Improved Bounds on the Restricted Isometry Constant for Orthogonal Matching Pursuit

In this letter, we first construct a counter example to show that for any given positive integer K ≥ 2 and for any 1 √ K+1 ≤ t < 1, there always exist a K−sparse x and a matrix A with the restricted isometry constant δK+1 = t such that the OMP algorithm fails in K iterations. Secondly, we show that even when δK+1 = 1 √ K+1 , the OMP algorithm can also perfectly recover every K−sparse vector x from y =Ax in K iteration. This improves the best existing results which were independently given by Mo et al. and Wang et al.