In this paper, the joint support recovery of several sparse signals whose supports exhibit similarities is examined. Each sparse signal is acquired using the same noisy linear measurement process, which returns fewer observations than the dimension of the sparse signals. The measurement noise is assumed additive, Gaussian, and admits different variances for each sparse signal that is measured. Using the theory of compressed sensing, the performance of simultaneous orthogonal matching pursuit (SOMP) is analyzed for the envisioned signal model. The cornerstone of this paper is a novel analysis method upper bounding the probability that SOMP recovers at least one incorrect entry of the joint support during a prescribed number of iterations. Furthermore, the probability of SOMP failing is investigated whenever the number of sparse signals being recovered simultaneously increases and tends to infinity. In particular, convincing observations and theoretical results suggest that SOMP committing no mistake in the noiseless case does not guarantee the absence of error in the noisy case whenever the number of acquired sparse signals scales to infinity. Finally, simulation results confirm the validity of the theoretical results.
@cite_15: We consider the orthogonal matching pursuit (OMP) algorithm for the recovery of a high-dimensional sparse signal based on a small number of noisy linear measurements. OMP is an iterative greedy algorithm that selects at each step the column, which is most correlated with the current residuals. In this paper, we present a fully data driven OMP algorithm with explicit stopping rules. It is shown that under conditions on the mutual incoherence and the minimum magnitude of the nonzero components of the signal, the support of the signal can be recovered exactly by the OMP algorithm with high probability. In addition, we also consider the problem of identifying significant components in the case where some of the nonzero components are possibly small. It is shown that in this case the OMP algorithm will still select all the significant components before possibly selecting incorrect ones. Moreover, with modified stopping rules, the OMP algorithm can ensure that no zero components are selected.
@cite_10: Orthogonal matching pursuit (OMP) algorithm is an efficient method for the recovery of a sparse signal in compressed sensing, due to its ease implementation and low complexity. In this paper, the robustness of the OMP algorithm under the restricted isometry property (RIP) is presented. It is shown that ( _K + K _ K,1 < 1 ) is sufficient for the OMP algorithm to recover exactly the support of arbitrary K-sparse signal if its nonzero components are large enough for both l 2 bounded and l ∞ bounded noises.
In a SMV setting, full support recovery guarantees for OMP with bounded noise signals as well as with Gaussian noises have been proposed in @cite_15 . This work also provides criteria on the stopping criteria to guarantee that OMP terminates after having picked all the correct atoms. This contribution has then been slightly refined in @cite_10 to provide conditions independent of the particular support that is to be recovered.