Compressed Sensing of Block-Sparse Signals: Uncertainty Relations and Efficient Recovery (0906.3173v2)

Abstract: We consider compressed sensing of block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block $k$-sparse signals in no more than $k$ steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed $\ell_2/\ell_1$-optimization approach. This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.

• The paper extends uncertainty relations by introducing block-coherence metrics to set recovery limits for block-sparse signals.
• The paper proposes Block Orthogonal Matching Pursuit and mixed ℓ2/ℓ1 Optimization algorithms for efficient signal recovery.
• The findings demonstrate that exploiting block structure improves recovery thresholds and guides the design of sensing matrices in compressed sensing.

Compressed Sensing of Block-Sparse Signals: Uncertainty Relations and Efficient Recovery

The paper "Compressed Sensing of Block-Sparse Signals: Uncertainty Relations and Efficient Recovery," authored by Yonina C. Eldar, Patrick Kuppinger, and Helmut Bölcskei, explores the theoretical and algorithmic advancements in the compressed sensing (CS) of block-sparse signals. Block-sparse signals are characterized by having their nonzero coefficients clustered in contiguous blocks, as opposed to being arbitrarily distributed.

Key Contributions

Uncertainty Relations

The authors extend existing uncertainty relations from conventional sparse signals to block-sparse signals. By introducing the concept of block-coherence, defined as the maximum block-wise spectral norm of the Gram matrix, the authors establish a generalized uncertainty relation: $$\frac{1}{2}(A+B) \geq \sqrt{AB} \geq \frac{1}{d(,)}$$ where $(,)$ represents the block-coherence between two orthonormal bases. This relation indicates the minimum combined sparsity across two bases in which a signal can be represented, showing how block-coherence naturally occurs in these contexts.

Efficient Recovery Algorithms

1. Block Orthogonal Matching Pursuit (BOMP): The paper proposes an extension of the OMP algorithm for block-sparse signals. By selecting entire blocks of coefficients at each iteration, BOMP can efficiently recover block-sparse signals under suitable conditions related to block-coherence.
2. Mixed $\ell_2/\ell_1$-Optimization (L-OPT): The authors extend the known $\ell_1$-minimization method to handle block-sparse signals by introducing an optimization problem that incorporates the block structure:

$$\min_{} \sum_{\ell=1}^M \|[\ell]\|_2 \quad \text{s.t.} \ = {\bf x}$$

where ${\bf D}$ is the sensing matrix and ${\bf y}$ are the measurements. The research demonstrates that under small block-restricted isometry constants or suitably bounded block-coherence, L-OPT guarantees recovery of block-sparse signals.

Theoretical Findings

The findings assert that explicitly leveraging block-sparsity can significantly improve the performance and recovery thresholds of CS algorithms. For instance, the recovery conditions provided by block-coherence are less stringent compared to conditions based on conventional coherence, enabling the recovery of signals with higher sparsity levels. Specifically, the paper shows that for matrices of block size $d$, if the block coherence $\mu_b$ and sub-coherence $\nu$ meet certain bounds, recovery algorithms can effectively reconstruct the original signal.

Practical and Theoretical Implications

1. Dictionary and Algorithm Design: The derived conditions and algorithms highlight the importance of considering the intrinsic structure of signals in designing both the measurement matrices and the recovery algorithms. This perspective can be particularly beneficial in applications like multi-band signal processing and gene expression data analysis.
2. Future Directions: While this paper addresses theoretical constructs and algorithmic efficiency, future research can delve into practical applications and empirical validations across various domains. Another potential area of development could be the exploration of different types of structured sparsity, such as tree-based or graph-based sparsity patterns.

Conclusion

The paper demonstrates significant advancements in the understanding and application of compressed sensing for block-sparse signals. By extending key concepts like uncertainty relations and introducing structured sparsity into recovery algorithms, the authors present a compelling case for the benefits of exploiting block structures. The implications of this research are profound, with potential improvements in both the hardware design of sensing matrices and the software implementation of recovery algorithms, further enhancing the efficiency and applicability of compressed sensing in real-world scenarios.