The Davis–Kahan–Wedin$\sin \Theta $theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis–Kahan–Wedin$\sin \Theta $theorem when the perturbation is a Gaussian random matrix. Under certain structural assumptions, we obtain an optimal bound that significantly improves upon the classic Davis–Kahan–Wedin$\sin \Theta $theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.

The Davis–Kahan–Wedin theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis–Kahan–Wedin theorem when the perturbation is a Gaussian random matrix. Under certain structural assumptions, we obtain an optimal bound that significantly improves upon the classic Davis–Kahan–Wedin theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.

The estimation of a covariance matrix from an insufficient amount of data is one of the most common problems in fields as diverse as multivariate statistics, wireless communications, signal processing, biology, learning theory, and finance. In a joint work of Marzetta, Tucci and Simon, a new approach to handle singular covariance matrices was suggested. The main idea was to use dimensionality reduction in conjunction with an average over the Stiefel manifold. In this paper, we continue with this research and we consider some new approaches to handle this problem. One of the methods is called the mean conjugate estimator under Ewens measure and uses a randomization of the sample covariance matrix over all the permutation matrices with respect to the Ewens measure. The techniques used to attack this problem are broad and run from random matrix theory to combinatorics.

The estimation of a covariance matrix from an insufficient amount of data is one of the most common problems in fields as diverse as multivariate statistics, wireless communications, signal processing, biology, learning theory and finance. In [13], a new approach to handle rank deficient covariance matrices was suggested. The main idea was to use dimensionality reduction in conjunction with an average over the Stiefel manifold. In this paper we further continue in this direction and consider a few innovative methods that show considerable improvements with respect to more traditional ones such as diagonal loading. One of the methods is called the Ewens estimator and uses a randomization of the sample covariance matrix over all the permutation matrices with respect to the Ewens measure. The techniques used to attack this problem are broad and run from random matrix theory to combinatorics.