Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Yanbin He, Geethu Joseph

TL;DR

This work tackles block-sparse recovery when block boundaries are unknown and isolated nonzeros may occur. It embeds a DoL-TV regularizer within sparse Bayesian learning and solves the resulting nonconvex objective via a novel majorization-minimization framework that uses an exponential reparameterization $\bm\gamma = e^{\bm z}$, $\lambda = e^{\beta}$ to reveal convex structure, with a majorized surrogate for a nonconvex trace term. An ADMM-based inner loop optimizes convex subproblems within each MM iteration, enabling efficient updates and joint estimation of the noise variance. Numerical results on synthetic data and extended-source DOA estimation show improved recovery accuracy and faster runtimes compared with state-of-the-art methods, highlighting robustness to unknown block boundaries and isolated entries. The approach enhances practical block-sparse recovery in real-world sensing problems where noise characteristics are not known a priori and block structures are not neatly aligned.

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.

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

TL;DR

This work tackles block-sparse recovery when block boundaries are unknown and isolated nonzeros may occur. It embeds a DoL-TV regularizer within sparse Bayesian learning and solves the resulting nonconvex objective via a novel majorization-minimization framework that uses an exponential reparameterization , to reveal convex structure, with a majorized surrogate for a nonconvex trace term. An ADMM-based inner loop optimizes convex subproblems within each MM iteration, enabling efficient updates and joint estimation of the noise variance. Numerical results on synthetic data and extended-source DOA estimation show improved recovery accuracy and faster runtimes compared with state-of-the-art methods, highlighting robustness to unknown block boundaries and isolated entries. The approach enhances practical block-sparse recovery in real-world sensing problems where noise characteristics are not known a priori and block structures are not neatly aligned.

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.
Paper Structure (13 sections, 2 theorems, 19 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 2 theorems, 19 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. DoL-TV SBL Cost
  3. An Iterative Algorithm for DoL-TV SBL
  4. An MM Procedure
  5. ADMM Procedure
  6. Step 1
  7. Step 2
  8. Step 3
  9. Numerical Evaluation
  10. Synthetic Signals
  11. Extended Sources DOA Estimation
  12. Conclusion
  13. Proof of Proposition \ref{['prop.log_cvx']}

Key Result

Proposition 1

Function $\log |e^{\beta} \bm I + \bm H \mathop{\mathrm{diag}}\nolimits(e^{\bm z}) \bm H^\mathsf{H}|$ is convex in $(\bm z,\beta)\in \mathbb{R}^{N+1}$.

Figures (2)

  • Figure 1: NSE and F1 score as functions of SNR.
  • Figure 2: Power of reconstructed sparse signals across different angles. Shaded area: location of extended source.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Lemma 1: Complex extension of Lemma 4 in wiesel2012geodesic
  • proof