High-dimensional phenomena

Fields of research

High dimensional phenomena are ubiquitous in modern applications, such as in telecommunications, genomics, and imaging. The mathematical and algorithmic analyses underlying these models involve deep tools from asymptotic geometric analysis, random matrices, and combinatorics. In particular, random matrix theory (RMT) appears as a central concept at the heart of mathematical and computational challenges.

Compressed Sensing is a recent framework that enables approximation and exact reconstruction of sparse signals from incomplete measurements. It is strongly related to other problems of different fields such as approximation theory (diameter of sections), high dimensional geometry (neighborliness and asymptotic geometry of convex bodies), harmonic analysis (trigonometric diameter and selection of characters) and random matrices (behavior of the largest and smallest singular values).

The behavior of large wireless networks depends on a large number of non observable parameters that are modeled as realizations of entries of certain large random matrices whose probability distributions are assumed to be known, at least up to certain parameters. Certain performance metrics, to be optimized, can be interpreted as functionals of the eigenvalues of large random matrices, and can, therefore, be approximated by deterministic terms that can be more or less evaluated. This potentially allows to develop new efficient and possibly decentralized optimization algorithms for large wireless networks.

The application of large random matrices to statistics is even more recent, dealing with the estimation of the eigenvalues of a population covariance matrix from the sample covariance matrix when the sample size is of the same order of magnitude as the number of variables. Many connections between large random matrices and statistical problems remain to be explored both from practical and methodological point of views.

Scientific program of the Bézout Labex

In Compressed sensing, the main property that should satisfy a matrix A to be a good decoder for sparse signals from incomplete measurements is called the Restricted Isometric Property (RIP). But very few matrices are known to satisfy this property. One of the Bézout Labex’s objectives is to address a constructive approach to solve this problem, like using a clear algorithmic description of the matrix that satisfies this property.

Large random matrix theory (RMT) is now recognized as a useful tool for the analysis of large wireless telecommunication networks. Although RMT tools have yet been used in several contexts, a lot of work remains to be done in this field.

Several classical statistical methodologies developed in the context of multivariate time series appear to be relevant when the number of observations is much larger than the dimension of observations. However, large dimension time series can be met in applicative contexts such as large sensor networks, large wireless networks, finance… Generally, the number available of observations is limited, and can be of the same order of magnitude than the dimension of the series. In this context, it is necessary to develop new practical estimation schemes as well as relevant large sample analysis technics in which the number of observations and their dimension converge towards infinity at the same rate. Although this framework appears important, it has not been addressed extensively in the past years. The Labex proposes to develop methodologies based on large random matrix theory, and to study large dimension «information plus noise model» in which the observation is a noisy version of a «useful signal». Important generic problems such as detection of the useful signal, or the estimation of parameters depending on its statistics (e.g. functionals of the eigenvalues/eigenvectors of its covariance matrix) will be addressed. Applications to statistical signal processing and wireless communications will be developed. Preliminary results have been obtained in the context of the ANR projet Sesame (conSitent EStimation and Large random MatricEs, end in December 2011).

Valorization of results, transfer, and expertise

Industrial collaborations on the theme of large random matrices have already been developed. This concerns, in particular, companies in the area of mobile communications for doctoral funding (CIFRE contracts), research projects in cognitive radio-communications as the DEMAIN ANR Telecom project, and to some extent, the European Fifth Framework Programme (FP5) ANTIUM. The Bézout Labex plans to continue the development of these results in the field of wireless communications in two directions.

The first applicative framework will be the optimisation of ressources of a network, which requires the maximisation of metrics that can be either highly complex or simply non computable, as dependent on random data with known statistics but non observable. In some contexts, these performance indicators may converge towards deterministic terms both simple to calculate and to optimize.

The second issue is related to cognitive radio in which mobile terminals may in the future, after analysis of their local electromagnetic environment, choose the bandwidth that best suits their needs. For this, the terminal will, without having much prior information, evaluate the quality of service it can obtain in each band, and then get an idea of nuisances that its presence will produce. This environment is rich in statistical problems in which large random matrices appear naturally. This is due to the random nature of propagation channels and to the growing numbers of users of mobile networks.