Rank Awareness in Joint Sparse Recovery (1004.4529v1)

Abstract: In this paper we revisit the sparse multiple measurement vector (MMV) problem where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem has received increasing interest as an extension of the single channel sparse recovery problem which lies at the heart of the emerging field of compressed sensing. However the sparse approximation problem has origins which include links to the field of array signal processing where we find the inspiration for a new family of MMV algorithms based on the MUSIC algorithm. We highlight the role of the rank of the coefficient matrix X in determining the difficulty of the recovery problem. We derive the necessary and sufficient conditions for the uniqueness of the sparse MMV solution, which indicates that the larger the rank of X the less sparse X needs to be to ensure uniqueness. We also show that the larger the rank of X the less the computational effort required to solve the MMV problem through a combinatorial search. In the second part of the paper we consider practical suboptimal algorithms for solving the sparse MMV problem. We examine the rank awareness of popular algorithms such as SOMP and mixed norm minimization techniques and show them to be rank blind in terms of worst case analysis. We then consider a family of greedy algorithms that are rank aware. The simplest such algorithm is a discrete version of MUSIC and is guaranteed to recover the sparse vectors in the full rank MMV case under mild conditions. We extend this idea to develop a rank aware pursuit algorithm that naturally reduces to Order Recursive Matching Pursuit (ORMP) in the single measurement case and also provides guaranteed recovery in the full rank multi-measurement case. Numerical simulations demonstrate that the rank aware algorithms are significantly better than existing algorithms in dealing with multiple measurements.

• The paper establishes that exploiting matrix rank can ensure unique recovery in the Multiple Measurement Vector (MMV) problem, requiring less sparsity for higher rank signals.
• The paper introduces rank-aware greedy algorithms, inspired by MUSIC, which achieve exact recovery for full-rank signals with fewer measurements and reduced computational cost.
• This research shows how rank awareness enhances recovery performance, reduces measurement requirements, and opens new avenues by connecting sparse recovery with matrix rank properties.

Rank Awareness in Joint Sparse Recovery: An Academic Perspective

The paper "Rank Awareness in Joint Sparse Recovery" by Mike E. Davies and Yonina C. Eldar revisits the sparse multiple measurement vector (MMV) problem, a cornerstone issue in compressed sensing with roots in array signal processing. The authors explore the role that the rank of an unknown signal matrix, denoted as $X$, plays in the recovery problem, proposing novel algorithmic strategies inspired by the MUSIC method traditionally used in array processing.

Problem Formulation

The MMV problem extends the single measurement vector (SMV) framework, aiming to recover a set of jointly sparse multichannel signals from incomplete measurements. Fundamentally, the challenge lies in reconstructing a sparse signal matrix $X$ from observed data $Y = \Phi X$, where $\Phi$ is the sensing matrix with $m < n$, implying undersampling. The intricacy of this problem is intertwined with ensuring uniqueness of the solution $X$, traditionally known to be NP-hard.

Theoretical Insights

The authors derive necessary and sufficient conditions for the uniqueness of the sparse MMV solution, establishing that the larger the rank of $X$, the less sparse it needs to be to ensure unique recovery. This revelation is pivotal; it implies that rank, a structural property of the matrix $X$, can be strategically exploited to simplify the recovery problem. The results extend to show that higher rank matrices reduce the computational complexity required for solving MMV through combinatorial approaches, such as exhaustive search.

Algorithmic Development

In practical scenarios, suboptimal algorithms like Simultaneous Orthogonal Matching Pursuit (SOMP) and mixed norm minimization are employed. However, the paper critiques these approaches as being rank blind in worst-case analyses, which means they do not benefit from the increased rank of $X$. The authors introduce rank-aware greedy algorithms, extending the MUSIC algorithm to a discrete setup. This paradigm shift guarantees recovery of sparse vectors in the full rank MMV setting under less stringent conditions compared to traditional methods. The developed algorithms achieve exact recovery with both minimal measurements and reduced computational efforts, as demonstrated by robust numerical simulations.

Implications and Future Work

The implications of this research are multifaceted. Practically, exploiting rank information allows for enhanced recovery performance and reduced measurement requirements. Theoretically, the paper interconnects sparse recovery with linear algebraic concepts like matrix rank, opening avenues for further exploration in rank-structured recovery problems. Given the prevalence of compressed sensing applications, such adjustments can positively impact fields like signal processing, medical imaging, and remote sensing, where efficient data acquisition is paramount.

Looking forward, it is worth exploring the integration of these rank-aware strategies with other modern compressed sensing techniques, such as model-based compression or blind compressed sensing. Additionally, extending these results to accommodate noise and outliers could significantly enhance the practical applicability of the proposed methods, enabling more robust and reliable recovery in real-world scenarios.

Conclusion

This paper provides a nuanced view of the MMV problem through the lens of rank, presenting theoretically grounded and computationally viable solutions that improve upon existing methodologies. While the paper draws from well-established theories in compressed sensing and array signal processing, it paves the way for further innovations by showcasing the untapped potential that lies in understanding and leveraging structural properties like matrix rank.