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

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 sparse estimators based on a modification of the expectation-maximization algorithm formulated for Type II estimation. The estimators include as special instances the algorithms proposed by Tipping and Faul [1] and Babacan et al. [2]. Numerical results show the superiority of the proposed estimators over these state-of-the-art algorithms in terms of convergence speed, sparseness, reconstruction error, and robustness in low and medium signal-to-noise ratio regimes.