Sparse Bayesian Learning (SBL) models are extensively used in signal processing and machine learning for promoting sparsity through hierarchical priors. The hyperparameters in SBL models are crucial for the model's performance, but they are often difficult to estimate due to the non-convexity and the high-dimensionality of the associated objective function. This paper presents a comprehensive framework for hyperparameter estimation in SBL models, encompassing well-known algorithms such as the expectation-maximization (EM), MacKay, and convex bounding (CB) algorithms. These algorithms are cohesively interpreted within an alternating minimization and linearization (AML) paradigm, distinguished by their unique linearized surrogate functions. Additionally, a novel algorithm within the AML framework is introduced, showing enhanced efficiency, especially under low signal noise ratios. This is further improved by a new alternating minimization and quadratic approximation (AMQ) paradigm, which includes a proximal regularization term. The paper substantiates these advancements with thorough convergence analysis and numerical experiments, demonstrating the algorithm's effectiveness in various noise conditions and signal-to-noise ratios.

翻译：稀疏贝叶斯学习（SBL）模型通过分层先验促进稀疏性，被广泛应用于信号处理和机器学习领域。SBL模型中的超参数对模型性能至关重要，但由于相应目标函数的非凸性和高维性，其估计往往存在困难。本文提出了一套SBL模型超参数估计的综合框架，涵盖了期望最大化（EM）、MacKay和凸边界（CB）等经典算法。这些算法在交替最小化与线性化（AML）范式中得到统一解释，其差异体现在特定的线性化替代函数上。此外，本文在AML框架内提出了一种新算法，该算法在低信噪比条件下展现出更高效率，并通过引入近端正则化项的新型交替最小化与二次逼近（AMQ）范式进一步优化。论文通过严谨的收敛性分析和数值实验，验证了该算法在不同噪声条件及信噪比下的有效性。