CURRICULUM VITAE

Romain COUILLETProfesseur des Universites (HDR) a CentraleSupelecTitulaire de la chaire IDEX GSTATS a Univ. Grenoble-AlpesTitulaire de la chaire MIAI LargeDATA a Univ. Grenoble-Alpes

Ne le 18 mars 1983 (36 ans). Nationalite francaise.

Table des Matieres

Informations Generales 2Experience professionnelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Langues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Titres universitaires 2Diplomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Cursus anterieur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Activites d’enseignement 3Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Niveaux these et master . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Niveaux pre-master . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Encadrement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Organisation de cours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Activites Peripheriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Activites de Recherche 4Informations generales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Distinctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Projets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Activites communautaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Organisation de Conferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Encadrement de theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Encadrement de postdocs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Visites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Articles avec comite de relecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Tutoriaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Presentation Analytique des Travaux 15Domaines de recherche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Resume des activites de recherche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Articles avec comite de relecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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Informations Generales

Universite Grenoble–Alpes, Saint-Martin d’Heres, France.Experienceprofessionnelle Titulaire de la chaire IDEX en datascience GSTATS Avril 2018 - Present

Titulaire de la chaire MIAI LargeDATA Septembre 2019 - Present

• Recherche en theorie des matrices aleatoires appliquees a l’apprentissage engrandes dimensions.

• Enseignement dans les masters en science des donnees.

CentraleSupelec, Universite ParisSaclay, Gif sur Yvette, France.

Professeur des Universites Janvier 2016 - PresentEnseignant chercheur Janvier 2011 - Present

• Recherche en probabilites, statistiques, apprentissage, traıtement des donnees etdes signaux.

• Enseignement en cycle ingenieur, Master (SAR et MVA), these.

ST-Ericsson, Sophia Antipolis, France.

Ingenieur R&D, etudiant en these Septembre 2007 - Decembre 2010

• Recherche en theorie des matrices aleatoires.• Applications aux standards 4G et MIMO.

Francais (langue maternelle), anglais (courant), allemand (scolaire).Langues

Titres universitaires

Universite d’Orsay, Saclay, France. Janvier 2011 - Fevrier 2015Diplomes

HDR en Physique

• Titre : Methodes d’estimation robuste dans le regime des grandes matrices aleatoires• Jury : A. Hero, L. Pastur, J-Y. Tourneret (rapporteurs), F. Benaych-Georges, P.

Bondon, M. McKay, E. Ollila.

CentraleSupelec, Gif sur Yvette, France. Janvier 2008 - Novembre 2010

These, specialite Physique (Telecommunications), Novembre 2010

• Titre : Application de la theorie des matrices aleatoires aux futurs reseauxflexibles de communications sans fils

• Directeur de these : Merouane Debbah• Jury : P. Loubaton, X. Mestre (rapporteurs), M. Debbah, P. Duhamel, W. Ha-

chem, A. Moustakas, J. Silverstein.

EURECOM, Sophia Antipolis, France. Septembre 2005 - Juin 2007

Diplome d’ingenieur, Ingenierie des telecommunications, Septembre 2007

Communications mobiles, systemes embarques, informatique.

Telecom ParisTech, Paris, France. Septembre 2004 - Juin 2007

Master Science, Systemes de Communications, (mention TB), Mars 2008

Communications sans fils, traitement d’image, techniques de detection aveugles.

Lycee Louis le Grand, Paris, France. Septembre 2001 - Juin 2004Cursusanterieur Classe preparatoire aux grandes ecoles

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Activites d’enseignement

Service [2011-2018] Enseignant-chercheur temps-plein a CentraleSupelec ∼240h ETD

[2018-2020] Enseignant-chercheur (20%) a CS, chaire UGA (80%) ∼80h ETD

Niveaux theseet master

ENS ParisSaclay (Saclay, France) depuis 2013

• Random matrix theory and machine learning applications (Master MVA, coursmagistraux, 24h ETD)

Universite Grenoble-Alpes (Grenoble, France) depuis 2018

• Introduction to Convex Optimization Theory (Master SIGMA, cours magistraux,20h ETD)

• Introduction to Scientific Writing (Master SIGMA, cours magistraux, 9h ETD)• Scientific Writing and Associated Softwares (Niveau these, cours & TPs, 24h ETD)

