Fast Convex Optimization Algorithms for Exact Recovery of a Corrupted Low-Rank Matrix


This paper studies algorithms for solving the problem of recovering a low-rank matrix with a fraction of its entries arbitrarily corrupted. This problem can be viewed as a robust version of classical PCA, and arises in a number of application domains, including image processing, web data ranking, and bioinformatic data analysis. It was recently shown that under surprisingly broad conditions, it can be exactly solved via a convex programming surrogate that combines nuclear norm minimization and L1-norm minimization. This paper develops and compares two complementary approaches for solving this convex program. The first is an accelerated proximal gradient algorithm directly applied to the primal; while the second is a gradient algorithm applied to the dual problem. Both are several orders of magnitude faster than the previous state-of-the-art algorithm for this problem, which was based on iterative thresholding. Simulations demonstrate the performance improvement that can be obtained via these two algorithms, and clarify their relative merits.

Technical Report UILU-ENG-09-2214, UIUC