Sparsity via Hyperpriors: A Theoretical and Algorithmic Study under Empirical Bayes Framework

TL;DR

This paper analyzes how hyperpriors within the empirical Bayes framework influence sparsity and solution stability, proposing algorithms and demonstrating improved image deblurring results through theoretical insights and numerical experiments.

Contribution

It provides a theoretical connection between hyperprior choices and sparsity, and develops a convergent algorithm for sparse inverse problems.

Findings

Hyperpriors like half-Laplace promote sparsity effectively.

The proposed PALM algorithm converges for both convex and concave hyperpriors.

Introducing hyperpriors improves image deblurring accuracy.

Abstract

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.