We study compressed sensing (CS) signal reconstruction problems where aninput signal is measured via matrix multiplication under additive whiteGaussian noise. Our signals are assumed to be stationary and ergodic, but theinput statistics are unknown; the goal is to provide reconstruction algorithmsthat are universal to the input statistics. We present a novel algorithmicframework that combines: (i) the approximate message passing (AMP) CSreconstruction framework, which solves the matrix channel recovery problem byiterative scalar channel denoising; (ii) a universal denoising scheme based oncontext quantization, which partitions the stationary ergodic signal denoisinginto independent and identically distributed (i.i.d.) subsequence denoising;and (iii) a density estimation approach that approximates the probabilitydistribution of an i.i.d. sequence by fitting a Gaussian mixture (GM) model. Inaddition to the algorithmic framework, we provide three contributions: (i)numerical results showing that state evolution holds for non-separable Bayesiansliding-window denoisers; (ii) an i.i.d. denoiser based on a modified GMlearning algorithm; and (iii) a universal denoiser that does not needinformation about the range where the input takes values from or require theinput signal to be bounded. We provide two implementations of our universal CSrecovery algorithm with one being faster and the other being more accurate. Thetwo implementations compare favorably with existing universal reconstructionalgorithms in terms of both reconstruction quality and runtime.
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