An Empirical-Bayes Approach to Recovering Linearly Constrained Non-Negative Sparse Signals

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 optimization, density estimation, and certain cases of compressive imaging. Our first approach solves a linearly constrained non-negative version of LASSO using the max-sum version of the generalized approximate message passing (GAMP) algorithm, where we consider both quadratic and absolute loss, and where we propose a novel approach to tuning the LASSO regularization parameter via the expectation maximization (EM) algorithm. Our second approach is based on the sum-product version of the GAMP algorithm, where we propose the use of a Bernoulli non-negative Gaussian-mixture signal prior and a Laplacian likelihood and propose 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.