An empirical-Bayes approach to recovering linearly constrained non-negative sparse signals

Abstract—We propose two novel approaches for the recovery of an (approximately) sparse signal from noisy linear measurements in the case that the signal is a priori known to be non-negative and obey given linear equality constraints, such as a simplex signal. This problem arises in, e.g., hyperspectral imaging, portfolio op-timization, density estimation, and certain cases of compressive imaging. Our first approach solves a linearly constrained non-neg-ative version of LASSO using the max-sum version of the gen-eralized approximate message passing (GAMP) algorithm, where we consider both quadratic and absolute loss, and where we pro-pose a novel approach to tuning the LASSO regularization param-eter via the expectation maximization (EM) algorithm. Our second approach is based on the sum–product version of the GAMP al-gorithm, where we propose the use of a Bernoulli non-negative Gaussian-mixture signal prior and a Laplacian likelihood and pro-pose an EM-based approach to learning the underlying statistical parameters. In both approaches, the linear equality constraints are enforced by augmenting GAMP’s generalized-linear observation model with noiseless pseudo-measurements. Extensive numerical experiments demonstrate the state-of-the-art performance of our proposed approaches. Index Terms—Belief propagation, compressed sensing, estima-tion, expectation maximization algorithms. I