Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem

TL;DR

This paper introduces a Bayesian framework using Rectified Gaussian Scale Mixtures for solving sparse non-negative least squares problems, improving accuracy and robustness over existing methods.

Contribution

It develops a novel R-GSM prior and four R-SBL variants, enhancing sparse solution estimation with improved performance and computational options.

Findings

Outperforms existing S-NNLS solvers in signal recovery

Demonstrates robustness against different design matrix structures

Provides multiple computational trade-offs for practical use

Abstract

In this paper, we develop a Bayesian evidence maximization framework to solve the sparse non-negative least squares (S-NNLS) problem. We introduce a family of probability densities referred to as the Rectified Gaussian Scale Mixture (R- GSM) to model the sparsity enforcing prior distribution for the solution. The R-GSM prior encompasses a variety of heavy-tailed densities such as the rectified Laplacian and rectified Student- t distributions with a proper choice of the mixing density. We utilize the hierarchical representation induced by the R-GSM prior and develop an evidence maximization framework based on the Expectation-Maximization (EM) algorithm. Using the EM based method, we estimate the hyper-parameters and obtain a point estimate for the solution. We refer to the proposed method as rectified sparse Bayesian learning (R-SBL). We provide four R- SBL variants that offer a range of options for computational complexity and the quality of the E-step computation. These methods include the Markov chain Monte Carlo EM, linear minimum mean-square-error estimation, approximate message passing and a diagonal approximation. Using numerical experiments, we show that the proposed R-SBL method outperforms existing S-NNLS solvers in terms of both signal and support recovery performance, and is also very robust against the structure of the design matrix.