Successful Recovery Performance Guarantees of SOMP Under the $\ell _{2}$-Norm of Noise
Successful Recovery Performance Guarantees of SOMP Under the $\ell _{2}$-Norm of Noise
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DOI：
10.1109/tvt.2023.3315325
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发表时间：
2021-08
影响因子：
6.8
通讯作者：
W. Zhang;Taejoon Kim
中科院分区：
文献类型：
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作者：
W. Zhang;Taejoon Kim
同时正交匹配追踪(SOMP)是一种流行的、贪婪的恢复行稀疏矩阵共同支持度的方法。然而，与无噪声场景相比，有噪声SOMP的性能分析还处于起步阶段，尤其是在无界噪声的场景下。在本文中，我们提出了一种新的基于互不相干特性(MIP)的研究，用于在测量矩阵和稀疏信号是确定性的情况下分析噪声SOMP的性能。具体地说，当噪声有界时，我们给出了以MIP的形式保证精确支持恢复的条件。当噪声是无界的时，我们得到一个关于成功恢复概率(SRP)的界，它取决于噪声矩阵的$\ell_{2}$-范数的具体分布。然后，针对噪声为随机高斯分布的典型情况，证明了SRP的下界服从Tracy-Widom定律分布。该分析揭示了保证预定义恢复性能所需的测量数量、噪声水平、稀疏向量数量和互相关值。理论上，我们证明了测量矩阵的互相关性必须与噪声标准差成正比地减小，而稀疏向量的数目需要与噪声方差成比例地增加。最后，我们通过数值模拟广泛地验证了所得到的分析。
The simultaneous orthogonal matching pursuit (SOMP) is a popular, greedy approach for common support recovery of a row-sparse matrix. However, compared to the noiseless scenario, the performance analysis of noisy SOMP is still nascent, especially in the scenario of unbounded noise. In this article, we present a new study based on the mutual incoherence property (MIP) for performance analysis of noisy SOMP when the measurement matrix and sparse signal are deterministic. Specifically, when noise is bounded, we provide the condition on which the exact support recovery is guaranteed in terms of the MIP. When noise is unbounded, we instead derive a bound on the successful recovery probability (SRP) that depends on the specific distribution of the $\ell _{2}$-norm of the noise matrix. Then we focus on the typical case when noise is random Gaussian, and show that the lower bound of SRP follows Tracy-Widom law distribution. The analysis reveals the number of measurements, noise level, the number of sparse vectors, and the value of mutual coherence that are required to guarantee a predefined recovery performance. Theoretically, we show that the mutual coherence of the measurement matrix must decrease proportionally to the noise standard deviation, and the number of sparse vectors needs to grow proportionally to the noise variance. Finally, we extensively validate the derived analysis through numerical simulations.