We consider the problem of generating pseudorandom matrices based on the similarity of their spectra to Wigner's semicircular law. We introduce the notion of an r-independent pseudo-Wigner matrix ensemble and prove the closeness of the spectra of its matrices to the semicircular density in the Kolmogorov distance. We give an explicit construction of a family of N × N pseudo-Wigner ensembles using dual BCH codes and show that the Kolmogorov complexity of the obtained matrices is of the order of log(N) bits for a fixed designed Kolmogorov distance precision. We compare our construction with the quasi-random graphs introduced by Chung et al. and demonstrate that the pseudo-Wigner matrices pass stronger randomness tests than the adjacency matrices of these graphs (lifted by the mapping 0 → 1 and 1 → -1) do. Finally, we provide numerical simulations verifying our theoretical results.

We consider the problem of generating symmetric pseudo-random sign (±1) matrices based on the similarity of their spectra to Wigner's semicircular law. Using binary m-sequences (Golomb sequences) of lengths n = 2m- 1, we give a simple explicit construction of circulant n x n sign matrices and show that their spectra converge to the semicircular law when n grows. The Kolmogorov complexity of the proposed matrices equals to that of Golomb sequences and is at most 2log2(n) bits.

We consider the problem of generating symmetric pseudo-random sign (±1) matrices based on the similarity of their spectra to Wigner's semicircular law. We introduce r-independent pseudo-Wigner ensembles and prove closeness of their spectra to the semicircular density in Kolmogorov distance. We give an explicit construction of a family of N × N pseudo-Wigner ensembles using dual BCH codes and show that the Kolmogorov complexity of the obtained matrices is of the order of log (N) bits for a fixed Kolmogorov distance precision. Finally, we provide numerical simulations verifying our theoretical results.

We consider the problem of generating symmetric pseudo-random sign (±1) matrices based on the similarity of their spectra to Wigner's semicircular law. Using binary m-sequences (Golomb sequences) of lengths n = 2m- 1, we give a simple explicit construction of circulant n × n sign matrices and show that their spectra converge to the semicircular law when n grows. The Kolmogorov complexity of the proposed matrices equals to that of Golomb sequences and is at most 2log2(n) bits.

We consider the joint estimation of structured covariance matrices. We assume the structure is unknown and perform the estimation using heterogeneous training sets. More precisely, we are given groups of measurements coming from centered normal populations with different covariance matrices. Assuming that all these covariance matrices span a low dimensional affine subspace in the space of symmetric matrices, our aim is to determine this structure. It is then utilized to improve the covariance estimation. We provide an algorithm discovering and exploring the underlying covariance structure and analyze its error bounds. Numerical simulations are presented to illustrate the performance benefits of the proposed algorithm.

This paper analyzes the performance of Tyler's M-estimator of the scatter matrix in elliptical populations. We focus on non-asymptotic performance analysis of Tyler's estimator. Given n samples of dimension p2/(1-c2)2n with high probability, where c is the coherence coefficient of the properly scaled estimator. Under additional group symmetry conditions we improve the obtained bound, utilizing the inherent sparsity properties of group symmetry.

We develop the General Method of Moments (GMM) Approach for estimating the covariance matrices of non-Gaussian distributions with convex structure. The GMM turns out to be a non-convex optimization problem, thus making the addition of prior knowledge in form of convex structure constraints cumbersome. We propose a different approach to this estimator and show that the Tyler's estimator can be obtained as a solution of a convexly relaxed GMM problem, thus making the imposition of convex constraints easier. This new framework provides consistent solutions which outperform the standard projection methods. As an application of this method we consider Gaussian Compound samples with Toeplitz and banded covariance matrices. We provide synthetic numerical data and demonstrate the performance advantages of our method.