Pattern-Coupled Sparse Bayesian Learning for Recovery of Block-Sparse Signals (1311.2150v1)

Abstract: We consider the problem of recovering block-sparse signals whose structures are unknown \emph{a priori}. Block-sparse signals with nonzero coefficients occurring in clusters arise naturally in many practical scenarios. However, the knowledge of the block structure is usually unavailable in practice. In this paper, we develop a new sparse Bayesian learning method for recovery of block-sparse signals with unknown cluster patterns. Specifically, a pattern-coupled hierarchical Gaussian prior model is introduced to characterize the statistical dependencies among coefficients, in which a set of hyperparameters are employed to control the sparsity of signal coefficients. Unlike the conventional sparse Bayesian learning framework in which each individual hyperparameter is associated independently with each coefficient, in this paper, the prior for each coefficient not only involves its own hyperparameter, but also the hyperparameters of its immediate neighbors. In doing this way, the sparsity patterns of neighboring coefficients are related to each other and the hierarchical model has the potential to encourage structured-sparse solutions. The hyperparameters, along with the sparse signal, are learned by maximizing their posterior probability via an expectation-maximization (EM) algorithm. Numerical results show that the proposed algorithm presents uniform superiority over other existing methods in a series of experiments.

• The paper proposes a Pattern-Coupled Sparse Bayesian Learning framework to recover block-sparse signals even when the block structures are not known beforehand.
• It introduces a hierarchical Gaussian prior that couples neighboring coefficients' hyperparameters and uses an EM algorithm to iteratively estimate hyperparameters and the sparse signal.
• Numerical evaluations demonstrate that the proposed method outperforms conventional sparse recovery techniques, showing improved accuracy and robustness to noise.

Pattern-Coupled Sparse Bayesian Learning for Recovery of Block-Sparse Signals

In the paper "Pattern-Coupled Sparse Bayesian Learning for Recovery of Block-Sparse Signals," the authors propose a novel framework aimed at enhancing the recovery of block-sparse signals, specifically when the block structures are not known a priori. Such situations are often encountered in compressive sensing scenarios, where signals exhibit natural clusters of non-zero coefficients that cannot be easily predefined.

The authors have developed a tailored sparse Bayesian learning algorithm that introduces a hierarchical Gaussian prior model to adeptly capture the statistical dependencies among adjacent coefficients. This is achieved by associating each coefficient not only with its own hyperparameter but also involving the hyperparameters of its neighboring coefficients. This coupling mechanism is designed to incentivize structured sparsity in the recovered signal, suppressing isolated non-zero coefficients that are inconsistent with the overall block pattern.

Key to the approach is the expectation-maximization (EM) algorithm employed to iteratively estimate the hyperparameters and sparse signal simultaneously by maximizing their posterior probability. In this context, the EM algorithm efficiently navigates the challenging landscape of sparse reconstruction by refining hyperparameter estimates, which govern the underlying sparsity patterns.

The paper provides comprehensive numerical evaluations, establishing the proposed method's superiority over existing sparse recovery techniques, such as conventional sparse Bayesian learning and state-of-the-art algorithms tailored for block-sparse signals. In particular, it is noted that the proposed method consistently outperformed other approaches across an array of simulated experiments which involved varying levels of block sparsity and noise conditions. The results illustrate not only improved accuracy in recovering the underlying sparse signals but also enhanced robustness to noise—transcending mere performance metrics to underline the practical utility of their methodology.

The implications of this research are significant for both theoretical exploration and practical applications in areas requiring sparse representations, such as signal processing and image reconstruction. The authors offer a pathway toward generalized sparse recovery techniques that do not necessarily require prior knowledge of signal structure, thus broadening the scope for adaptive and real-time applications. Moreover, the theoretical development of coupled hierarchical models sheds light on potentially deeper insights into modeling dependencies in sparse contexts, a key consideration that could inform future enhancements in the field of compressive sensing and beyond.

Future work could explore optimizing computational efficiency further or extending the current hierarchical model to more complex interdependencies beyond immediate neighbors, which could facilitate even broader applications. Additionally, assessing the impact of different sparse priors and their parameterization on recovery quality could offer further depth and adaptiveness to the framework. Overall, this paper marks an important contribution to the evolution of sparse signal processing and Bayesian methodologies.