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 is called effective sparsity. We also present a condition on the so-called restricted isometry constants, \deltas, ensuring sparse recovery via \ell^1-minimization. We show that \delta2s<4/\sqrt{41} is sufficient and that this can be improved further to almost allow for a sufficient condition of the type \delta2s<2/3. [ABSTRACT FROM AUTHOR]