CentraleSupelec (Gif sur Yvette, France) depuis 2011

• Techniques of scientific writing (Niveau these, cours et TP, 24h ETD)• Introduction to random matrix theory (Master SAR, seminaires, 18h ETD)• Theoretical foundations of flexible radio networks (Master SAR, seminaires, 18h ETD)

Niveauxpre-master

CentraleSupelec (Gif sur Yvette, France) 2011-2018

• Representation statistique des signaux (TD, 2×12h ETD)• Signaux et systemes (TD, 24h ETD)• Filtrages numerique et analogique (TP, 32h ETD)• Introduction a la redaction scientifique (electif, 18h ETD)

Encadrement ENS Paris Saclay depuis 2011

• Stages de master (4 a 6 mois, jusqu’a 4 etudiants/an)

CentraleSupelec (Gif sur Yvette, France) 2011-2018

• Projets longs de master (master SAR, projets de 3 mois, 2 etudiants/an)• Projets majeure Telecom (etudiants de 3e annee, projets de 3 mois, 2-3 etudiants/an)• Projets de conception (etudiants de 1ere annee, projets de 2 mois, 6 etudiants/an)• Projets de synthese (etudiants de 2e annee, projets de 2 mois, 4 etudiants/an)

Administration CentraleSupelec (Gif sur Yvette, France) 2015-2018

• Definition du nouveau programme ingenieur CentraleSupelec.• Participation au chantier “strategie et image de marque” de CentraleSupelec.

Organisation decours

Grenoble INP (Grenoble, France) depuis 2018

• Refonte du cours d’optimisation convexe (niveau master ; cours, TPs, examens)• Creation d’un cours de communication scientifique (niveau master et doctorat).

CentraleSupelec (Gif-sur-Yvette, France) depuis 2012

• Mise en place des 2 seminaires du master SAR (cours, examens)• Mise en place des modules de redaction scientifique

ENS ParisSaclay (Saclay, France) depuis 2013

• Mise en place du cours du master MVA (cours, TP, examen)

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ActivitesPeripheriques

Concours CPGE Centrale–Supelec 2016-2018

• Evaluateur au Concours Centrale–Supelec (80h)

Activites de Recherche

Sujets de recherche : statistique, theorie des matrices aleatoires, apprentissage au-Informationsgenerales tomatique, statistiques robustes, graphes, traitement des donnees et du signal, commu-

nications mobiles, finance statistique, applications marginales (traitement naturel dulangage, virologie, donnees medicales).

Resume des publications (chiffres Google Scholar de mars 2020)

Global 1 livre, 3 chapitres, 50+ revues, 85+ conferences, 6 brevets/idees innovantes.Citations 3400+ (cinq meilleures : 723, 542, 145, 110, 88)h-index 27i10-index 58

Medaille de bronze du CNRS. 2013Distinctions

Medaille de Bronze 2013 du CNRS dans la subsection INS2I.

Recompense mes travaux de jeune chercheur en statistiques, traitement des donnees,du signal et communications mobiles depuis 2008.

Prix de jeune chercheur IEEE ComSoc 2013

2013 IEEE ComSoc Outstanding Young Researcher Award for the EMEA Region

Recompense mes travaux de jeune chercheur en matrices aleatoires et communi-cations mobiles depuis 2008.

Prix de meilleure these. 2011

Prix EEA/GdR ISIS/GRETSI 2011 de la meilleure these 2010

Prix pour ma these de doctorat, “Application de la theorie des matrices aleatoiresaux futurs reseaux flexibles de communications sans fil”

Prix de meilleure these (d’un etudiant). 2020

Prix EEA/GdR ISIS/GRETSI 2020 de la meilleure these 2019

Prix pour ma these de doctorat, “Methodes des matrices aleatoires pour l’ap-prentissage en grande dimension”

Meilleur article etudiant. 2019Best student paper award de laconference EUSIPCO 2019Meilleur article etudiant. 2013

Second prix du 2012-2013 IEEE Australia Council Student Paper Contest

Meilleur article etudiant. 2011

Finaliste du Best Student Paper Award de la conference Asilomar 2011

Meilleur article etudiant. 2008

Meilleur article etudiant de la conference ValueTools 2008

Orateur en session pleniere. 2016

Conference ACM RACS 2016 a Odense, Danemark.Projets en

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Projetscours.

