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
2168494960 1996023858 2951810887 2899347127
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 Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially after the pioneering works of Emmanuel Cand e s and collaborators. This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has "a little bit of structure." Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to solve problems related to low rank matrix estimation, blockmodels, distance matrix completion, latent space models, positive definite matrix completion, graphon estimation and generalized Bradley--Terry models for pairwise comparison. This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and Oh. The reconstruction is accomplished by minimizing the nuclear norm, or sum of the singular values, of the hidden matrix subject to agreement with the provided entries. If the underlying matrix satisfies a certain incoherence condition, then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition. The proof of this assertion is short, self contained, and uses very elementary analysis. The novel techniques herein are based on recent work in quantum information theory. Given an m ×n matrix A and an integer k less than the rank of A, the “best” rank k approximation to A that minimizes the error with respect to the Frobenius norm is Ak, which is obtained by projecting A on the top k left singular vectors of A. While Ak is routinely used in data analysis, it is difficult to interpret and understand it in terms of the original data, namely the columns and rows of A. For example, these columns and rows often come from some application domain, whereas the singular vectors are linear combinations of (up to all) the columns or rows of A. We address the problem of obtaining low-rank approximations that are directly interpretable in terms of the original columns or rows of A. Our main results are two polynomial time randomized algorithms that take as input a matrix A and return as output a matrix C, consisting of a “small” (i.e., a low-degree polynomial in k, 1 e, and log(1 δ)) number of actual columns of A such that ||A–CC+A||F ≤(1+e) ||A–Ak||F with probability at least 1–δ. Our algorithms are simple, and they take time of the order of the time needed to compute the top k right singular vectors of A. In addition, they sample the columns of A via the method of “subspace sampling,” so-named since the sampling probabilities depend on the lengths of the rows of the top singular vectors and since they ensure that we capture entirely a certain subspace of interest.