New Bounds for Restricted Isometry Constants

This paper discusses new bounds for restricted isometry constants in compressed sensing. Let Phi be an n x p real matrix and k be a positive integer with k <= n. One of the main results of this paper shows that if the restricted isometry constant delta(k) of Phi satisfies delta(k) < 0.307 then k-sparse signals are guaranteed to be recovered exactly via l(1) minimization when no noise is present and k-sparse signals can be estimated stably in the noisy case. It is also shown that the bound cannot be substantially improved. An explicit example is constructed in which delta(k) = k-1/2k-1 < 0.5, but it is impossible to recover certain k-sparse signals.