed data matrix. The aim is to recover the data matrix as a sum of a low-rank matrix and a sparse matrix so as to eliminate erratic noise (outliers). This problem is known to be NP-hard in general. A classical approach to solving RPCA is to consider convex relaxations. One such heuristic involves the minimization of the (weighted) sum of a nuclear norm part, that promotes a low-rank component, with an \(\ell _1\) norm part, to promote a sparse component. This results in a well-structured convex problem that can be efficiently solved by modern first-order methods. However, first-order methods often yield low accuracy solutions. Moreover, the heuristic of using a norm consisting of a weighted sum of norms may lose some of the advantages that each norm had when used separately. In this paper, we propose a novel nonconvex and nonsmooth reformulation of the original NP-hard RPCA model. The new model adds a redundant semidefinite cone constraint and solves small subproblems using a PALM algorithm. Each subproblem results in an exposing vector for a facial reduction technique that is able to reduce the size significantly. This makes the problem amenable to efficient algorithms in order to obtain high-level accuracy. We include numerical results that confirm the efficacy of our approach.

中文翻译：

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使用面部缩小功能进行稳健的主成分分析

我们介绍了一种用于部分观测数据矩阵的鲁棒主成分分析（RPCA）的新方法。目的是将数据矩阵恢复为低秩矩阵和稀疏矩阵的和，以消除不稳定的噪声（异常值）。通常，已知此问题是NP难题。解决RPCA的经典方法是考虑凸松弛。一种这样的启发式方法涉及将核规范部分的（加权）总和最小化，从而促进低阶成分的\（\ ell _1 \）规范部分，以促进稀疏的组成部分。这导致结构良好的凸问题，可以通过现代一阶方法有效解决。但是，一阶方法通常会产生低精度的解决方案。此外，使用由加权的规范和组成的规范的试探法可能会失去单独使用每个规范所具有的一些优势。在本文中，我们提出了一种新颖的非凸且非平滑的原始NP-hard RPCA模型的公式。新模型增加了冗余的半定锥约束，并使用PALM算法解决了小的子问题。每个子问题都会导致一个暴露向量能够大幅缩小尺寸的面部缩小技术。这使得该问题适合于高效算法以获得高级别的准确性。我们包含的数值结果证实了我们方法的有效性。