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Sparse Bayesian Learning for Robust PCA

In this paper, we propose a new Bayesian model to solve the Robust PCA problem - recovering the underlying low-rank matrix and sparse matrix from their noisy compositions. We first derive and analyze a new objective function, which is proven to be equivalent to the fundamental minimizing "rank+sparsity" objective. To solve this objective, we develop a concise Sparse Bayesian Learning (SBL) method that has minimum assumptions and effectively deals with the crux of the problem. The concise modeling allows simple and effective Empirical Bayesian inference via MAP-EM. Simulation studies demonstrate the superiority of the proposed method over the existing state-of-the-art methods. The efficacy of the method is further verified through a text extraction image processing task.