We propose two novel approaches to 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
simplex signals. 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.

本文提出了两种新颖的方法，分别基于 max-sum GAMP 算法和 sum-product GAMP 算法，用于从有噪声线性测量中恢复稀疏信号，且信号为非负且受给定线性等式约束，例如 simplex 信号。两种方法的实验表明，提出的方法具有最先进的性能。