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.

In this work we examine the behavior of the minimum singular value of random Vandermonde matrices. In particular, we prove that the minimum singular value s1(N) is at most N exp(-C√N) where N is the dimension of the matrix and C is a constant. Furthermore, the value of the constant C is determined explicitly. The main result is obtained in two different ways. One approach uses techniques from stochastic processes and in particular, a construction related to the Brownian bridge. The other one is a more direct analytical approach involving combinatorics and complex analysis. As a consequence, we obtain a lower bound on the maximum absolute value of a random polynomial on the unit circle, which may be of independent mathematical 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 [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.

Random geometric graphs have proven to be extremely useful in modeling static wireless ad-hoc and sensor networks. The study of these graphs and their spectral properties is a very active field with many interesting applications. In this work, we study the spectral properties of these graphs and in particular we prove the lack of spectral gap in random geometric graphs under a wide range of cases. We also present some results on the spectral properties of generalized weighted versions of these graphs. We later focus on mobile geometric graphs or dynamic Boolean models as were introduced by van den Berg, Meester, and White in 1997. These mobile models have been used to model mobile ad-hoc and sensor networks. Our results are aimed at understanding the dynamics of message spreading in a real-world dynamical network. We simulate a message spreading process and present some results related to the total propagation time under different scenarios. We call total propagation time the time it takes for a certain message to reach all the nodes in the network. This message could be a packet, a virus, a rumor in a social network, or something more general. In particular, we study how the metric, the mobility properties, the presence or absence of obstacles and finally the propagation characteristics affect the dynamics of the message spreading.

In many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data. More specifically, if we have a set ofNindependent, identically distributed measurements of anMdimensional random vector the maximum likelihood estimate is the sample covariance matrix. Here we consider the case whereNMsuch that this estimate is singular (noninvertible) and therefore fundamentally bad. We present a radically new approach to deal with this situation based on the idea of dimensionality reduction through an ensemble of isotropically random unitary matrices. We obtain two estimates cov and invcov which are estimates for the covariance matrix and the inverse covariance matrix respectively. Both estimates retain the original eigenvectors while altering the eigenvalues. We have a closed form analytical expression for cov and invcov in terms of the eigenvector/eigenvalue decomposition of the sample covariance. We motivate the use of invcov through applications to linear estimation, supervised learning, and high-resolution spectral estimation. We also compare the performance of these estimators with other more conventional methods.

This paper centers on the limit eigenvalue distribution for random Vandermonde matrices with unit magnitude complex entries. The phases of the entries are chosen independently and identically distributed from the interval [-π,π] . Various types of distribution for the phase are considered and we establish the existence of the empirical eigenvalue distribution in the large matrix limit on a wide range of cases. The rate of growth of the maximum eigenvalue is examined and shown to be no greater thanO(logN) and no slower thanO(logN/loglogN) whereNis the dimension of the matrix. Additional results include the existence of the capacity of the Vandermonde channel (limit integral for the expected log determinant).