  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, dubbed accelerated alternating projections, is introduced for robust
PCA which significantly improves the computational efficiency of the existing
alternating projections proposed in [Netrapalli, Praneeth, et al., 2014] when
updating the low rank factor. The acceleration is achieved by first projecting
a matrix onto some low dimensional subspace before obtaining a new estimate of
the low rank matrix 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.
