Restricted isometry properties and nonconvex compressive sensing

In previous work, numerical experiments showed that ιp minimization with 0 < p < 1 recovers sparse signals from fewer linear measurements than does ι1 minimization. It was also shown that a weaker restricted isometry property is sufficient to guarantee perfect recovery in the ιp case. In this work, we generalize this result to an ιp variant of the restricted isometry property, and then determine how many random Gaussian measurements are sufficient for the condition to hold with high probability.