Sparse Signal Recovery with Temporally Correlated Source Vectors Using Sparse Bayesian Learning (1102.3949v2)

Abstract: We address the sparse signal recovery problem in the context of multiple measurement vectors (MMV) when elements in each nonzero row of the solution matrix are temporally correlated. Existing algorithms do not consider such temporal correlations and thus their performance degrades significantly with the correlations. In this work, we propose a block sparse Bayesian learning framework which models the temporal correlations. In this framework we derive two sparse Bayesian learning (SBL) algorithms, which have superior recovery performance compared to existing algorithms, especially in the presence of high temporal correlations. Furthermore, our algorithms are better at handling highly underdetermined problems and require less row-sparsity on the solution matrix. We also provide analysis of the global and local minima of their cost function, and show that the SBL cost function has the very desirable property that the global minimum is at the sparsest solution to the MMV problem. Extensive experiments also provide some interesting results that motivate future theoretical research on the MMV model.

• The paper presents a block sparse Bayesian learning (bSBL) framework to recover sparse signals by modeling temporal correlations in multiple measurement vectors.
• It introduces T-SBL and T-MSBL algorithms that achieve superior performance in underdetermined systems with high noise levels.
• Extensive analysis demonstrates that leveraging covariance modeling through a Mahalanobis distance enhances signal sparsity and robustness.

Sparse Signal Recovery with Temporally Correlated Source Vectors Using Sparse Bayesian Learning: An Overview

In the context of signal processing, the task of recovering sparse signals from a set of measurements has drawn significant attention, particularly with the advent of Compressed Sensing (CS). This work by Zhang and Rao introduces a novel approach to tackle sparse signal recovery under the Multiple Measurement Vectors (MMV) model while accounting for temporal correlations within the source vectors.

The Problem and Its Relevance

Sparse signal recovery aims to estimate a sparse source vector from measurements modeled as $\mathbf{y} = \mathbf{\Phi} \mathbf{x} + \mathbf{v}$, where $\mathbf{y}$ is the measurement vector, $\mathbf{\Phi}$ is the dictionary matrix, and $\mathbf{v}$ represents noise. While the Single Measurement Vector (SMV) model has been extensively studied, practical scenarios often involve sequences of measurements, necessitating the MMV model. The MMV model can be described as $\mathbf{Y} = \mathbf{\Phi} \mathbf{X} + \mathbf{V}$, where $\mathbf{Y}$ and $\mathbf{X}$ encapsulate multiple measurement and source vectors, respectively.

The challenge addressed by this paper is the degradation of recovery performance in existing algorithms due to temporal correlations in the source vectors. Such correlations are prevalent in numerous applications like EEG/MEG source localization and DOA estimation, where fixed sparsity patterns only hold for short durations.

Proposed Approach: Block Sparse Bayesian Learning (bSBL)

The core of the proposed method is the block sparse Bayesian learning framework, which transforms the MMV problem into a block sparse model and leverages the Bayesian framework to incorporate temporal correlations.

Key Components and Methodology

1. Model Formulation: The MMV model is converted to a block SMV model given by $\mathbf{y} = \mathbf{D} \mathbf{x} + \mathbf{v}$ where $\mathbf{D} = \mathbf{\Phi} \otimes \mathbf{I}_L$ and $\mathbf{x} = \mathrm{vec}(\mathbf{X}^T)$. This block structure allows the model to capture the temporal correlation via a covariance matrix $\mathbf{B}$.
2. Sparse Bayesian Learning (SBL): Using the Bayesian framework, the algorithm infers the posterior distribution of the source vectors. Two algorithms—T-SBL and T-MSBL—are derived. T-SBL operates in a higher-dimensional space, capturing precise temporal correlations at the cost of computational efficiency. T-MSBL, derived through approximations, balances performance with computational tractability by operating in the original problem space.
3. Learning Rules: Hyperparameters including $\gamma_i$, $\mathbf{B}$, and noise variance $\lambda$ are estimated using Expectation-Maximization (EM) steps. The algorithms incorporate a Mahalanobis distance measure, replacing the standard $\ell_2$ norm to account for correlations.

Analysis and Theoretical Insights

The paper rigorously analyzes the global and local minima of the proposed algorithms' cost functions. Key findings include:

• The uniqueness and sparsity of the global minimum.
• Analytical forms for local minima, underscoring the role of the covariance matrix $\mathbf{B}$ in whitening the sources and aiding sparse recovery.
• Proofs confirm that the algorithms retain desirable properties ensuring robustness and accuracy even in the presence of highly temporally correlated sources.

Experimental Results and Implications

Extensive experiments demonstrate that the proposed algorithms significantly outperform existing methods, particularly in scenarios with high temporal correlations and low SNR environments. Key empirical observations include:

• Superior performance in handling highly underdetermined systems.
• Efficacy in recovering sources with diverse temporal structures.
• Robustness in varying noise levels, where exploiting temporal correlation proves beneficial.

Future Directions

This work opens several avenues for future research, including:

• Exploring different parameterizations and estimates of the covariance matrix $\mathbf{B}$.
• Investigating the interplay between temporal correlations and measurement noise further.
• Extending the framework to more complex signal models and larger datasets.

In conclusion, the introduction of the bSBL framework and its extensions, T-SBL and T-MSBL, marks a significant step in sparse signal recovery, particularly for applications requiring the modeling of temporal correlations. By providing strong theoretical guarantees and demonstrating empirical robustness, this work lays a solid foundation for future advancements in this domain.