Editorial Board

JMI2010B-5 Convex optimization techniques for the efficient recovery of a sparsely corrupted low-rank matrix (pp.147-156)

Author(s): Silvia Gandy and Isao Yamada

J. Math-for-Ind. 2B (2010) 147-156.

We address the problem of recovering a low-rank matrix that has a small fraction of its entries arbitrarily corrupted. This problem is recently attracting attention as nontrivial extension of the classical PCA (principal component analysis) problem with applications in image processing and model/system identification. It was shown that the problem can be solved via a convex optimization formulation when certain conditions hold. Several algorithms were proposed in the sequel, including interior-point methods, iterative thresholding and accelerated proximal gradients. In this work we address the problem from two completely different sides. First, we propose an algorithm based on the Douglas-Rachford splitting technique which has inherent convergence guarantees. Second, we propose, based on algorithms from rank minimization and sparse vector recovery, a computationally efficient greedy algorithm that scales better to large problem sizes than existing algorithms. We compare the performance of these proposed algorithms to the accelerated proximal gradients algorithm.

Keyword(s).  PCA, rank minimization, nuclear norm minimization, sparse error, Douglas-Rachford splitting, greedy algorithms