On the gap between restricted isometry properties and sparse recovery conditions

We consider the problem of recovering sparse vectors from underdetermined linear measurements via `pconstrained basis pursuit. Previous analyses of this problem based on generalized restricted isometry properties have suggested that two phenomena occur if p 6= 2. First, one may need substantially more than s log(en/s) measurements (optimal for p = 2) for uniform recovery of all s-sparse vectors. Second, the matrix that achieves recovery with the optimal number of measurements may not be Gaussian (as for p = 2). We present a new, direct analysis which shows that in fact neither of these phenomena occur. Via a suitable version of the null space property we show that a standard Gaussian matrix provides `q/`1-recovery guarantees for `p-constrained basis pursuit in the optimal measurement regime. Our result extends to several heavier-tailed measurement matrices. As an application, we show that one can obtain a consistent reconstruction from uniform scalar quantized measurements in the optimal measurement regime.