基于正则化REAPER和无矩阵近似算法的鲁棒PCA
Robust PCA via Regularized REAPER with a Matrix-Free Proximal Algorithm
摘要：
已知主成分分析（PCA）对异常值敏感，因此在文献中提出了各种鲁棒PCA变体。最近的一个模型，称为REAPER，旨在通过解决一个凸优化问题来找到主成分。通常，主成分的数量必须事先确定，并且在具有数据大小的对称半正定矩阵上执行最小化，尽管主成分的数量基本上较小。如果数据的维度很大，这在图像处理中经常发生，则禁止使用它。在本文中，我们提出了一个正则化版本的REAPER，它通过惩罚相应的正交投影仪的核范数来加强主成分数量的稀疏性。这样做的优点是只需要主成分数量的上界。我们的第二个贡献是一个无矩阵算法，找到一个最小的正则化的REAPER，这也是适合于高维数据。该算法将原始-对偶极小化方法与厚重启动Lanczos过程相结合。作为一个附带的结果，我们讨论了稳健PCA中的偏差问题。数值例子证明了我们的算法的性能。
英文摘要：
Principal component analysis (PCA) is known to be sensitive to outliers, so that various robust PCA variants were proposed in the literature. A recent model, called REAPER, aims to find the principal components by solving a convex optimization problem. Usually the number of principal components must be determined in advance and the minimization is performed over symmetric positive semi-definite matrices having the size of the data, although the number of principal components is substantially smaller. This prohibits its use if the dimension of the data is large which is often the case in image processing. In this paper, we propose a regularized version of REAPER which enforces the sparsity of the number of principal components by penalizing the nuclear norm of the corresponding orthogonal projector. This has the advantage that only an upper bound on the number of principal components is required. Our second contribution is a matrix-free algorithm to find a minimizer of the regularized REAPER which is also suited for high dimensional data. The algorithm couples a primal-dual minimization approach with a thick-restarted Lanczos process. As a side result, we discuss the topic of the bias in robust PCA. Numerical examples demonstrate the performance of our algorithm.
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