On the derivation of RIP for random Gaussian matrices and binary sparse signals

The number of measurements, M, sufficient for successful recovery via the L1 minimization is well known to be M = O(Klog(N/K)) M [7] with Gaussian measurement matrices used for sensing K sparse signals of ambient dimension N. We aim to shed a light on the source of the log(N) factor, and see if the bound can be improved by considering it for simplest possible K-sparse signals -0/1 binary K sparse signals. Previous work exists with which it is reasonable to expect reduction in the number of measurements when the signal has smaller degrees of freedom. We derive an upper bound on the probability that any set of K randomly selected Gaussian column vectors are mutually independent; we use this to find an upper bound on the probability that a Gaussian sensing matrix satisfies the restricted isometry condition. Using this result, a sufficient condition for good signal recovery is found. Surprisingly, the result remains the same, i.e., M = O(Klog(N/K)), which may suggest the log(N) factor is generic for Gaussian measurements.