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 paper, we present a new study based on the mutual incoherence property (MIP) for performance analysis of noisy SOMP. 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 common 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.

翻译：同步正交匹配追踪（SOMP）是一种用于行稀疏矩阵公共支撑恢复的经典贪婪算法。然而，与无噪声场景相比，含噪声SOMP的性能分析仍处于起步阶段，尤其是在非有界噪声场景下。本文基于互不相关性（MIP）提出了一种新的含噪SOMP性能分析方法。具体而言，当噪声有界时，我们给出了基于MIP的精确支撑恢复保证条件；当噪声无界时，我们推导了依赖于噪声矩阵$\ell_2$范数特定分布的成功恢复概率（SRP）上界。接着针对随机高斯噪声这一常见情形，证明SRP下界服从Tracy-Widom律分布。该分析揭示了保证预设恢复性能所需的测量次数、噪声水平、稀疏向量数量以及互不相关性取值。理论上证明了测量矩阵的互不相关性必须随噪声标准差成比例减小，而稀疏向量数量需随噪声方差成比例增长。最后通过数值仿真全面验证了所推导的分析结果。