Multivariate moment problems with applications to spectral estimation and physical layer security in wireless communications

This thesis focuses on generalized moment problems and their applications in the framework of information engineering. Its contribution is twofold. The first part of this dissertation proposes two new techniques for tackling multivariate spectral estimation, which is a key topic in system identification: Relative entropy rate estimation and multivariate circulant rational covariance extension. The former provides a very natural multivariate extension of a state-of-the-art approach for scalar parametric spectral estimation with a complexity bound, known as THREE (Tunable High-Resolution Estimator). It allows to take into account available a priori information on the spectral density. It exhibits high resolution features and it is robust in case of short data records. As for multivariate circulant rational covariance extension, it is a new convex optimization approach to spectral estimation for periodic multivariate processes, in which the computation of the solution can be tackled efficiently by means of Fast Fourier Transform. Numerical examples show that this procedure also provides an efficient tool for approximating regular covariance extension for multivariate processes. The second part of this dissertation considers the problem of deriving a universal performance bound for a message source authentication scheme based on channel estimates in a wireless fading scenario, where an attacker may have correlated observations available and possibly unbounded computational power. Under the assumption that the channels are represented by multivariate complex Gaussian variables, it is proved that the tightest bound corresponds to a forging strategy that produces a zero mean signal that is jointly Gaussian with the attacker observations. A characterization of their joint covariance matrix is derived through the solution of a system of two nonlinear matrix equations. Based upon this characterization, the thesis proposes an efficient iterative algorithm for its computation: The solution to the matricial system appears as fixed point of the iteration. Numerical examples suggest that this procedure is effective in assessing worst case channel authentication performance

Scalable iterative methods for sampling from massive Gaussian random vectors

Sampling from Gaussian Markov random fields (GMRFs), that is multivariate Gaussian ran- dom vectors that are parameterised by the inverse of their covariance matrix, is a fundamental problem in computational statistics. In this paper, we show how we can exploit arbitrarily accu- rate approximations to a GMRF to speed up Krylov subspace sampling methods. We also show that these methods can be used when computing the normalising constant of a large multivariate Gaussian distribution, which is needed for both any likelihood-based inference method. The method we derive is also applicable to other structured Gaussian random vectors and, in particu- lar, we show that when the precision matrix is a perturbation of a (block) circulant matrix, it is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure

Identification of Sparse Reciprocal Graphical Models

In this paper we propose an identification procedure of a sparse graphical model associated to a Gaussian stationary stochastic process. The identification paradigm exploits the approximation of autoregressive processes through reciprocal processes in order to improve the robustness of the identification algorithm, especially when the order of the autoregressive process becomes large. We show that the proposed paradigm leads to a regularized, circulant matrix completion problem whose solution only requires computations of the eigenvalues of matrices of dimension equal to the dimension of the process