Projet Contribution Periode

ANR MIAI LargeDATA chair 50% (PI) 2019-2023IDEX GSTATS chair 100% (PI) 2018-2020ANR DARLING 50% (co-PI) 2019-2022HUAWEI RMT4AI 100% (PI) 2019-2021

Projets precedents.

Projet Contribution Periode

ANR RMT4GRAPH 100% (PI) 2014-2019ERC MORE 50% 2012-2017ParisSaclay RMT4ML 100% (PI) 2017-2020Fondation Supelec DeepRMT 100% (PI) 2017-2020Singapore MERLION 100% (PI) 2015-2017Mastodons AGADIR 20% 2017HUAWEI RMTin5G 100% (PI) 2015-2016ANR DIONISOS 25% (co-PI) 2012-2016ANR SESAME 20% 2008-2012FP7 NEWCOM# 10% 2012-2015FP7 NEWCOM++ 10% 2009-2011

Jury CRCN-INRIA depuis 2018Activitescommunautaires Membre du jury CRCN INRIA-GRA (centre de Grenoble)

en 2018, 2019, 2020 Agence Nationale pour la Recherche(ANR, France) depuis 2015

Relecteur de projets ANR (environ 2/an depuis 2015)Academy of Finland 2015

Evaluateur pour l’academie de Finlande (relecturede projets et comite sur 2 jours a Helsinki) IEEE(membre senior) depuis 2007

Editeur associe pour IEEE TSP (2015–2019)

GRETSI (membre) depuis 2011

Membre de l’association GRETSI (depuis 2011)

Organisateur des sessions speciales de la conferenceOrganisation deConferences IEEE CAMSAP 2019.

Technical Area Chair de la conference Asilo-mar 2016.

Organisateur de sessions speciales et journeesscientifiques :

• International : Session speciale “RandomMatrices in Signal Processing and MachineLearning” (IEEE SSP, 2016), Session speciale“Random Matrix Advances in Signal Pro-cessing” (IEEE SSP, 2014), Session speciale

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“Random Matrices and Applications” (IEEEAsilomar, 2013).

Theses en cours.Encadrement detheses Charles Sejourne 2020-2023

Encadrement depostdocs

Henrique Goulart depuis 2019

Jack W. Silverstein, Professeur a NorthCarolina State University, expert entheorie des matrices aleatoires.

Alfred O. Hero, Professeur a l’Uni-versity of Michigan, expert en sta-tistiques, science des donnees et trai-tement du signal.

Florent Benaych-Georges, Professeura l’Universite de Paris Descartes eta l’Ecole Polytechnique, expert entheorie des graphes et matrices aleatoires.

Gilles Wainrib, Professeur assistanta l’ENS Paris, expert en apprentis-sage et reseaux de neurones aleatoires.

Walid Hachem, DR CNRS a l’Uni-versite de Marne la Vallee, experten theorie des matrices aleatoires etapplications.

Merouane Debbah, Professeur at Cen-traleSupelec, expert en matrices aleatoireset communications mobiles.

Abla Kammoun, Research Scientista KAUST University, experte en ma-trices aleatoires, traitement du si-gnal et statistiques.

Hong-Kong University of ScienceVisitesand Technology, Hong-Kong. Juin2014

Collaboration avec le ProfesseurM. McKay,

Articles aveccomite derelecture

J1. C. Louart, R. Couillet “A Concen-tration of Measure Approachto Large Dimensional RobustStatistics”, (submitted to) An-nals of Applied Probability,2020.

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J2. K. Elkhalil, A. Kammoun,R. Couillet, T. Al-Naffouri,M-S. Alouini, “A Large Di-mensional Analysis of Regu-larized Discriminant Analy-sis Classifiers” (to appear) IEEETransactions on Signal Pro-cessing, 2020.

J3. L. Dall’Amico, R. Couillet,N. Tremblay, “A unified fra-mework for spectral cluste-ring in sparse graphs”, (sub-mitted to) Journal of MachineLearning Research, 2020.

J4. C. Louart, R. Couillet, “Concen-tration of Measure and LargeRandom Matrices with an ap-plication to Sample CovarianceMatrices”, (submitted to) Ran-dom Matrix Theory and Ap-plications, 2019.

