23 avril 2016

Catégorie : Doctorant

Available Ph.D position: Estimators for sparse hybrid/paramaters based on Bayesian inference over graphical models

Laboratories :

- L2S - UMR 8506, Signals and Statistics Dep, CentraleSupelec, Université Paris-Sud, CNRS, Univeristé Paris-Saclay
- IRIT - UMR 5505, ENSEEIHT, Toulouse

Supervisors :

- Direction: Rémy Boyer, Maitre de Conférences, HDR : remy.boyer@l2s.centralesupelec.fr
- Co-direction: Jean-Yves Tourneret, Professeur des Universités : jean-yves.tourneret@enseeiht.fr

Keywords: Mixed/hybrid parameters, sparse signals, statistical estimation, Bayesian inference, graphical models.

Consider a noisy model y=f(θ,ψ)+n where (θ,ψ) are vectors composed of discrete and continuous parameters, that we want to estimate based on the knowledge of the measurement vector y. The noise-free signal f(θ,ψ) models a mathematically ill-posed and non-identifiable system, meaning that several vectors of parameters of interest (θ,ψ) can generate the same function f(θ,ψ) and that there are more parameters of interest than the available number of measurements. In this difficult scenario, a convenient way to alleviate this uncertainty is to structure the vector θ using a sparse model. In this case, the vector θ consists of a set of binary entries and the unknown positions ofthe entries taking the value « 1 » are collected in a support set denoted by F. An estimator of (θ,ψ) has to face a twofold objective: estimate(i) the discrete positions contained in the support set F and (ii) the continuous amplitudes in vector ψ being associated with the support set F (as well as possible nuisance parameters contained in ψ). This type of problem is called hybrid (as it is defined with parameters taking discrete and continuous values) has received recent interest in several applications such as the change point detection [1], spectralanalysis for astrophysical data [2] or source localization in EEG signals [3]. However, currently, there is no general and satisfactory mathematical formalism to describe / predict the performance of an estimator of (θ,ψ) in the case of hybrid parameters. Note that the model can often be re-parametrized using quantities taking discrete and continuous values, for example leading to the Bernoulli-Gaussian models [1] [2]. Again, determining the estimation performance for the mixed parameters scenario is an issue that has received few attention in the literature (we can however mention the recent work of F. Xaver [4]).

It seems important to propose a new approach to describe and analyze the performance of hybrid or mixed parameter estimators in the framework of the theory of the lower bounds of the mean squared error (MSE) [7]. This is the scientific axis of the first part of this thesis. The second axis of this exploratory project will focus on the study of Bayesian inference methods adapted to the hybrid scenario. Such methods have already been the subject of several works in the literature [1] - [3]. However, they require the use of Monte Carlo methods which may result in heavy computational cost. This computational cost can be reduced by using recent work conducted on "message passing" methods. We believe that these methods can have a great interest for the problems considered in [1] - [3]. It is the second objective of this PhD thesis.

- Proposal for a new theory of MSE lower bounds integratingsparsity constraints and adapted to estimators of hybrid or mixed parameter estimators.
- Statistical analysis of "message passing" based estimators in the context of hybrid or mixed parameters for the change point detection problem. The methodology is based on the interpretation of the estimation problem based on a graphical model and the derivation of MAP or MMSE estimators using the "message passing" algorithm [5]-[6].

An outstanding, self and highly motivated candidate is sollicited with a strong interest in the field of advanced mathematical and statistical methods applied to signal processing. Candidates with

- a solid background in probability, statistics and estimation theory.
- excellent analytical, technical and problem solving skills,
- good written, oral communication and organization skills,

are encouraged to apply. Above all, the applicants must be motivated to learn quickly and work effectively on challenging research problems.

Please send your CV, transcripts of grades with qualifications and pertinent information, as soon as possible, to R. Boyer (remy.boyer@l2s.centralesupelec.fr).

[1] G. Kail, J.-Y. Tourneret, N. Dobigeon and F. Hlawatsch, "Blind Deconvolution of Sparse Pulse Sequences under a Minimum Distance Constraint: A Partially Collapsed Gibbs Sampler Method," IEEE Trans. Sig. Process., vol. 60, no. 6, pp. 2727-2743, June 2012.

[2] S. Bourguignon, H. Carfantan, and J. Idier, "A sparsity-based method for the estimation of spectral lines from irregularly sampled data, " IEEE Journal of Selected Topics in Signal Processing, Special Issue: Convex Optimization Methods for Signal Processing, vol. 1, no. 4, Dec. 2007.

[3] F. Costa, H. Batatia, L. Chaari and J.-Y. Tourneret, "Sparse EEG Source Localization Using Bernoulli Laplacian Priors", IEEE Trans. Biomed. Eng., vol. 62, no. 12, pp. 2888-2898, 2015.

[4] F. Xaver, "Mixed discrete-continuous Bayesian inference: censored measurements of sparse signals, " IEEE Trans. Sig. Process., vol. 63, no. 21, pp. 5609-5620, Sep. 2015.

[5] J. P. Vila and P. Schniter, "Expectation-maximisation Gaussian-mixture approximate message passing, " IEEE Trans. Sig. Process., vol. 61, no. 19, pp. 4658-4672, Oct. 2013.

[6] S. Rangan, "Generalized approximate message passing for estimation with random linear mixing," in Proc. IEEE Int. Symp. Inf. Theory, St. Petersburg, Russia, Aug. 2011, pp. 2168–2172 (see also Technical Report on ArXiv at http://arxiv.org/pdf/1010.5141.pdf).

[7] M. N. El korso, R. Boyer, P. Larzabal and B. H. Fleury, "Estimation Performance for the Bayesian Hierarchical Linear Model", IEEE Trans. Sig. Process., vol. 23, no. 4, pp. 488-492, 2016.

(c) GdR 720 ISIS - CNRS - 2011-2015.