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 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.

💡 Research Summary

This paper introduces a novel Bayesian hierarchical prior—the Bessel K model—for sparse Bayesian learning (SBL) that works uniformly for both real‑valued and complex‑valued linear regression problems. The authors start by recalling that SBL typically employs a two‑layer hierarchical prior: a conditional Gaussian prior p(w | γ) and a hyper‑prior p(γ). By choosing the mixing distribution p(γ) as a Gamma distribution with shape ε and rate η, the marginal prior p(w) becomes a product of Bessel K functions, a family that includes many well‑known sparsity‑inducing densities as special cases.

The paper distinguishes between Type I (MAP estimation of w) and Type II (evidence maximization, i.e., MAP estimation of γ followed by the posterior mean of w) approaches. For Type I, the induced penalty term is q_I(w_i)=−log