Noisy compressive sampling limits in linear and sublinear regimes

The authors have recently established a set of results that characterize the number of measurements required to recover a sparse signal in ℂ^M with L non-zero coefficients from compressed samples in the presence of noise. These results indicate that for a number of different recovery criteria, O(L) (an asymptotically linear multiple of L) measurements are necessary and sufficient for signal recovery, whenever L grows linearly as a function of M. We review these results that improve on the existing literature, which are mostly derived for a specific recovery algorithm based on convex programming, where O(L log(M-L)) measurements are required. The results discussed here also show that O(L log(M-L)) measurements are required in the sublinear regime (L = o(M)).