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Accelerated Alternating Projections for Robust Principal Component Analysis

Abstract

We study robust PCA for the fully observed setting, which is about separating a low rank matrix L\boldsymbol{L} and a sparse matrix S\boldsymbol{S} from their sum D=L+S\boldsymbol{D}=\boldsymbol{L}+\boldsymbol{S}. In this paper, a new algorithm, termed accelerated alternating projections, is introduced for robust PCA which accelerates existing alternating projections proposed in [Netrapalli, Praneeth, et al., 2014]. Let Lk\boldsymbol{L}_k and Sk\boldsymbol{S}_k be the current estimates of the low rank matrix and the sparse matrix, respectively. The algorithm achieves significant acceleration by first projecting DSk\boldsymbol{D}-\boldsymbol{S}_k onto a low dimensional subspace before obtaining the new estimate of L\boldsymbol{L} via truncated SVD. Exact recovery guarantee has been established which shows linear convergence of the proposed algorithm. Empirical performance evaluations establish the advantage of our algorithm over other state-of-the-art algorithms for robust PCA.

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