Multilevel approximate robust principal component analysis

Robust principal component analysis (RPCA) is currently the method of choice for recovering a low-rank matrix from sparse corruptions that are of unknown value and support by decomposing the observation matrix into low-rank and sparse matrices. RPCA has many applications including background subtraction, learning of robust subspaces from visual data, etc. Nevertheless, the application of SVD in each iteration of optimisation methods renders the application of RPCA challenging in cases when data is large. In this paper, we propose the first, to the best of our knowledge, multilevel approach for solving convex and non-convex RPCA models. The basic idea is to construct lower dimensional models and perform SVD on them instead of the original high dimensional problem. We show that the proposed approach gives a good approximate solution to the original problem for both convex and non-convex formulations, while being many times faster than original RPCA methods in several real world datasets.

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