Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem
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In this paper, we develop a Bayesian evidence maximization framework to solve the sparse non-negative least squares problem (S-NNLS). We introduce a family of scale mixtures referred as to Rectified Gaussian Scale Mixture (R-GSM) to model the sparsity enforcing prior distribution for the signal of interest. Through proper choice of the mixing density, the R-GSM prior encompasses a wide variety of heavy-tailed distributions such as the rectified Laplacian and rectified Student-t distributions. Utilizing the hierarchical representation induced by the scale mixture prior, an evidence maximization or Type II estimation method based on the expectation-maximization (EM) framework is developed to estimate the hyper-parameters and to obtain a point estimate of the parameter of interest. In the proposed method, called rectified Sparse Bayesian Learning (R-SBL), we provide four alternative approaches that offer a range of options to trade-off computational complexity to quality of the E-step computation. The methods include the Markov Chain Monte Carlo EM, linear minimum mean square estimation, approximate message passing and a diagonal approximation. Through numerical experiments, we show that the proposed R-SBL method outperforms existing S-NNLS solvers in terms of both signal and support recovery.
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