A Variational Approach for Sparse Component Estimation and Low-Rank Matrix Recovery

Abstract

We propose a variational Bayesian based algorithm for the estimation of the sparse component of an outliercorrupted low-rank matrix, when linearly transformed composite data are observed. The model constitutes a generalization of robust principal component analysis. The problem considered herein is applicable in various practical scenarios, such as foreground detection in blurred and noisy video sequences and detection of network anomalies among others. The proposed algorithm models the low-rank matrix and the sparse component using a hierarchical Bayesian framework, and employs a variational approach for inference of the unknowns. The effectiveness of the proposed algorithm is demonstrated using real life experiments, and its performance improvement over regularization based approaches is shown.

Automated Recovery of Compressedly Observed Sparse Signals From Smooth Background

“…Smoothness of is resulted when most of the outer products in are zeros. To achieve this, common sparsity promoting priors are simultaneously assigned to the columns of and , as in [20] (2)…”

“…Omitting the details that can be found in [20], the posterior means of and are given by (6) where denotes the posterior mean of , and the covariance matrices and are inter-related as…”

“…All hyperparameters in the model are set to a small value (e.g., ), making the algorithm fully automated. As a comparison, we show the performance on the same test data of two alternative algorithms, namely the Variational Bayes Sparse Estimator (VBSE) proposed in [20], and the state-of-the-art Alternating Directions Method of Multipliers (ADMM) proposed in [1]. For the problem considered herein, VBSE is the only available Bayesian approach to our knowledge and ADMM is reported to have high recovery accuracy in the literature.…”

“…The optimization-based algorithms usually require proper selection of user parameters. In contrast, [20] takes a Bayesian perspective and automatically estimates all model parameters. A limitation of [20], though, is the memory and computational requirement incurred by the use of Hessian information.…”

“…As explained in [21], this prior has been shown to promote sparsity to a higher level than the Sparse Bayesian Learning (SBL) prior used in [20]. In addition, in order to reduce memory consumption, we develop a constructive approach based on the principles in [22] and [12] that essentially replaces the demanding matrix inversion with a sequence of efficient rank-one updates.…”