Signal-Dependent Performance Analysis of Orthogonal Matching Pursuit for Exact Sparse Recovery

Exact recovery of K-sparse signals x ∈ ℝ
ⁿ
from linear measurements y = Ax, where A ∈ ℝ
^m×n
is a sensing matrix, arises from many applications. The orthogonal matching pursuit (OMP) algorithm is widely used for reconstructing x based on y and A due to its excellent recovery performance and high efficiency. A fundamental question in the performance analysis of OMP is the characterizations of the probability of exact recovery of x for random matrix A and the minimal m to guarantee a target recovery performance. In many practical applications, in addition to sparsity, x also has some additional properties (for example, the nonzero entries of x independently and identically follow a Gaussian distribution, or x has exponentially decaying property). This paper shows that these properties can be used to refine the answer to the above question. Toward this end, we first show that the prior information of the nonzero entries of x can be used to provide an upper bound on ||x||
₁
²
/||x||
₂
²
. Then, we use this upper bound to develop a lower bound on the probability of exact recovery of x using OMP in K iterations. Furthermore, we develop a lower bound on the number of measurements m to guarantee that the exact recovery probability using K iterations of OMP is no smaller than a given target probability. Finally, we show that when K = O(√ln n), as both n and K go to infinity, sufficient to ensure that the probability of exact recovering any K-for any 0 <; ζ ≤ 1/√π, m = 2K ln(n/ζ) measurements are ln n)sparse x is no lower than 1 - ζ with K iterations of OMP. This improves the m = 4K ln(2n/ζ) result of Tropp et al. For K-sparse α-strongly decaying signals and for K-sparse x whose nonzero entries independently and identically follow the Gaussian distribution, the number of measurements sufficient for exact recovery with probability no lower than 1 - ζ reduces further to m = (√K + 4√α+1/α-1 ln(n/ζ))
²
and to asymptotically m ≈ 1.9K ln(n/ζ), respectively.