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 ≤ cn^(2/3) / log^2 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.

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