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Thesis

Title. Applications of large random matrices to high dimensional statistical signal processing

Abstract. This thesis focuses on statistical problems involving a multivariate time series yn of large dimension M defined as the sum of gaussian white noise temporally and spatially and a useful signal defined as the output of an unknown finite impulse response single input multiple outputs system driven by a deterministic scalar nonobservable sequence. Supposing (yn)n=1,...,N is available, a number of existing methods are based on the functionals of empirical covariance matrix ˆRL of ML--dimensional vectors (y(L)n)n=1,...,N obtained by stacking the vectors (yk)k=n,...,n+L1, where L is a relevant parameter. In the case where the number of observations N is much larger than ML the dimension of vectors (y(L)n)n=1,...,N, ˆRL behaves as its mathematical expectation in the sense of spectral norm. This allows us to study the inference technique based on ˆRL via classical techniques of asymptotic statistics. In this thesis, we interested in the case where ML and N have the same order of magnitude, we call this the asymptotic regimes in which M and N converge towards infinity, such that the ratio MLN converges towards a strictly positive constant, given that L may scale with M,N. To solve the problems in this work, it is necessary to investigate the behaviour of the eigenvalues and eigenvectors of the random matrix ˆRL. Taking account of the particular structure of vectors (y(L)n)n=1,...,N, ˆRL coincides with the Gram matrix of a block-Hankel matrix ΣL, and this specificity requires the development of appropriate techniques.

We interested to the case where the number of coefficients P of the finite impulse response generated the useful signal and the parameter L remain fixed when M,N grow large. As a consequence, the matrix ΣL is a finite rank perturbation of block-Hankel matrix WL composed of additive noise. We prove that eigenvalues and eigenvectors of ˆRL behave as if the entries of matrix WL are independent and identically distributed. This allows us to construct detection tests of useful signal based on largest eigenvalues of ˆRL and to develop new estimation strategies of the regularization parameter of the spatio-temporal Wiener filter estimated from a training sequence. This approach is characterized by the asymptotic behaviour of the resolvent of matrix ˆRL. We also prove that these results provide consistent subspace estimation methods for source localization using spatial-smoothing scheme.

Motivated by the case where P and L may converge towards infinity, we move off somewhat the initial model. We suppose that the matrix ΣL is the sum of the block-Hankel random matrix WL with a deterministic matrix without particular structure. Using the approaches based on Stieltjes transform and tools adapted to gaussian noise, we prove that the empirical eigenvalue distribution of ˆRL has deterministic behaviour which we shall describe. Provided L2MN converges towards 0, we do likewise for the elements of the resolvent of ˆRL.

Finally, we return to the initial model, but we suppose that P,L converge towards infinity with the same rate. In this context, matrix ΣL is a matrix whose rank goes to infinity, and thus the techniques employed in chapter 2 are not applicable. Using the results obtained in chapter 3, we establish that when L2MN goes to 0, the elements of the resolvent of ˆRL behave as the elements of a deterministic matrix which coincides with the deterministic equivalent of the resolvent of the information plus noise model in which entries of the noise matrix are independent and identically distributed. In the case where LM goes to 0, this allows us to extend the results of chapter 2 related to the determination of the regularization parameter of the spatial-temporal Wiener filter estimated from a training sequence.

Key words : Large Random Matrix Theory, DoA estimation, Wiener filter, GLRT test, Spatial Smoothing.

Jury composition : The diaporama of the thesis defense can be found here.

The manuscript is here.

The thesis defense took place at l'université Paris-Est Marne-la-Vallée on February 28th, 2017.