Nearly-optimal Robust Matrix Completion

Abstract

In this paper, we consider the problem of Robust Matrix Completion (RMC) where the goal is to recover a low-rank matrix by observing a small number of its entries out of which a few can be arbitrarily corrupted. We propose a simple projected gradient descent method to estimate the low-rank matrix that alternately performs a projected gradient descent step and cleans up a few of the corrupted entries using hard-thresholding. Our algorithm solves RMC using nearly optimal number of observations as well as nearly optimal number of corruptions. Our result also implies significant improvement over the existing time complexity bounds for the low-rank matrix completion problem. Finally, an application of our result to the robust PCA problem (low-rank+sparse matrix separation) leads to nearly linear time (in matrix dimensions) algorithm for the same; existing state-of-the-art methods require quadratic time. Our empirical results corroborate our theoretical results and show that even for moderate sized problems, our method for robust PCA is an order of magnitude faster than the existing methods.

A Unified Framework for Low-Rank plus Sparse Matrix Recovery

“…When considering the noiseless case, only the optimization error term exists. It is worth noting that our robustness guarantee required for the gradient descent phase matches the best-known results O(1/r) in Hsu et al (2011); Chen et al (2013); Cherapanamjeri et al (2016).…”

“…• The gradient descent phase of our proposed algorithm matches the best-known robustness guarantee O(1/r) (Hsu et al, 2011;Chen et al, 2013). Compared with existing robust PCA algorithms (Yi et al, 2016;Cherapanamjeri et al, 2016), our algorithm achieves improved computational complexity O r 3 d log d log(1/ ) , while matching the optimal sample complexity O(r 2 d log d) for Burer-Monteiro factorization-based low-rank matrix recovery (Zheng and Lafferty, 2016) under the incoherence condition.…”

“…However, most of the aforementioned work are based on convex relaxation, which involves a computationally expensive SVD step in each iteration. To address such computational barrier, various nonconvex optimization algorithms (Netrapalli et al, 2014;Chen and Wainwright, 2015;Yi et al, 2016;Cherapanamjeri et al, 2016) have been carried out to solve low-rank plus sparse matrix recovery with provable guarantees. For example, Netrapalli et al (2014) proposed alternating projection to simultaneously estimate the low-rank and sparse structure, while Chen and Wainwright (2015) showed that projected gradient descent based algorithm will linearly converge to the unknown matrix decomposition under suitable initialization procedure.…”

“…Their approach allows for O(1/r 1.5 ) sparsity with improved computational efficiency upon previous work. Most recently, Cherapanamjeri et al (2016) further improved the existing work in terms of robustness guarantee from O(1/r 1.5 ) to O(1/r). It is worth noting that these several pieces of work are limited to robust PCA, thus unable to deal with more general problem settings, such as robust matrix sensing.…”

Robust Wirtinger Flow for Phase Retrieval with Arbitrary Corruption

“…The remainder of this paper is organized as follows: in Section 3, we review the problem formulation in detail. We present the algorithm in Section 4, and the main theory in Section 2016), robust low-rank matrix completion (Agarwal et al, 2012;Goldfarb and Qin, 2014;Klopp et al, 2014;Cherapanamjeri et al, 2016), robust tensor decomposition (Gu et al, 2014;Anandkumar et al, 2015). The key idea of these methods is to estimate the unknown low-rank matrix/tensor and the sparse corruption matrix/tensor simultaneously.…”