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

We consider the recovery of an (approximately) sparse signal from noisy linear measurements, in the case that the signal is apriori known to be non-negative and obeys certain linear equality constraints. For this, we propose a novel empirical-Bayes approach that combines the Generalized Approximate Message Passing (GAMP) algorithm with the expectation maximization (EM) algorithm. To enforce both sparsity and non-negativity, we employ an i.i.d Bernoulli non-negative Gaussian mixture (NNGM) prior and perform approximate minimum mean-squared error (MMSE) recovery of the signal using sum-product GAMP. To learn the NNGM parameters, we use the EM algorithm with a suitable initialization. Meanwhile, the linear equality constraints are enforced by augmenting GAMP's linear observation model with noiseless pseudo-measurements. Numerical experiments demonstrate the state-of-the art mean-squared-error and runtime of our approach.