[Paper Review] Iterative Reweighted Algorithms for Sparse Signal Recovery with Temporally Correlated Source Vectors

Zhilin Zhang, Bhaskar D. Rao|arXiv (Cornell University)|Apr 28, 2011

Sparse and Compressive Sensing Techniques7 references4 citations

TL;DR

This paper proposes an iterative reweighted sparse Bayesian learning (SBL) algorithm that exploits temporal correlation in source vectors for improved multiple measurement vector (MMV) recovery. By replacing row norms with Mahalanobis distance in reweighted ℓ₂ algorithms, it achieves superior performance over existing methods, especially under high temporal correlation, with demonstrated gains in recovery accuracy and robustness to noise.

ABSTRACT

Iterative reweighted algorithms, as a class of algorithms for sparse signal recovery, have been found to have better performance than their non-reweighted counterparts. However, for solving the problem of multiple measurement vectors (MMVs), all the existing reweighted algorithms do not account for temporal correlation among source vectors and thus their performance degrades significantly in the presence of correlation. In this work we propose an iterative reweighted sparse Bayesian learning (SBL) algorithm exploiting the temporal correlation, and motivated by it, we propose a strategy to improve existing reweighted $\ell_2$ algorithms for the MMV problem, i.e. replacing their row norms with Mahalanobis distance measure. Simulations show that the proposed reweighted SBL algorithm has superior performance, and the proposed improvement strategy is effective for existing reweighted $\ell_2$ algorithms.

Motivation & Objective

• To address the degradation in sparse signal recovery performance when source vectors exhibit temporal correlation, a common issue in MMV problems.
• To develop a new iterative reweighted SBL algorithm that explicitly models temporal correlation in source vectors.
• To derive a general strategy for enhancing existing reweighted ℓ₂ algorithms by incorporating temporal correlation via Mahalanobis distance.
• To validate the effectiveness of the proposed algorithms through simulations under varying correlation levels and system conditions.

Proposed method

• Derives a new cost function in the block sparse Bayesian learning (bSBL) framework that incorporates temporal correlation through a row-wise correlation matrix Bi.
• Proposes an iterative reweighted ℓ₂ SBL algorithm (ReSBL-QM) that uses Mahalanobis distance to weight rows based on their temporal correlation structure.
• Introduces a simplified variant (ReSBL-L2) that replaces row norms with Mahalanobis distance in the reweighting scheme.
• Develops a general strategy to improve existing reweighted ℓ₂ algorithms by replacing row norms with Mahalanobis distance, using the correlation matrix Bi estimated from previous iterates.
• Employs a diagonal weighting matrix W(k) with weights derived from the Mahalanobis distance of each source row, updated iteratively.
• Uses an ε-decreasing strategy for numerical stability in the reweighting rule, ensuring convergence and robustness.

Experimental results

Research questions

• RQ1 How does temporal correlation among source vectors affect the performance of existing reweighted MMV algorithms?
• RQ2 Can a reweighted SBL algorithm be designed to explicitly exploit temporal correlation in source vectors for improved recovery?
• RQ3 Can the proposed reweighting strategy based on Mahalanobis distance be generalized to enhance existing reweighted ℓ₂ algorithms?
• RQ4 How does the performance of the proposed algorithms compare to state-of-the-art methods under varying levels of temporal correlation and system underdetermination?

Key findings

• The proposed ReSBL-QM algorithm achieves superior recovery performance compared to all baseline methods, especially under high temporal correlation.
• When temporal correlation is high (βi ∈ [0.5,1)), standard reweighted ℓ₂ and SBL algorithms suffer significant performance degradation, while the proposed methods maintain high accuracy.
• The modified algorithms tMFOCUSS and tIter-L2 outperform their original counterparts (M-FOCUSS and Iter-L2) by incorporating temporal correlation via Mahalanobis distance.
• The proposed strategy of replacing row norms with Mahalanobis distance is effective in improving existing reweighted ℓ₂ algorithms, demonstrating broad applicability.
• The ReSBL-QM algorithm is particularly effective when source row norms are similar, showing improved robustness in such scenarios.