Accelerated Alternating Projections for Robust Principal Component Analysis

We study robust PCA for the fully observed setting, which is about separating a low rank matrix $\boldsymbol{L}$ and a sparse matrix $\boldsymbol{S}$ from their sum $\boldsymbol{D}=\boldsymbol{L}+\boldsymbol{S}$. 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 $\boldsymbol{L}_k$ and $\boldsymbol{S}_k$ be the current estimates of the low rank matrix and the sparse matrix, respectively. The algorithm achieves significant acceleration by first projecting $\boldsymbol{D}-\boldsymbol{S}_k$ onto a low dimensional subspace before obtaining the new estimate of $\boldsymbol{L}$ 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.