The restricted isometry property for time-frequency structured random matrices

Abstract

This paper establishes the restricted isometry property for a Gabor system generated by n 2 time-frequency shifts of a random window function in n dimensions. The sth order restricted isometry constant of the associated n × n 2 Gabor synthesis matrix is small provided that s ≤ c n 2/3 / log2 n. 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. © 2012 Springer-Verlag.