Sparse Signal Recovery with Temporally Correlated Source Vectors Using Sparse Bayesian Learning

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.

💡 Research Summary

This paper tackles the sparse signal recovery problem in the multiple‑measurement‑vector (MMV) setting when the entries of each non‑zero row of the solution matrix exhibit strong temporal correlation. Conventional MMV algorithms (greedy, ℓ₂,₁ minimization, re‑weighted, Bayesian) typically assume that the source vectors are independent and identically distributed (i.i.d.) across time, an assumption that is violated in many practical scenarios such as EEG/MEG source localization or direction‑of‑arrival (DOA) tracking, where the underlying sources evolve smoothly and adjacent samples are highly correlated.

The authors introduce a block Sparse Bayesian Learning (bSBL) framework that converts the MMV model
(Y = \Phi X + V) (with (Y\in\mathbb{R}^{N\times L}), (\Phi\in\mathbb{R}^{N\times M})) into a block‑SMV model
(y = D x + v) where (y = \operatorname{vec}(Y^{!T})), (D = \Phi\otimes I_L), and (x = \operatorname{vec}(X^{!T})). In this representation each row (X_{i\cdot}) becomes a block (x_i\in\mathbb{R}^L).

A Gaussian prior is placed on each block:
(p(x_i;\gamma_i,B) = \mathcal{N}(0,\gamma_i B)),
where (\gamma_i\ge 0) controls row sparsity (as in standard SBL) and (B\in\mathbb{R}^{L\times L}) is a positive‑definite matrix that captures the temporal covariance of the source. To avoid over‑parameterization, a single shared matrix B is used for all rows, so the overall prior covariance is (\Sigma_0 = \Gamma\otimes B) with (\Gamma = \operatorname{diag}(\gamma_1,\dots,\gamma_M)).

Using a Gaussian likelihood with noise variance (\lambda), the marginal likelihood (p(y;\Theta)) (where (\Theta={\gamma,B,\lambda})) is maximized via Type‑II maximum likelihood (evidence maximization). An Expectation‑Maximization (EM) scheme is derived: the E‑step computes the posterior mean (\mu_x) and covariance (\Sigma_x) of (x); the M‑step updates the hyper‑parameters. Closed‑form updates are obtained:

• (\gamma_i \leftarrow \frac{1}{L}\operatorname{Tr}!\big(B^{-1}(\Sigma_x^{(i)}+\mu_x^{(i)}\mu_x^{(i)T})\big))
• (B \leftarrow \frac{1}{M}\sum_{i=1}^M \frac{1}{\gamma_i}\big(\Sigma_x^{(i)}+\mu_x^{(i)}\mu_x^{(i)T}\big))
• (\lambda) is updated by the average residual energy.

Two concrete algorithms are derived:

1. T‑SBL (Temporal‑SBL) implements the EM updates directly on the high‑dimensional block‑SMV model. It achieves the best possible reconstruction accuracy but incurs a computational cost of (O(NL M^2)) due to the large Kronecker‑structured dictionary.
2. T‑MSBL (Temporal‑MSBL) rewrites the updates in the original MMV parameter space, exploiting the identity ((\lambda\Gamma^{-1}\otimes B^{-1}+D^TD)^{-1}D^T = \Gamma D^T(\lambda I + D\Gamma D^T)^{-1}). This reduces the per‑iteration complexity to (O(NML)) while preserving the performance of T‑SBL. The only difference from the classic MSBL algorithm is the replacement of the Euclidean row norm (|X_{i\cdot}|2^2) by the Mahalanobis distance (X{i\cdot} B^{-1} X_{i\cdot}^T).

Theoretical contributions include:

• Global optimality: Theorem 1 proves that, in the noiseless case, the global minimizer of the cost function (\mathcal{L}(\Theta)=y^T\Sigma_y^{-1}y+\log|\Sigma_y|) coincides with the sparsest solution of the MMV problem. This property is inherited from the original SBL framework and is retained despite the introduction of the temporal covariance matrix B.
• Local minima and pruning: When a hyper‑parameter (\gamma_i) falls below a small threshold, the corresponding block is pruned (set to zero). The analysis shows that such pruning is a natural consequence of the EM updates and does not trap the algorithm in sub‑optimal dense solutions.

Experimental validation spans synthetic data, simulated EEG/MEG, and DOA scenarios. Key findings:

• With increasing temporal correlation (ρ ≥ 0.9), T‑MSBL’s reconstruction error remains low, whereas MSBL, M‑FOCUSS, and ℓ₂,₁ minimization deteriorate sharply.
• In highly under‑determined settings (e.g., N = 30, M = 200, K = 3 non‑zero rows), T‑MSBL successfully recovers the correct support with far fewer measurements than competing methods.
• Real EEG data (10 ms windows, L ≈ 5) show a 15 % improvement in source localization accuracy over MSBL.
• Runtime comparisons indicate that T‑MSBL is roughly twice as fast as MSBL in MATLAB while achieving virtually identical NMSE performance.

Conclusions: The paper presents a principled Bayesian framework that explicitly models temporal correlation in MMV problems, derives efficient algorithms with strong theoretical guarantees, and demonstrates superior empirical performance. Future work suggested includes learning row‑specific covariance matrices (B_i) without over‑fitting, extending the model to non‑Gaussian priors (e.g., heavy‑tailed or spike‑and‑slab), and developing real‑time hardware implementations.