Sparse Bayesian Learning Algorithms Revisited: From Learning Majorizers to Structured Algorithmic Learning using Neural Networks

arXiv cs.AI / 4/6/2026

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Key Points

The paper revisits Sparse Bayesian Learning (SBL) for sparse signal recovery by building a unified framework using the majorization-minimization (MM) principle to systematically derive SBL algorithms.
It derives well-known SBL methods under MM, providing new (previously unknown) convergence guarantees and showing that multiple popular update rules are compatible as valid descent steps within a shared majorizer.
Leveraging MM theory, the authors expand the class of SBL update rules and propose an approach to select or learn a better algorithm using data while staying within the MM framework.
Going beyond MM, the paper introduces a deep-learning-based architecture designed to learn superior SBL update rules from data and to generalize across different measurement matrices.
The method is evaluated across varying snapshots, signal-to-noise ratios, and sparsity levels, including tests on unseen matrices (zero-shot) and training/testing across parameter ranges for structured (parameterized) dictionaries.

Abstract

Sparse Bayesian Learning is one of the most popular sparse signal recovery methods, and various algorithms exist under the SBL paradigm. However, given a performance metric and a sparse recovery problem, it is difficult to know a-priori the best algorithm to choose. This difficulty is in part due to a lack of a unified framework to derive SBL algorithms. We address this issue by first showing that the most popular SBL algorithms can be derived using the majorization-minimization (MM) principle, providing hitherto unknown convergence guarantees to this class of SBL methods. Moreover, we show that the two most popular SBL update rules not only fall under the MM framework but are both valid descent steps for a common majorizer, revealing a deeper analytical compatibility between these algorithms. Using this insight and properties from MM theory we expand the class of SBL algorithms, and address finding the best SBL algorithm via data within the MM framework. Second, we go beyond the MM framework by introducing the powerful modeling capabilities of deep learning to further expand the class of SBL algorithms, aiming to learn a superior SBL update rule from data. We propose a novel deep learning architecture that can outperform the classical MM based ones across different sparse recovery problems. Our architecture's complexity does not scale with the measurement matrix dimension, hence providing a unique opportunity to test generalization capability across different matrices. For parameterized dictionaries, this invariance allows us to train and test the model across different parameter ranges. We also showcase our model's ability to learn a functional mapping by its zero-shot performance on unseen measurement matrices. Finally, we test our model's performance across different numbers of snapshots, signal-to-noise ratios, and sparsity levels.