Sparsity via Hyperpriors: A Theoretical and Algorithmic Study under Empirical Bayes Framework
This work addresses sparse learning in inverse problems like image deblurring, offering incremental improvements in hyperprior selection and algorithm design.
The paper tackled the problem of hyperparameter estimation in empirical Bayes for sparse learning, establishing a theoretical link between hyperpriors and sparsity, and demonstrated that certain hyperpriors like half-Laplace improve sparsity and stability, with numerical tests on image deblurring showing enhanced restoration accuracy.
This paper presents a comprehensive analysis of hyperparameter estimation within the empirical Bayes framework (EBF) for sparse learning. By studying the influence of hyperpriors on the solution of EBF, we establish a theoretical connection between the choice of the hyperprior and the sparsity as well as the local optimality of the resulting solutions. We show that some strictly increasing hyperpriors, such as half-Laplace and half-generalized Gaussian with the power in $(0,1)$, effectively promote sparsity and improve solution stability with respect to measurement noise. Based on this analysis, we adopt a proximal alternating linearized minimization (PALM) algorithm with convergence guaranties for both convex and concave hyperpriors. Extensive numerical tests on two-dimensional image deblurring problems demonstrate that introducing appropriate hyperpriors significantly promotes the sparsity of the solution and enhances restoration accuracy. Furthermore, we illustrate the influence of the noise level and the ill-posedness of inverse problems to EBF solutions.