On the Theorem of Uniform Recovery of Random Sampling Matrices
  We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an $s$-sparse signal from linear measurements (with high probability) is known to be $m\gtrsim s(\ln s)^3\ln N$. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of $m\times N$-matrices, by considering what we call \emph{low entropy}. We also present an improved condition on the so-called restricted isometry constants, $\delta_s$, ensuring sparse recovery via $\ell^1$-minimization. We show that $\delta_{2s}<4/\sqrt{41}$ is sufficient and that this can be improved further to almost allow for a sufficient condition of the type $\delta_{2s}<2/3$. 
Joel Andersson and Jan-Olov Str\"omberg
cs.IT cs.NA math.IT