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.
@cite_3: This paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data. The adjective "unsupervised" is justified by two properties of the algorithm: 1) it is capable of selecting the number of components and 2) unlike the standard expectation-maximization (EM) algorithm, it does not require careful initialization. The proposed method also avoids another drawback of EM for mixture fitting: the possibility of convergence toward a singular estimate at the boundary of the parameter space. The novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models; instead, we seamlessly integrate estimation and model selection in a single algorithm. Our technique can be applied to any type of parametric mixture model for which it is possible to write an EM algorithm; in this paper, we illustrate it with experiments involving Gaussian mixtures. These experiments testify for the good performance of our approach.
Fitting Gaussian mixture models: Figueiredo and Jain @cite_3 propose an algorithm that fits a given data sequence with a GM model. The algorithm employs a cost function that resembles the minimum message length (MML) criterion, and the parameters are learned using a modified expectation-maximization (EM) algorithm.