Analysis of generalised orthogonal matching pursuit using restricted isometry constant

Abstract

In compressive sensing, the generalised orthogonal matching pursuit (gOMP) algorithm is one kind of sparse signal recovery algorithm, which generalises the OMP algorithm by selecting a fixed number of atoms at each iteration. Restricted isometry constant-based sufficient conditions to guarantee the correct support identification and the successful recovery of a sparse signal using the gOMP algorithm in a noiseless case are proposed. The proposed sufficient bounds are more relaxed compared with the existing ones.Introduction: Compressive sensing [1] deals with the sparse signal recovery from its insufficient linear measurement. The measurement signal can be expressed as y = Φx, where x ∈ R n × 1 , y ∈ R m × 1 and Φ ∈ R m × n are an original sparse signal, measurement signal and measurement matrix, respectively. There exists an infinite number of solutions to the measurement equation as m ≪ n which means that the linear equation is underdetermined. However, with the prior knowledge that x is sparse, the solution to the linear equation is unique. Various algorithms were successively proposed to recover sparse signal x. Basically, conditions to guarantee the success of the algorithms are expressed via the 'restricted isometry constant' (RIC), which could be defined as the smallest value δ K (0 < δ K < 1) that makes the following inequalities true [1]:

A sharp condition for exact support recovery of sparse signals with orthogonal matching pursuit

“…Lemma 4 ( [25]): Let sets S 1 , S 2 satisfy |S 2 \ S 1 | ≥ 1 and matrix A satisfy the RIP of order |S 1 ∪S 2 |, then for any vector x ∈ R |S2\S1| ,…”

A Novel Sufficient Condition for Generalized Orthogonal Matching Pursuit

“…Clearly, our sufficient condition given by Corollary 1 is less restrictive than that given by [14, Theorem 3.1] in terms of both RIC and SNR. Notice that gOMP may terminate after performing k 0 with 0 < k 0 < K iterations, and in this case Ω is not guaranteed to be recovered by gOMP under (7) and (8). However, we have:…”

Theoretical results for sparse signal recovery with noises using generalized OMP algorithm

“…In [11,15,16], the success of the gOMP algorithm is based on this assumption. In the gOMP algorithm, if one or more correct atoms are selected in an iteration, we view this iteration as being effective.…”

“…Þ with the restriction that N Z 2 [15]. In the noisy case, [11] gives an error bound between the estimated signal by the gOMP algorithm and the original sparse signal x.…”