Sparse Estimation using Bayesian Hierarchical Prior Modeling for Real and Complex Linear Models

TL;DR

This paper introduces the Bessel K model, a new GSM-based prior for complex sparse signal estimation, providing a unified framework that improves convergence, sparsity, and robustness over existing methods.

Contribution

It proposes the Bessel K model for complex-valued sparse Bayesian learning, analyzing its properties and deriving a new estimator that outperforms existing algorithms.

Findings

Superior convergence speed compared to state-of-the-art methods

Enhanced sparsity and lower reconstruction error

Robust performance in low and medium SNR regimes

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.