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.

本文提出了一个综合框架，用于稀疏贝叶斯学习模型中的超参数估计，包括著名的算法如期望最大化、MacKay 和凸包算法。通过交替最小化和线性化范式来统一解释这些算法，并引入了一种新颖的算法，表现出增强的效率，尤其是在低信噪比条件下。此外，还通过收敛性分析和数值实验来证实这些进展，在各种噪声条件和信噪比下证明了算法的有效性。