Subspace Methods for Joint Sparse Recovery (1004.3071v4)

Abstract: We propose robust and efficient algorithms for the joint sparse recovery problem in compressed sensing, which simultaneously recover the supports of jointly sparse signals from their multiple measurement vectors obtained through a common sensing matrix. In a favorable situation, the unknown matrix, which consists of the jointly sparse signals, has linearly independent nonzero rows. In this case, the MUSIC (MUltiple SIgnal Classification) algorithm, originally proposed by Schmidt for the direction of arrival problem in sensor array processing and later proposed and analyzed for joint sparse recovery by Feng and Bresler, provides a guarantee with the minimum number of measurements. We focus instead on the unfavorable but practically significant case of rank-defect or ill-conditioning. This situation arises with limited number of measurement vectors, or with highly correlated signal components. In this case MUSIC fails, and in practice none of the existing methods can consistently approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC), which improves on MUSIC so that the support is reliably recovered under such unfavorable conditions. Combined with subspace-based greedy algorithms also proposed and analyzed in this paper, SA-MUSIC provides a computationally efficient algorithm with a performance guarantee. The performance guarantees are given in terms of a version of restricted isometry property. In particular, we also present a non-asymptotic perturbation analysis of the signal subspace estimation that has been missing in the previous study of MUSIC.

• The paper introduces subspace-augmented MUSIC (SA-MUSIC) and subspace-based greedy algorithms (SS-OMP, SS-OMSP) to improve joint sparse recovery, particularly for rank-deficient signals.
• Performance guarantees for the proposed methods are provided through the weak-1 restricted isometry property (RIP), which is less demanding than uniform RIP.
• Empirical evaluations demonstrate that SA-MUSIC algorithms achieve higher recovery rates and lower computational costs compared to existing methods like M-BP.

Overview of Subspace Methods for Joint Sparse Recovery

The paper "Subspace Methods for Joint Sparse Recovery" by Lee, Bresler, and Junge addresses the problem of joint sparse recovery in compressed sensing. This problem involves recovering the supports of sparse signals from multiple measurement vectors processed through a common sensing matrix. The authors focus on circumstances where the original signals form a rank-deficient matrix, which is a common and practical scenario in applications suffering from correlated signal components or limited measurements.

The presented work builds upon the MUSIC (MUltiple SIgnal Classification) algorithm, initially developed for direction of arrival (DOA) problems and adapted for joint sparse recovery. MUSIC offers guarantees when the signal matrix has full row rank but fails under rank-deficient conditions. The authors introduce a variant called subspace-augmented MUSIC (SA-MUSIC), which extends the capability of MUSIC through subspace augmentation, improving performance in rank-deficient scenarios. By combining SA-MUSIC with subspace-based greedy algorithms, the authors propose a method that achieves computational efficiency alongside robust performance guarantees.

Key Contributions

1. Introduction of Subspace Augmentation: The paper proposes SA-MUSIC, which leverages additional subspaces to complement the signal subspace, ensuring it covers the entire range of the sparse components even in the rank-deficient case. This involves constructing an augmented subspace from partial support sets, enabling the recovery process to approximate the conditions favorable to the original MUSIC procedure.
2. Subspace-Based Algorithms: Two novel greedy algorithms, SS-OMP (Signal Subspace Orthogonal Matching Pursuit) and SS-OMSP (Signal Subspace Orthogonal Matching Subspace Pursuit), were introduced for partial support recovery. These methods iteratively augment the support set, using subspace information to improve selection criteria in each iteration.
3. Performance Guarantees via Weak-1 RIP: The authors provide a performance analysis based on the weak restricted isometry property (weak RIP), focusing specifically on the weak-1 version. This is less demanding than uniform RIP, allowing for broader applicability and analysis for different sensing matrices like Gaussian and partial Fourier matrices.
4. Empirical Evaluation and Runtime Analysis: Through experiments, the paper demonstrates that SA-MUSIC algorithms achieve higher recovery rates and lower computational costs compared to contemporary methods like M-BP (Mixed Basis Pursuit), even in the presence of noise.
5. Non-Asymptotic Analysis of Subspace Estimation: By providing a detailed analysis of signal subspace estimation from finite snapshots, the authors establish conditions under which the estimated subspace remains accurately close to the true signal subspace. This forms a basis to quantify the robustness of the proposed methods against perturbations.

Implications and Future Directions

This paper significantly influences the field of joint sparse recovery, particularly in signal processing applications with correlated signals and high noise. The robustness of SA-MUSIC under suboptimal conditions, such as rank deficiency, represents an important advancement over traditional methods like MUSIC and BP. The introduction of subspace augmentation techniques highlights the potential for exploiting additional structure in sparse recovery problems.

In future research, exploring more efficient implementations and variants of SS-OMP and SS-OMSP could advance the practical utility of SA-MUSIC. Moreover, a deeper integration of subspace estimation techniques with adaptive sensing strategies remains an avenue for further exploration, potentially enhancing performance against varying noise levels and dynamic signal environments.

The theoretical framework laid out through weak-1 RIP could be extended to accommodate additional signal characteristics or constraints arising from new applications, thus broadening the adaptability of joint sparse recovery methodologies.

In sum, this paper establishes a solid foundation for future developments in subspace methods and sparse signal processing, promoting innovative techniques that balance computational efficiency with algorithmic robustness.