J5. X. Mai, R. Couillet, “ConsistentSemi-Supervised Graph Re-gularization for High Dimen-sional Data”, (submitted) Jour-nal of Machine Learning Re-search, 2019.

J6. R. Couillet, M. Tiomoko, S.Zozor, E. Moisan, “Randommatrix-improved estimation ofcovariance matrix distances”,Journal of Multivariate Ana-lysis, vol. 174, pp. 104531, 2019.

J7. X. Mai, R. Couillet, “A Ran-dom Matrix Analysis and Im-provement of Semi-SupervisedLearning for Large Dimensio-nal Data”, Journal of MachineLearning Research, vol. 19,no. 79, pp. 1-27, 2018.

J8. A. Kammoun, R. Couillet,“Subspace Kernel Clusteringof Large Dimensional Data”(submitted to) Annals of Ap-plied Probability, 2017.

J9. L. Yang, M. McKay, R. Couillet,“High-Dimensional MVDR Beam-forming: Optimized Solutionsbased on Spiked Random Ma-

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trix Models”, IEEE Transac-tions on Signal Processing, vol.66, no. 1, pp. 1933-1947, 2018.

J10. A. Karadimitrakis, A. L. Mous-takas, R. Couillet, “GallagerBound for MIMO Channels:Large-N Asymptotics” IEEETransactions on Wireless Com-munications, vol. 17, no. 2,pp. 1323-1330, 2018.

J11. N. Auguin, D. Morales, M.McKay, R. Couillet, “Large-dimensional behavior of re-gularized Maronna’s M-estimatorsof covariance matrices” IEEETransactions on Signal Pro-cessing, vol. 66, no. 13, pp.3529–3542, 2018.

J12. C. Louart, Z. Liao, R. Couillet,“A Random Matrix Approachto Neural Networks” Annalsof Applied Probability, vol.28, no. 2, pp. 1190–1248, 2018.

J13. Z. Liao, R. Couillet, “A LargeDimensional Analysis of LeastSquare Support Vector Ma-chines” IEEE Transactions onSignal Processing, vol. 67, no.4, pp. 1065-1074, 2018.

J14. R. Couillet, H. Tiomoko Ali,“Improved spectral commu-nity detection in large hete-rogeneous networks” Journalof Machine Learning Resarch,vol. 18, no. 225, pp. 1–49, 2018.

J15. R. Couillet, M. McKay, “Op-timal block-sparse PCA forhigh dimensional correlatedsamples” (submitted to) Jour-nal of Multivariate Analysis,2016.

J16. R. Couillet, G. Wainrib, H.Sevi, H. Tiomoko Ali, “Theasymptotic performance of li-near echo state neural net-works” Journal of Machine Lear-ning Research, vol. 17, no. 178,pp. 1-35, 2016.

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J17. R. Couillet, F. Benaych-Georges,“Kernel Spectral Clusteringof Large Dimensional Data”Electronic Journal of Statis-tics, vol. 10, no. 1, pp. 1393-1454, 2016.

J18. F. Benaych-Georges, R. Couillet,“Spectral Analysis of the GramMatrix of Mixture Models”ESAIM : Probability and Sta-tistics, DOI http ://dx.doi.org/10.1051/ps/2016007,2016.

J19. R. Couillet, Estimation ro-buste et matrices aleatoires,revue Traitement du Signal,vol. 33, no. 2-3, pp. 273-320,2016.

J20. R. Couillet, G. Wainrib, Pers-pectives en matrices aleatoireset grands reseaux, revue Trai-tement du Signal, vol. 33, no.2-3, pp. 351-376, 2016.

J21. M. Sadeghi, L. Sanguinetti,R. Couillet, Y. Chau, “LargeSystem Analysis of Power Nor-malization Techniques in Mas-sive MIMO”, IEEE Transac-tions on Vehicular Technolo-gies, vol. 66, no. 10, pp. 9005-9017, 2017.

J22. M. Sadeghi, L. Sanguinetti,R. Couillet, Y. Chau, “Redu-cing the Computational Com-plexity of Multicasting in Large-Scale Antenna Systems”, IEEETransactions on Wireless Com-munications, vol. 16, no. 5,pp. 2963-2975, 2017.

