The restricted isometry property for time-frequency structured random matrices

Kurzfassung/Abstract

This paper establishes the restricted isometry property for a Gabor system generated by n2 time–frequency shifts of a random window function in n dimensions. The sth order restricted isometry constant of the associated n × n2 Gabor synthesis matrix is small provided that s ≤ cn2/3 / log2n. This bound provides a qualitative improvement over previous estimates, which achieve only quadratic scaling of the sparsity s with respect to n. The proof depends on an estimate for the expected supremum of a second-order chaos.