Extension of SBL Algorithms for the Recovery of Block Sparse Signals with Intra-Block Correlation (1201.0862v5)

Abstract: We examine the recovery of block sparse signals and extend the framework in two important directions; one by exploiting signals' intra-block correlation and the other by generalizing signals' block structure. We propose two families of algorithms based on the framework of block sparse Bayesian learning (BSBL). One family, directly derived from the BSBL framework, requires knowledge of the block structure. Another family, derived from an expanded BSBL framework, is based on a weaker assumption on the block structure, and can be used when the block structure is completely unknown. Using these algorithms we show that exploiting intra-block correlation is very helpful in improving recovery performance. These algorithms also shed light on how to modify existing algorithms or design new ones to exploit such correlation and improve performance.

• The paper introduces an extension of the sparse Bayesian learning framework that exploits intra-block correlation to improve block sparse signal recovery.
• It proposes two algorithm families—one for known block partitions and one for unknown—demonstrating robust performance across various conditions.
• Empirical evaluations reveal enhanced phase transitions, recovery accuracy, and computational efficiency, rivaling near-oracle performance in noisy settings.

Overview of Block Sparse Signal Recovery Using Bayesian Approaches

This paper presents an advancement in the recovery of block sparse signals by extending the sparse Bayesian learning (SBL) framework to include intra-block correlation exploitation. The authors propose two families of algorithms: one that requires knowledge of the block structure and another that operates under a weaker assumption without prior knowledge of the block partition.

Block Sparse Bayesian Learning (BSBL) Framework

The authors build upon the traditional SBL framework to address block sparse signals, which are characterized by nonzero elements grouped into blocks. They incorporate intra-block correlation, a feature often overlooked in existing algorithms. The BSBL framework models blocks as multivariate Gaussian distributions, allowing the algorithm to learn the correlation structure within each block and improve the recovery process.

Algorithm Families

1. BSBL with Known Block Partition: The authors introduce three algorithms for scenarios where the block partition is known: BSBL-EM, BSBL-BO, and BSBL-$\ell_1$.

• BSBL-EM: Utilizes the expectation-maximization approach for parameter estimation, offering robust recovery in noisy environments.
• BSBL-BO: Employs the bound-optimization method to enhance convergence speed, balancing performance with computational efficiency.
• BSBL-$\ell_1$: Integrates BSBL with Group Lasso techniques, promising rapid processing for large-scale data through iterative reweighting.

1. EBSBL with Unknown Block Partition: The authors develop EBSBL-EM, EBSBL-BO, and EBSBL-$\ell_1$, effectively extending the BSBL framework to adapt even when block partitions are unspecified. They use an expanded model creatively to accommodate overlapping blocks, ensuring accurate block sparse signal recovery without predefined partition knowledge.

Empirical Evaluation

The paper presents extensive experiments to demonstrate the efficacy of their proposed algorithms. Key findings include:

• Phase Transition Analysis: Identifies scenarios where BSBL algorithms achieve superior recovery rates compared to existing methods, particularly when exploiting strong intra-block correlations. Notably, the algorithms demonstrate impressive capability to recover signals nearly equal to the number of measurements.
• Intra-Block Correlation Benefit: Establishes that the newly integrated correlation exploitation significantly enhances performance, particularly as correlation strength increases.
• Robustness in Noisy Environments: The algorithms show notable resilience and accuracy across a range of signal-to-noise ratios, with performance closely matching the 'oracle' solution.
• Success Without Known Block Partitions: Even without block partition knowledge, the authors’ algorithms outperform contemporary approaches, highlighting the practicality and robustness of their method.

Implications and Future Directions

The introduction of intra-block correlation in sparse signal models opens new avenues for improving compressed sensing frameworks, especially in fields dealing with physiological and image data where such correlations are prevalent. The adaptability in the block partition knowledge extends application possibilities and broadens the scope of these algorithms.

Future research could focus on further optimizing the computational efficiency of these methods and exploring their application across different domains where structured sparse signals occur. Additionally, the iterative reweighted framework offers potential for advancements in Group-Lasso type algorithms, suggesting a fertile ground for continued exploration in signal processing.