On the reconstruction of block-sparse signals with an optimal number of measurements (0804.0041v1)

Abstract: Let A be an M by N matrix (M < N) which is an instance of a real random Gaussian ensemble. In compressed sensing we are interested in finding the sparsest solution to the system of equations A x = y for a given y. In general, whenever the sparsity of x is smaller than half the dimension of y then with overwhelming probability over A the sparsest solution is unique and can be found by an exhaustive search over x with an exponential time complexity for any y. The recent work of Cand\'es, Donoho, and Tao shows that minimization of the L_1 norm of x subject to A x = y results in the sparsest solution provided the sparsity of x, say K, is smaller than a certain threshold for a given number of measurements. Specifically, if the dimension of y approaches the dimension of x, the sparsity of x should be K < 0.239 N. Here, we consider the case where x is d-block sparse, i.e., x consists of n = N / d blocks where each block is either a zero vector or a nonzero vector. Instead of L_1-norm relaxation, we consider the following relaxation min x | X_1 |2 + | X_2 |_2 + ... + | X_n |_2, subject to A x = y where X_i = (x{(i-1)d+1}, x_{(i-1)d+2}, ..., x_{i d}) for i = 1,2, ..., N. Our main result is that as n -> \infty, the minimization finds the sparsest solution to Ax = y, with overwhelming probability in A, for any x whose block sparsity is k/n < 1/2 - O(\epsilon), provided M/N > 1 - 1/d, and d = \Omega(\log(1/\epsilon)/\epsilon). The relaxation can be solved in polynomial time using semi-definite programming.

• The paper introduces a convex relaxation using a mixed l2/l1 norm to efficiently recover block-sparse signals.
• It employs a novel null-space characterization and probabilistic arguments that bypass traditional restricted isometry requirements.
• Numerical simulations confirm enhanced recoverable sparsity thresholds with increasing block sizes, validating its practical application.

Analysis of Block-Sparse Signal Reconstruction with Minimal Measurements

The paper "On the reconstruction of block-sparse signals with an optimal number of measurements" by Mihailo Stojnic, Farzad Parvaresh, and Babak Hassibi explores advancements in the field of compressed sensing, specifically focusing on the efficient recovery of block-sparse signals using random real Gaussian matrices. This work extends the methods of compressed sensing by addressing the unique challenges posed by block sparsity.

Background and Problem Statement

Compressed sensing involves recovering sparse signals from an under-determined set of linear measurements. The authors focus on signals that are not merely sparse but exhibit block sparsity—meaning non-zero elements are grouped into blocks, a trait found in various practical applications such as DNA microarrays and sparse communication channels. The primary question addressed is whether such signals can be efficiently reconstructed using a small number of measurements.

Prior Work

Previous studies demonstrated that $\ell_1$ norm minimization can yield the sparsest solution if signal sparsity remains below a certain threshold. However, those approaches did not adequately exploit block structures within the signals. The authors reference seminal work by Candes and Tao, which set foundational results for sparse recovery via $\ell_1$ minimization but indicate a gap for block-sparse cases.

Main Contributions

The authors introduce a convex relaxation tailored to block-sparse signals, formulated as an optimization problem using a mixed $\ell_2/\ell_1$ type norm. The principal result indicates that for large values of block size $d$, as the measurement-to-dimension ratio approaches one, the method can successfully recover block-sparse signals with a sparsity threshold approaching half the number of measurements. The relaxation can be solved using semi-definite programming, offering a polynomial time complexity.

Methodology

The solution approach bypasses traditional prerequisites like the restricted isometry property, instead leveraging a novel null-space characterization technique and probabilistic arguments. The key innovation lies in exploiting the structure of block-sparse signals directly within optimization frameworks, pushing beyond prior $\ell_1$ methods.

Numerical Results

Simulations demonstrate that the proposed method significantly enhances the recoverable sparsity thresholds when compared to traditional $\ell_1$ methods, especially with increasing block length. The empirical results conform closely to theoretical expectations, supporting the robustness of this approach in practical settings.

Discussion

The implications for compressed sensing are substantial, especially in domains where block sparsity is inherent. This paradigm shift from treating sparse signals with isolated non-zero entries to recognizing structured sparsity aligns with real-world signals, potentially enhancing applications in communications and bioinformatics.

Future Directions

One avenue for further exploration is refining the probabilistic bounds to obtain tighter thresholds. Additionally, adapting the algorithm for different random matrix ensembles beyond Gaussian might widen its applicability. Investigating the method's resilience to noise and other practical imperfections could also yield significant insights.

Conclusion

This work provides a compelling methodology for block-sparse signal reconstruction, significantly enhancing the efficiency and scalability of compressed sensing techniques. By addressing block structures, the paper contributes a meaningful advancement in the theoretical and practical aspects of signal processing.