J23. L. Sanguinetti, R. Couillet,M. Debbah, “Large SystemAnalysis of Base Station Co-operation for Power Minimi-zation” IEEE Transactions onWireless Communications, vol.15, no. 8, pp. 5480-5496, 2016.

J24. A. Abboud, F. Iutzeler, R.Couillet, H. Siguerdidjane, M.Debbah, “Distributed Production-Sharing Optimization and Ap-

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plication to Power Grid Net-works,” IEEE Transactions onSignal and Information Pro-cessing over Networks, vol. 2,no. 1, pp. 1628, 2016.

J25. A. Kammoun, R. Couillet,F. Pascal, M.-S. Alouini, “Op-timal Design of the AdaptiveNormalized Matched Filter De-tector using Regularized Ty-ler Estimator” IEEE Transac-tions on Aerospace and Elec-tronic Systems, vol. 54, no. 2,pp. 755–769, 2018.

J26. A. Kammoun, R. Couillet,F. Pascal, M.-S. Alouini, “Conver-gence and Fluctuations of Re-gularized Tyler Estimators”IEEE Transactions on SignalProcessing, vol. 64, no. 4, pp.1048-1060, 2016.

J27. D. Morales-Jimenez, R. Couillet,M. McKay, “Large Dimensio-nal Analysis of Robust M-Estimatorsof Covariance with Outliers”IEEE Transactions on SignalProcessing, vol. 63, no. 21,pp. 5784-5797, 2015.

J28. L. Yang, R. Couillet, M. McKay,“A Robust Statistics Approachto Minimum Variance Port-folio Optimization” IEEE Tran-sactions on Signal Processing,vol. 63, no. 24, pp. 6684–6697,2015.

J29. R. Couillet, A. Kammoun,F. Pascal, “Second order sta-tistics of robust estimators ofscatter. Application to GLRTdetection for elliptical signals”Elsevier Journal of Multiva-riate Analysis, vol. 143, pp.249-274, 2015.

J30. A. Muller, R. Couillet, E.Bjørnson, S. Wagner, M. Deb-bah, “Interference-Aware RZFPrecoding for Multi-Cell Down-link Systems” IEEE Transac-tions on Signal Processing, vol.63, no. 15, pp. 3959-3973 2015.

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J31. R. Couillet, “Robust spikedrandom matrices and a ro-bust G-MUSIC estimator” El-sevier Journal of MultivariateAnalysis, vol. 140, pp. 139-161, 2015.

J32. R. Couillet, M. McKay, “LargeDimensional Analysis and Op-timization of Robust Shrin-kage Covariance Matrix Es-timators” Elsevier Journal ofMultivariate Analysis, vol. 131,pp. 99-120, 2014.

J33. Y. Chitour, R. Couillet, F.Pascal “On the convergenceof Maronna’s M-estimators ofscatter” IEEE Signal Proces-sing Letters, vol. 22, no. 6,pp. 709-712, 2014.

J34. R. Couillet, F. Pascal, J.W. Silverstein, “The RandomMatrix Regime of Maronna’sM-estimator with ellipticallydistributed samples”, vol. 139,pp. 56-78, Elsevier Journal ofMultivariate Analysis, 2015.

J35. J. Vinogradova, R. Couillet,W. Hachem, “Estimation ofToeplitz covariance matricesin large dimensional regimewith application to source de-tection large”, IEEE Transac-tions on Signal Processing, vol.63, no. 18, pp. 4903-4913, 2015.

J36. R. Couillet, W. Hachem, “Ana-lysis of the limiting spectralmeasure of large random ma-trices of the separable cova-riance type”, Random MatrixTheory and Applications, vol.3, pp. 1-23, 2014.

J37. J. Hoydis, R. Couillet, P.Piantanida, “The Second-OrderCoding Rate of the MIMORayleigh Block-Fading Chan-nel,” IEEE Transactions onInformation Theory, vol. 61,no. 12, pp. 6591-6622, 2015.

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J38. J. Vinogradova, R. Couillet,W. Hachem, “Statistical In-ference in Large Antenna Ar-rays under Unknown Noise Pat-tern,” IEEE Transactions onSignal Processing, vol. 61, no.22, pp. 5633-5645, 2013.

