High dimensional statistical signal processing – DIONISOS

Large random matrices have been proved to be of fundamental importance in mathematics (high dimensional probability and statistics, operator algebras, combinatorics, number theory,...) and in physics (nuclear physics, quantum fields theory, quantum chaos,..) for a long time. The introduction of large random matrix theory in electrical engineering is more recent. It was introduced at the end of the nineties in the context of digital communications in order to analyse the performance of large CDMA and MIMO systems. Except some pionneering works of Girko, the use of large random matrices is even more recent in statistical signal processing. The corresponding tools turn out to be useful when the observation is a large dimension (say M) multivariate time series (y(n)), n=1,...N, and that the sample size N is of the same order of magnitude than M. This context poses a number of new difficult statistical problems that are intensively studied by the high-dimensional statistics community. The most significant example is related to the fundamental problem of estimating of the covariance matrix of the observation because the standard empirical covariance matrix defined as the empirical mean of the y(n)y(n)* is known to perform poorly if N is not significantly larger than M.

In the context of this project, the dimension of the observation corresponds to the number of elements of a large sensor network, and the components of vector y(n) represent the signal received at time n on the various sensors. It turns out that a number of fundamental statistical signal processing schemes such as source detection, source localisation, independent source separation, estimation of linear prediction filters...fail in the case where M and N are large and of the same order of magnitude, a context modelled in the following by the asymptotic regime M and N both converge to infinity in such a way that the ratio M/N converges to a non zero constant c. The purpose of this project is to develop new mathematical tools and algorithms which allow to enhance the performance of classical methods in the above regime.

In Task 1, the literature of large random matrices and of high-dimensional statistics is reviewed in order to identify the situations in which new mathematical results are needed.

Task 2 is devoted to the study of fundamental statistical inference problems related to the so-called narrow-band sources antenna array model.
Although a few topics related to this model were addressed quite recently in the above asymptotic regime, important problems still deserve to be studied. Task 2-1 first revisits classical detection and estimation problems related to narrowband sources in the case where M and N are of the same order of magnitude. The goal of Task 2-2 is to adapt certain blind source separation algorithms to the above asymptotic regime. For this, it is necessary to study, among others, the behaviour of large random matrices such as the empirical covariance matrix associated to vector z(n) defined as the Kronecker product of y(n) with itself.

Task 3 addresses the case of wide band sources for which the contribution of each source to the observation is the output of an unknown a 1 input / M outputs filter driven by the source signal. In this context, it is crucial to estimate the covariance matrix of the extended vector obtained by stacking L consecutive observations. The goal of Task 3-1 is to develop the corresponding new mathematical results. The behaviour of the spectrum of the empirical covariance matrix of the augmented vector is first studied when ML and N are of the same order of magnitude. Then, consistent operator norm estimators based on banding, thresholding or tapering are developed when these schemes are relevant. The new results are used to address the detection of wideband sources (Task 3-2) and the trained and blind spatio-temporal equalization (Task 3-3).