Doctorant sous la direction de A. Renaux

Lucien Bacharach
Former Ph.D Student

Now with ISAE Supaéro, Dept. of Electronics, Optronics and Signals.

lucien (dot) bacharach (at) isae-supaero (dot) fr

Thesis Directors: Alexandre Renaux, Mohammed Nabil El Korso

Thesis title: Characterization of mean squared error fundamental limitations in parameter estimation of signals with change-points

Started on October 1st, 2015. Defended on September 28th, 2018.

Thesis abstract: This thesis deals with the study of estimation performance in signal processing, and focuses on the analysis of lower bounds on the Mean Squared Error (MSE) for abrupt change-point estimation. Those tools contribute to characterizing the estimation behavior for estimators such as the maximum likelihood estimator (in the frequentist context), as well as the maximum a posteriori and the conditional mean estimators (in the Bayesian context). The main difficulty comes from the fact that, when dealing with sampled signals, the parameters of interest (i.e., the change-point locations) lie on a discrete space. Consequently, the classical large sample theory results (e.g., the asymptotic normali ty of the maximum likelihood estimator) or the Cramér-Rao bound do not apply. Some results concerning the asymptotic distribution of the maximum likelihood estimator are available in the mathematical literature but are currently of limited interest for practical signal processing problems. Focusing on the 2nd-order moments of the estimators , lower bounds on the MSE make up essential tools, which ha ve been the subject of many studies in the last years. As a result, new inequalities have been proposed, leading to tighter lower bounds in comparison with the Cramér-Rao bound. These new lower bounds have less regularity conditions and enable the prediction of estimators' behavior in terms of MSE, both in asymptotic and non-asymptotic regimes. The goal of this thesis is to complete previous results on lower bounds in the asymptotic area (i.e. when the number of samples and/or the signal-to-noise ratio is high) for change-point estimation but, also, to provide an analysis in the non-asymptotic region. The tools used here are the lower bounds of the Weiss- Weinstein family, which are already known in signal processing to out perform the Cramér-Rao bound in applications such as spectral analysis or array processing. A closed-form expression of this fa mily is provided for a single and multiple change-points, and some extensions are given when the distribution parameters are unknown. An analysis in terms of robustness regarding the prior distribution on the change locations is also provided. Finally, we apply our results to specific problems, such as Gaussian data, Poisson data and exponentially distributed data.

Keywords: Estimation theory, performance bound analysis, Bayesian lower bound, change-point, mean square error

Short bio: I studied aeronautical engineering at ISAE-ENSICA (Toulouse, France), where I got graduated in 2014. Then I received the Master Research degree in Signal and Image Processing from Paris-Sud University (Orsay, France) in 2015. I am currently pursuing the Ph.D. degree in signal processing with Paris-Sud University, at Laboratory of Signals and Systems (L2S, Gif-sur-Yvette, France). My research interests and activities include estimation and detection theory in statistical signal processing, with an emphasis on performance analysis using tools such as lower bounds on the mean square error.


Journal papers

  • L. Bacharach, M. N. El Korso, A. Renaux and J.-Y. Tourneret, "A hybrid lower bound for parameter estimation of signals with multiple change-points", IEEE Transactions on Signal Processing, Volume: 67, Issue: 5, March 2019, pp. 1267-1279 [pdf]
  • L. Bacharach, A. Renaux, M. N. El Korso and E. Chaumette, "Weiss-Weinstein bound on multiple change-points estimation",  IEEE Transactions on Signal Processing, Volume: 65, Issue: 10,  May 2017, pp.  2686-2700 [pdf]

Conference papers

  • L. Bacharach, C. Fritsche, U. Orguner and E. Chaumette, "A tighter Bayesian Cramér-Rao bound", accepted, ICASSP 2019
  • A. Mian, L. Bacharach}, G. Ginolhac, A. Renaux, M. N. El Korso, J.-P. Ovarlez, ``Designing SAR Images Change-Point Estimation Strategies Using an MSE Lower Bound,'' accepted, ICASSP 2019.
  • L. Bacharach, M. N. El Korso and A. Renaux, "Prior influence on Weiss-Weinstein bounds for multiple change-point estimation", in Proc. of the 26th European Signal Processing Conference (EUSIPCO 2018), Rome, Italy, September 2018 [pdf]
  • L. Bacharach, G. Bibiche, A. Renaux and M. N. El Korso, "Bornes bayésiennes pour la localisation d'un point de rupture : Application à des processus exponentiels", in Proc. of Colloque GRETSI (GRETSI 2017), Juan-les-Pins, France, September 2017 [pdf]
  • L. Bacharach, M. N. El Korso, A. Renaux and J.-Y. Tourneret, "A Bayesian lower bound for parameter estimation of Poisson data including multiple changes", in Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2017), New Orleans, LA, USA, pp. 4486–4490 [pdf] (extended version with proofs: [pdf]).
  • L. Bacharach, M. N. El Korso and A. Renaux, "Weiss-Weinstein Bound for an unknown abrupt frequency change", in Proc. of the IEEE Workshop on Statistical Signal Processing (SSP 2016), Palma de Mallorca, Spain, June 2016, pp. 420–424 [pdf]
  • L. Bacharach, A. Renaux, M. N. El Korso and E. Chaumette, "Weiss-Weinstein bound for change-point estimation", in Proc. of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP 2015), Cancun, Mexico, December 2015, pp. 477–480 [pdf]

Administrative responsibility: former PhD Students delegate at the L2S Laboratory Council (from 2017 to 2018).

Teaching (in French):

Durant les trois années de ma thèse j'ai effectué une mission d'enseignement avec l'Université Paris-Sud : j'ai donné des cours (principalement des TD et des TP) dans la formation par apprentissage de Polytech Paris-Sud, pour les élèves de 3e et 4e année.

2015-2018 :

  • Électronique (3A Polytech Paris-Sud, formation par apprentissage) :
    • 48h. Travaux Pratiques : amplification, limitations d'un amplificateur opérationnel, filtrage analogique.
  • Langage Java (4A Polytech Paris-Sud, formation par apprentissage) :
    • 32h. Travaux Pratiques : introduction à la programmation orientée objet.
  • Mathématiques de spécialité (3A Polytech Paris-Sud, formation par apprentissage) :
    • 12h. Cours/Travaux Dirigés : rappels de mathématiques (développements limités, algèbre linéaire, matrices de rotations), introduction à l'algorithme CORDIC.
    • 20h. Travaux Pratiques : programmation du calcul de fonctions trigonométriques par l'algorithme CORDIC.
  • Microcontrôleurs (4A Polytech Paris-Sud, formation par apprentissage) :
    • 6h. Cours/Travaux Dirigés : introduction aux microcontrôleurs, étude de divers périphériques.
    • 48h. Travaux Pratiques : entrées/sorties tout ou rien, timers, convertisseur analogique/numérique, liaison série UART.
    • 28h. Projet : programmation de microcontrôleurs (famille dsPIC30F de Microchip®) pour le fonctionnement de robots hexapodes (entrées/sorties tout ou rien, timers, convertisseur analogique/numérique, liaison série UART, bus CAN).