Successful Recovery Performance Guarantees of SOMP Under the $\ell _{2}$-Norm of Noise

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.