In this paper, we investigate extreme singular values of the analysis matrix of a Gabor frame @math with a random window @math . Columns of such matrices are time and frequency shifts of @math , and @math is the set of time-frequency shift indices. Our aim is to obtain bounds on the singular values of such random time-frequency structured matrices for various choices of the frame set @math , and to investigate their dependence on the structure of @math , as well as on its cardinality. We also compare the results obtained for Gabor frame analysis matrices with the respective results for matrices with independent identically distributed entries.
Before we study the case when @math is a Gabor frame, we include here a short overwiew of the results on the singular values for random frames with independent entries, such as Gaussian matrices. The largest singular value of the analysis matrix @math of a random frame with independent entries can be estimated using Latala’s theorem @cite_8 . The following result implies that, with high probability, @math .
@cite_8
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Let @math denote a random symmetric @math matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values @math and @math with probability @math each). It is widely conjectured that @math is singular with probability at most @math . On the other hand, the best known upper bound on the singularity probability of @math , due to Vershynin (2011), is @math , for some unspecified small constant @math . This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of @math is at most @math for all sufficiently large @math . The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.