J39. F. Chapon, R. Couillet, W.Hachem, X. Mestre, “The out-liers among the singular va-lues of large rectangular ran-dom matrices with additivefixed rank deformation,” Mar-kov Processes and Related Fields,vol. 20, pp. 183-228, 2014.

J40. R. Couillet, F. Pascal, J.W. Silverstein, “Robust Es-timates of Covariance Matricesin the Large Dimensional Re-gime,” IEEE Transactions onInformation Theory, vol. 60,no. 11, 2014.

J41. G. Geraci, R. Couillet, J.Yuan, M. Debbah, I. B. Col-lings, “Large System Analy-sis of Linear Precoding in MISOBroadcast Channels with Confi-dential Messages,” IEEE Jour-nal on Selected Area in Com-munications, vol. 31, no. 9,pp. 1660-1671, 2013. Secondprize of the 2012-2013 IEEEAustralia Council StudentPaper Contest.

J42. J. Hoydis, R. Couillet, M.Debbah, “Iterative Determi-nistic Equivalents for the Ca-pacity Analysis of Commu-nication Systems,” TechnicalReport.

J43. R. Couillet, S. Medina Per-laza, H. Tembine, M. Deb-bah, “Electrical Vehicles inthe Smart Grid: A Mean FieldGame Analysis,” IEEE Jour-nal on Selected Areas in Com-munications : Smart Grid Com-munications Series, vol. 30,no. 6, pp. 1086-1096, 2012.

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J44. J. Yao, R. Couillet, J. Na-jim, M. Debbah, “Fluctuationsof an Improved Population Ei-genvalue Estimator in SampleCovariance Matrix Models,”IEEE Transactions on Infor-mation Theory, vol. 59, no.2, pp. 1149-1163, 2013.

J45. R. Couillet, M. Debbah,“SignalProcessing in Large Systems:a New Paradigm,” IEEE Si-gnal Processing Magazine, vol.30, no. 1, pp. 24-39, 2013.

J46. R. Couillet, W. Hachem,“Fluctuationsof spiked random matrix mo-dels and failure diagnosis insensor networks,” IEEE Tran-sactions on Information Theory,vol. 59, no. 1, pp. 509-525,2013.

J47. A. Kammoun, R. Couillet,J. Najim, M. Debbah,“Performanceof capacity inference methodsunder colored interference,”IEEE Transactions on Infor-mation Theory, vol. 59, no.2, pp. 1129-1148, 2013.

J48. R. Couillet, J. Hoydis, M.Debbah, “Random beamfor-ming over quasi-static and fa-ding channels: A determinis-tic equivalent approach,” IEEETransactions on InformationTheory, vol. 58, no. 10, pp.6392-6425, 2012.

J49. S. Wagner, R. Couillet, M.Debbah, D. T. M. Slock, “LargeSystem Analysis of Linear Pre-coding in MISO Broadcast Chan-nels with Limited Feedback”,IEEE Transactions on Infor-mation Theory, vol. 58, no.7, pp. 4509-4537, 2012.

J50. R. Couillet, J. W. Silver-stein, Z. Bai, M. Debbah, “Eigen-Inference for Energy Estima-tion of Multiple Sources”, IEEETransactions on InformationTheory, vol. 57, no. 4, pp. 2420-2439, 2011.

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J51. R. Couillet, M. Debbah, J.W. Silverstein, “A Determi-nistic Equivalent for the Ana-lysis of Correlated MIMO Mul-tiple Access Channels”, IEEETransactions on InformationTheory, vol. 57, no. 6, pp. 3493-3514, 2011.

J52. R. Couillet, M. Debbah, “ABayesian Framework for Col-laborative Multi-Source SignalSensing”, IEEE Transactionson Signal Processing, vol. 58,no. 10, pp. 5186-5195, 2010.

J53. R. Couillet, A. Ancora, M.Debbah, “Bayesian Founda-tions of Channel Estimationfor Cognitive Radios”, Advancesin Electronics and Telecom-munications, vol. 1, no. 1, pp.41-49, 2010.

J54. R. Couillet, M. Debbah,“Letelephone du futur : plus in-telligent pour une exploita-tion optimale des frequences”Revue de l’Electricite et del’Electronique, no. 6, pp. 71-83, 2010.

