通过超先验控制稀疏性，提升图像去模糊的精度与稳定性
Sparsity via Hyperpriors: A Theoretical and Algorithmic Study under Empirical Bayes Framework
- 基于经验贝叶斯框架，用超先验调节稀疏学习中的参数
- 半拉普拉斯等超先验可显著提升解的稀疏性与抗噪能力
- 适用于图像恢复等不适定逆问题，尤其对噪声敏感场景有效
本文系统研究了经验贝叶斯框架下超参数估计对稀疏学习的影响。通过分析超先验对解的影响，建立了其与解的稀疏性及局部最优性之间的理论联系。研究表明，某些严格递增的超先验（如半拉普拉斯、幂次在(0,1)区间的半广义高斯）能有效促进稀疏性，并增强解对测量噪声的稳定性。基于此分析，采用具有收敛保证的近端交替线性化最小化（PALM）算法处理凸与凹超先验。在二维图像去模糊问题上的大量数值实验表明，引入合适的超先验可显著提升解的稀疏性并改善恢复精度。此外，还揭示了噪声水平和逆问题不适定性对经验贝叶斯解的影响。
原文摘要 · Abstract (English)
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.
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