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 ≳ s(ln s)<;sup>3<;/sup>lnN. 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 × N-matrices, by considering what is called effective sparsity. We also present a condition on the so-called restricted isometry constants, δ<;sub>s<;/sub>, ensuring sparse recovery via ℓ<;sup>1<;/sup>-minimization. We show that is sufficient and that this can be improved further to almost allow for a sufficient condition of the type .