J55. R. Couillet, M. Debbah, “Ma-thematical foundations of cog-nitive radios”, Journal of Te-lecommunications and Infor-mation Technologies, no. 4,2009.

J56. R. Couillet, M. Debbah, “Ou-tage performance of flexibleOFDM schemes in packet-switchedtransmissions”, Eurasip Jour-nal on Advances on Signal Pro-cessing, Volume 2009, ArticleID 698417, 2009.

B2. R. Couillet, M. Debbah, Ma-thematical Foundations forSignal Processing, Commu-nications and Networking, CRCPress, Taylor & Francis Group,2011 [chapitre de livre]

Chapitre “Random matrix theo-ry” sur les matrices aleatoires

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et plus precisement les methodesd’inference statistique.

B3. R. Couillet, M. Debbah, Or-thogonal Frequency Divi-sion Multiple Access Fun-damentals and Applications,Auerbach Publications, CRCPress, Taylor & Francis Group,2010 [chapitre de livre]

Chapitre “Fundamentals ofOFDMA Synchronization”sur les considerations theoriqueset les outils appliques ensynchronisation pour l’OFDMet l’OFDMA.

• R. Couillet, M. Debbah,No. 08368023.1 “Methodfor short-time OFDM trans-mission and apparatus forperforming flexible OFDMmodulation”

• R. Couillet, S. Wagner, No.09368025.4 “Precoding pro-cess for a transmitter of aMU-MIMO communicationsystem”

• R. Couillet, No. 09368030.4“Process for estimating thechannel in an OFDM com-munication system, and re-ceiver for doing the same”

II1. R. Couillet, Idee In-novante “User SubspaceClustering”

Tutoriaux T1. R. Couillet, M. Debbah,“Eigen-Inference Statis-tical methods for Cogni-tive Radio”, European Wi-reless, Lucca, Italy, 2010.

PresentationAnalytique des

Travaux

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Mathematiques appliquees :Domaines derecherche • theorie des matrices aleatoires :

modeles hermitiens, modelesspiked, modeles non stan-dard a entrees non independantesou non lineaires, matricesa noyaux, etc.

• mathematiques : probabi-lites, analyse complexe, algebre

• statistique : estimation ro-buste, methodes a noyau,analyse en composantes prin-cipales, etc.

Graphes et apprentissageautomatise :

• graphes : methodes spec-trales de detection de com-munaute

• reseaux de neurones : reseaux“echo-state”, “extreme lear-ning machines” et “ran-dom feature maps”, reseauxprofonds, retropropagationdu gradient

• apprentissage automatise :methodes spectrales a noyaux(algorithmes type Ng–Weiss–Jordan), support vector ma-chines, methodes semi-supervisees

Traitement du signal :

• traitement d’antennes : detectionet estimation, methodes sous-espaces, estimation et regressionrobustes

• reseaux de capteurs : detectionde changements d’etat, al-gorithmes distribues

Telecommunications mo-biles :

• traitement des signaux etdonnees : inference statis-tique pour les reseaux cog-nitifs, classification non su-pervisee d’utilisateurs

• theorie de l’information :performances de systemesMIMO larges, communi-cations MIMO a paquetscourts

• communications mobiles :

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design de precodeurs mas-sive MIMO, multi-cellulaires,systemes realistes

Finances statistiques :

• optimisation de porte-feuilles :probleme de Markowtiz etses derives en grandes di-mensions

• inference de series tem-porelles financieres : modelesde reseaux de neurones recurrents(echo-state) pour les donneesfinancieres

• apprentissage et series fi-nancieres : methodes declassification et regroupe-ment d’actifs par methodesa noyaux modernes.

Mobile communications :

• information theory : largeMIMO system performance,finite blocklength commu-nications

Mes activites de rechercheResume desactivites derecherche

sont centrees sur l’analysetheorique des performanceset l’amelioration d’algorithmeset methodes pour les systemesde grandes dimensions. L’ou-til au cœur de ma rechercheest la theorie des matricesaleatoires, et plus generalementles probabilites, l’analysecomplexe et l’algebre pourdes vecteurs et matricesde grandes dimensions.

