  We study compressed sensing (CS) signal reconstruction problems where an
input signal is measured via matrix multiplication under additive white
Gaussian noise. Our signals are assumed to be stationary and ergodic, but the
input statistics are unknown; the goal is to provide reconstruction algorithms
that are universal to the input statistics. We present a novel algorithm that
combines: (i) the approximate message passing (AMP) CS reconstruction
framework, which converts the matrix channel recovery problem into scalar
channel denoising; (ii) a universal denoising scheme based on context
quantization, which partitions the stationary ergodic signal denoising into
independent and identically distributed (i.i.d.) subsequence denoising; and
(iii) a density estimation approach that approximates the probability
distribution of an i.i.d. sequence by fitting a Gaussian mixture (GM) model. In
addition to the algorithmic framework, we provide three contributions: (i)
numerical results showing that state evolution holds for non-separable Bayesian
sliding-window denoisers; (ii) a universal denoiser that does not require the
input signal to be bounded; and (iii) we modify the GM learning algorithm, and
extend it to an i.i.d. denoiser. Our universal CS recovery algorithm compares
favorably with existing reconstruction algorithms in terms of both
reconstruction quality and runtime, despite not knowing the input statistics of
the stationary ergodic signal.
