Multipath Matching Pursuit (1308.4791v6)

Abstract: In this paper, we propose an algorithm referred to as multipath matching pursuit that investigates multiple promising candidates to recover sparse signals from compressed measurements. Our method is inspired by the fact that the problem to find the candidate that minimizes the residual is readily modeled as a combinatoric tree search problem and the greedy search strategy is a good fit for solving this problem. In the empirical results as well as the restricted isometry property (RIP) based performance guarantee, we show that the proposed MMP algorithm is effective in reconstructing original sparse signals for both noiseless and noisy scenarios.

• The paper introduces MMP, which uses a combinatorial tree search to explore multiple candidate paths for robust sparse signal recovery.
• It establishes RIP-based recovery conditions and demonstrates superior exact recovery ratios and lower mean squared errors than traditional greedy methods.
• Empirical results show that MMP and its depth-first variant enhance computational efficiency and robustness, making them valuable for compressive sensing applications.

Multipath Matching Pursuit: A Robust Approach to Sparse Signal Recovery

This paper presents an innovative algorithm called Multipath Matching Pursuit (MMP) for the recovery of sparse signals from compressed measurements, which addresses the inherent weaknesses of existing greedy algorithms, such as Orthogonal Matching Pursuit (OMP). While OMP is known for its simplicity and efficiency, it is highly sensitive to incorrect index selection during iterations, which hampers its effectiveness in noisy environments. The proposed MMP algorithm mitigates these shortcomings by examining multiple promising candidates in a combinatorial tree search approach, increasing the likelihood of selecting the true support indices.

Key Contributions

1. Algorithm Design:
   • MMP operates by investigating multiple candidate paths in a search tree, contrasting with OMP's single-path exploration. This multi-path approach enhances the algorithm's robustness, especially in scenarios where noise can mislead the index selection process.
   • The tree search problem is efficiently addressed by exploiting greedy strategies to minimize residuals in signal reconstruction.
   • A variant called Multipath Matching Pursuit with Depth-First Search (MMP-DF) is introduced to strictly control computational complexity, where the number of candidate paths investigated is limited.
2. Theoretical Analysis:
   • The paper provides RIP-based conditions guaranteeing perfect recovery of sparse signals in noiseless and noisy scenarios, establishing the superiority of MMP's recovery performance over existing algorithms.
   • A significant finding is that MMP can accurately recover K-sparse signals if the sensing matrix satisfies $\delta_{K+L} < \frac{\sqrt{L}}{\sqrt{K}+2\sqrt{L}}$, with enhanced performance metrics against variations like BP and CoSaMP.
   • In noisy scenarios, MMP’s capability to identify true supports is analyzed, with conditions stating that MMP achieves the Oracle LS estimator performance under defined SNR thresholds.
3. Empirical Results:
   • MMP outperforms existing algorithms in terms of exact recovery ratio and mean squared error across various sparsity levels and signal-to-noise ratios.
   • The computational complexity comparison reveals MMP-DF’s competitive efficiency despite its robustness, maintaining adequate performance with constraints on search breadth.
4. Potential Improvements:
   • Suggested enhancements include integrating probabilistic tree pruning to reduce computational costs further or hybrid approaches incorporating OMP paths initially followed by branching to MMP operations.

Implications and Future Research

MMP represents a significant stride in compressive sensing, where sparse signal recovery is crucial for applications ranging from telecommunications to medical imaging. Its robustness in noisy scenarios aligns well with real-world applications where perfect knowledge of the signal and noise properties is unattainable. The improvements in accuracy and adaptability to varying system dimensions without substantial computational penalties position MMP as a valuable tool in signal processing.

This research sets a pathway for future developments in sparse recovery methods, encouraging exploration of hybrid algorithms and complex tree pruning mechanisms to enhance computational efficiency further. Given the ubiquity of compressed sensing problems, these advancements may lead to more widespread applications in machine learning and data analytics where sparse modeling is vital.

The paper's in-depth theoretical analysis and simulation results provide a strong foundation for exploring new domains utilizing compressive sensing techniques, making significant contributions to both theoretical development and practical applications in signal processing.