Exact Joint Sparse Frequency Recovery via Optimization Methods (1405.6585v2)

Abstract: Frequency recovery/estimation from discrete samples of superimposed sinusoidal signals is a classic yet important problem in statistical signal processing. Its research has recently been advanced by atomic norm techniques which exploit signal sparsity, work directly on continuous frequencies, and completely resolve the grid mismatch problem of previous compressed sensing methods. In this work we investigate the frequency recovery problem in the presence of multiple measurement vectors (MMVs) which share the same frequency components, termed as joint sparse frequency recovery and arising naturally from array processing applications. To study the advantage of MMVs, we first propose an $\ell_{2,0}$ norm like approach by exploiting joint sparsity and show that the number of recoverable frequencies can be increased except in a trivial case. While the resulting optimization problem is shown to be rank minimization that cannot be practically solved, we then propose an MMV atomic norm approach that is a convex relaxation and can be viewed as a continuous counterpart of the $\ell_{2,1}$ norm method. We show that this MMV atomic norm approach can be solved by semidefinite programming. We also provide theoretical results showing that the frequencies can be exactly recovered under appropriate conditions. The above results either extend the MMV compressed sensing results from the discrete to the continuous setting or extend the recent super-resolution and continuous compressed sensing framework from the single to the multiple measurement vectors case. Extensive simulation results are provided to validate our theoretical findings and they also imply that the proposed MMV atomic norm approach can improve the performance in terms of reduced number of required measurements and/or relaxed frequency separation condition.

• The paper proposes advanced optimization methods, particularly an MMV atomic norm approach, for exact joint sparse frequency recovery by leveraging joint sparsity inherent in multiple measurement vectors.
• It introduces a convex relaxation technique via the atomic norm, solvable by semidefinite programming (SDP), as a practical method extending super-resolution to multiple measurement vectors.
• This approach provides provable guarantees for exact frequency recovery, enhances signal recovery in applications like DOA estimation, and suggests future research directions in array design.

Exact Joint Sparse Frequency Recovery via Optimization Methods

This paper addresses the significant problem of frequency recovery and estimation from discrete samples of superimposed sinusoidal signals, with a particular focus on scenarios involving Multiple Measurement Vectors (MMVs). The authors propose advanced optimization methods to enhance the joint sparse frequency recovery (JSFR), extending traditional techniques by leveraging the joint sparsity inherent in MMVs. This approach offers a distinct alternative to previously dominant compressed sensing methods.

Key Contributions and Methodology

The paper introduces an innovative MMV atomic norm approach that operates as a continuous counterpart to the widely known discrete $\ell_{2,1}$ norm method. This atomic norm approach is significant because it allows frequency components shared across multiple measurement vectors to be exactly recovered under certain conditions. It circumvents the grid mismatch issues typical of compressed sensing by working directly with continuous frequencies, utilizing the semidefinite programming (SDP) frameworks for computational tractability.

Essentially, the authors propose two main approaches:

1. MMV Atomic $\ell_0$ Norm Approach: This method theoretically shows that MMVs can improve the frequency recovery performance by increasing the number of recoverable frequencies. However, it culminates in a rank minimization problem that is NP-hard to solve, making practical implementation impractical.
2. Convex Relaxation via MMV Atomic Norm: Offered as a practical solution, this method applies a convex relaxation technique using the atomic norm, which can be efficiently solved by SDP. This approach successfully extends the recent advances in super-resolution and continuous compressed sensing from single to multiple measurement vectors.

Theoretical and Practical Implications

The theoretical underpinnings of this work are robust, providing provable guarantees for the exact recovery of frequencies under appropriate conditions. Notably, the conditions involve frequency separation and the number of available snapshots, akin to those established by prior single measurement vector studies. The MMV framework substantially enhances the efficacy of signal recovery by exploiting the peri-source longitudinal correlation, which is especially beneficial in array processing applications.

This approach has profound implications for direction-of-arrival (DOA) estimation and various other signal processing scenarios where sensor array measurements are constrained due to physical or economic limitations. The proposed methods mitigate the traditional restricted focus on uncorrelated sources, thus broadening the potential application fields.

Future Prospects

Looking forward, this paper opens a path for new research directions in AI and signal processing, especially in designing sensor arrays where the geometry—represented by sampling index sets—could be optimized to maximize the effectiveness of JSFR. Moreover, further exploration into average case analysis could uncover nuanced benefits of utilizing MMV beyond the worst-case scenarios analyzed herein.

The paper also suggests potential advancements in algorithmic design that could extend the practical applicability of these methods to more consistent and scalable frameworks for real-world applications, including robust noise resilience in practice.

In summary, this work contributes significantly to the field of signal processing by advancing our understanding and capabilities in exact joint sparse frequency recovery, leveraging both theoretical innovation and practical optimization methods.