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.
The following optimal estimate of the smallest singular value of the analysis matrix for a random subgaussian frame with independent entries is due to Rudelson and Vershynin @cite_1 .
@cite_1
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We prove an optimal estimate of the smallest singular value of a random sub- Gaussian matrix, valid for all dimensions. For an Nn matrix A with inde- pendent and identically distributed sub-Gaussian entries, the smallest singular value of A is at least of the order p Np n � 1 with high probability. A sharp The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to information theory operate with random matrices in fixed dimensions. This survey addresses the non-asymptotic theory of extreme singular values of random matrices with independent entries. We focus on recently developed geometric methods for estimating the hard edge of random matrices (the smallest singular value). 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. We present deviation inequalities of random operators of the form 1 N ∑N i=1 Xi ⊗ Xi from the average operator E(X ⊗ X), where Xi are independent random vectors distributed as X, which is a random vector in R or in 2. We use these inequalities to estimate the singular values of random matrices with independent rows (without assuming that the entries are independent).