Total Variation Sparse Bayesian Learning for Block Sparsity via Majorization-Minimization

TL;DR

This paper introduces a new optimization framework for total variation sparse Bayesian learning that effectively handles unknown block boundaries and isolated non-zero entries, improving accuracy and efficiency in sparse recovery tasks.

Contribution

It develops a majorization-minimization based approach for DoL-TV SBL with exponential reparameterization, extending to unknown noise variance estimation.

Findings

Enhanced accuracy in sparse recovery on synthetic data

Faster runtime compared to benchmark methods

Effective handling of unknown block boundaries

Abstract

Block sparsity is a widely exploited structure in sparse recovery, offering significant gains when signal blocks are known. Yet, practical signals often exhibit unknown block boundaries and isolated non-zero entries, which challenge traditional approaches. A promising method to handle such complex sparsity patterns is the difference-of-logs total variation (DoL-TV) regularized sparse Bayesian learning (SBL). However, due to the complex form of DoL-TV term, the resulting optimization problem is hard to solve. This paper develops a new optimization framework for the DoL-TV SBL cost function. By introducing an exponential reparameterization of the SBL hyperparameters, we reveal a novel structure that admits a majorization-minimization formulation and naturally extends to unknown noise variance estimation. Sparse recovery results on both synthetic data and extended source direction-of-arrival estimation demonstrate improved accuracy and runtime performance compared to benchmark methods.