Sparse Recovery with Pre-Gaussian Random Matrices

For an m × N underdetermined system of linear equations with independent pre-Gaussian random coefficients satisfying simple moment conditions, it is proved that the s-sparse solutions of the system can be found by `1-minimization under the optimal condition m ≥ c s ln(eN/s). The main ingredient of the proof is a variation of a classical Restricted Isometry Property, where the inner norm becomes the `1-norm and where the outer norm depends on the probability distributions.