Sparse Signal Recovery via Correlated Degradation Model

Sparse signal recovery aims to recover an unknown signal x ∈ R from few non-adaptive, possibly noisy, linear measurements y∈R using a nonlinear sparsity-promoting algorithm, under the assumption that x is sparse or compressible with respect to a known basis or frame [1]. Specifically, y = Ax+ e, where A ∈ Rm×n is the measurement matrix, e ∈ R is the measurement error, and m n. Many of the sparse recovery approaches proposed during the past decade can be regarded as iterative estimation of a signal from a degraded observation [2]–[6]. A common feature of these approaches is that the degradation is modeled as independent and identically distributed (i.i.d.) additive noise that has to be alleviated by a denoising filter at each iteration. For instance, Approximate Message Passing (AMP) [2] enforces an i.i.d.additive Gaussian noise denoising problem at each iteration through the Onsager correction term [2]–[4]. Plug&Play-Prior (P) frameworks [5], [6] use denoising algorithms as regularizers (priors) for model-based inversion via the alternating direction method of multipliers (ADMM) [7]. The idea of ADMM is to convert an unconstrained optimization problem x̂ = argminx f(x) + λg(x) into its equivalent constrained form which is then decoupled into two separate optimizations: