Some Bounds and the Conditional Maximum Bound for Restricted Isometry Constants

Compressed sensing seeks to recover an unknown sparse signal with entries by making far fewer than measurements. The restricted isometry Constants (RIC) has become a dominant tool used for such cases since if RIC satisfies some bound then sparse signals are guaranteed to be recovered exactly when no noise is present and sparse signals can be estimated stably in the noisy case. During the last few years, a great deal of attention has been focused on bounds of RIC, see, e. g., Candes (2008), Foucart et al (2009), Foucart (2010), Cai et al (2010), Mo et al (2011), Ji et al (2012). Finding bounds of RIC has theoretical and applied significance. In this paper, we obtain a bound of RIC. It improves the results by Cai et al (2010) and Ji et al (2012). Further, we discuss the problems related larger bound of RIC, and give the conditional maximum bound.