Sparse estimation using Bayesian hierarchical prior modeling for real and complex linear models

Abstract

In sparse Bayesian learning (SBL), Gaussian scale mixtures (GSMs) have been used to model sparsity-inducing priors that realize a class of concave penalty functions for the regression task in real-valued signal models. Motivated by the relative scarcity of formal tools for SBL in complex-valued models, this paper proposes a GSM model -the Bessel K model -that induces concave penalty functions for the estimation of complex sparse signals. The properties of the Bessel K model are analyzed when it is applied to Type I and Type II estimation. This analysis reveals that, by tuning the parameters of the mixing pdf different penalty functions are invoked depending on the estimation type used, the value of the noise variance, and whether real or complex signals are estimated. Using the Bessel K model, we derive a sparse estimator based on a modification of the expectation-maximization algorithm formulated for Type II estimation. The estimator includes as a special instance the algorithms proposed by Tipping and Faul [1] and by Babacan et al. [2]. Numerical results show the superiority of the proposed estimator over these state-of-the-art estimators in terms of convergence speed, sparseness, reconstruction error, and robustness in low and medium signal-to-noise ratio regimes.

Radar coincidence imaging with phase error using Bayesian hierarchical prior modeling

“…27, the approximated posterior of β can be expressed as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 7 ; 6 3 ; 5 9 7 qðβÞ ∝ expfhln pðyjβ; α 0 ; φÞi qðα 0 Þ hpðβjαÞi qðαÞ g:…”

“…Here, we set ε ¼ 0, a ¼ 1, and b ¼ 10 −6 , which strongly promotes a sparse estimate and make the prior for η noninformative. 27 Compared with the two-layer (2-L) prior model, the 3-L prior model considers η as random variable specified by a distribution and incorporates it into the inference framework. This leads to additional degrees of freedom in controlling the sparsity property of the resulting inference scheme.…”

“…This leads to additional degrees of freedom in controlling the sparsity property of the resulting inference scheme. 27 In addition, the measurement noise is assumed to be complex Gaussian with zero-mean and variance α −1 0 , to allow conjugate-exponential analysis. Then, the following 2-L hierarchical Gaussian-Gamma (G-Ga) prior is assigned:…”

“…27, the derivation detail is shown in Appendix A. It can be concluded that the posterior distributions of β and α k are complex Gaussian distribution and generalized inverse Gaussian (GIG) distribution, respectively, while both η and α 0 obey a Gamma distribution.…”

Fast L0-based image deconvolution with variational Bayesian inference and majorization-minimization

“…T has a strong influence on s. So, as suggested in [10], we further assume b is a vector associated with a Gamma prior:…”