We address the problem of efficient sparse fixed-rank (S-FR) matrix decomposition, i.e., splitting a corrupted matrix $M$ into an uncorrupted matrix $L$ of rank $r$ and a sparse matrix of outliers $S$. Fixed-rank constraints are usually imposed by the physical restrictions of the system under study. Here we propose a method to perform accurate and very efficient S-FR decomposition that is more suitable for large-scale problems than existing approaches. Our method is a grateful combination of geometrical and algebraical techniques, which avoids the bottleneck caused by the Truncated SVD (TSVD). Instead, a polar factorization is used to exploit the manifold structure of fixed-rank problems as the product of two Stiefel and an SPD manifold, leading to a better convergence and stability. Then, closed-form projectors help to speed up each iteration of the method. We introduce a novel and fast projector for the $text{SPD}$ manifold and a proof of its validity. Further acceleration is achieved using a Nystrom scheme. Extensive experiments with synthetic and real data in the context of robust photometric stereo and spectral clustering show that our proposals outperform the state of the art.
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