Chronologiquement, moninteret s’est d’abord portesur l’analyse des perfor-mances de systemes com-plexes de communicationsmobiles (de 2007 a 2012principalement). Parmi mestravaux notables, (J ?( ?) ?51)etablit pour la premierefois la region de capaciteergodique d’un systeme multi-utilisateur MIMO en voiemontante, grace a des resultats

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nouveaux en theorie desmatrices aleatoires. Les per-formances de precodeurslineaires de ce meme systememais en voie descendanteont ete ensuite etudiees fi-nement et sous des hypothesesrealistes dans (J ?( ?) ?49),qui generalise a cette oc-casion de nombreux tra-vaux. D’autres travaux, tech-niquement plus pousses,etendent ces resultats, no-tamment dans (J ?( ?) ?48,J ?( ?) ?42).Un travail bien plus recentet techniquement plus ardu(J ?( ?) ?37) analyse les pro-babilites d’erreurs asymp-totiques d’un systeme MIMOdans le cas de communi-cations a paquets courts.Sur la periode 2010–2013,l’interet de mes recherchess’est deplace vers le trai-tement du signal pour lescommunications mobiles,avec en particulier des ar-ticles sur l’estimation ra-pide de debits et positionsd’utilisateurs dans un contextede radio intelligente (J ?( ?) ?52,J ?( ?) ?50,J ?( ?) ?47,J ?( ?) ?44).Tous ces travaux, effectuesdans le cadre de ma theseet au-dela, ont notammentete menes en collaborationavec Jack Silverstein et Zhi-dong Bai, professeurs enmathematiques et expertsen theorie des matrices aleatoires.

Suite a cette premiere vaguede travaux, restreints mathematiquementa l’analyse de fonctionnellesdu spectre de matrices deGram, j’ai deplace mes centresd’interets vers l’analyse sta-tistique de grands systemespour des applications autraitement du signal, etnotamment au traitementd’antennes. Ici on s’interesse,non plus a des fonction-nelles de spectres, mais ades couples valeurs-vecteurs

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propres isoles dans le spectre(modeles dits spiked). Mestravaux dans cette direc-tion ont permis d’etablirdes resultats mathematiquementplus generaux pour des modelesen traitement d’antennesplus realistes, comme enparticulier les travaux (J ?( ?) ?38,J ?( ?) ?39,J ?( ?) ?35)sur les modeles avec correlationspatiale et temporelle ouencore (J ?( ?) ?46) dans uncontexte de reseau de cap-teurs.

Neanmoins, toutes ces etudes,aussi utiles et technique-ment poussees soient-elles,venaient naturellement ala suite d’etudes prealables.Il m’a alors tenu a cœurd’attaquer des problemesnouveaux qui requierentune plus grande ouvertured’esprit. C’est ainsi quej’ai engage, sur une periodede trois ans, l’etude pu-rement statistique d’esti-mateurs robustes dans leregime des grands systemes.Les objets et outils mathematiquesimpliques dans ce travaildifferent fortement des tech-niques de matrices aleatoiresclassiques qui ne sont pasadaptables ici. Parmi lesresultats marquants, les ar-ticles successifs (J ?( ?) ?40,J ?( ?) ?34,J ?( ?) ?32,J ?( ?) ?29)ont permis d’etablir unecomprehension fine du com-portement de ces estima-teurs et d’en deduire unemethode generique pourune utilisation amelioreedans les problemes de grandestailles ; des applications specifiquesau traitement d’antennes(J ?( ?) ?31) ou encore al’analyse robuste de donneesfinancieres (J ?( ?) ?28) enont decoule. Une grandepartie de ces travaux a etemenee conjointement avecMatthew McKay et Frederic

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Pascal, professeurs specialisesen traitement du signal.L’ensemble des travaux men-tionnes jusqu’ici a donnelieu a l’ecriture du livre(B??), a l’obtention desbrevets (P??–P ?( ?) ? ?) eta ete recompense du prixIEEE Outstanding YoungResearcher Award et dela medaille de Bronze duCNRS en 2013. Ces resultatsont egalement constitue lecœur de ma these d’HDRavec pour rapporteurs lesprofesseurs Leonid Pastur,mathematicien expert enmatrices aleatoires, ainsique les professeurs Jean-Yves Tourneret et AlfredHero tout deux experts mon-diaux en traitement desdonnees et signaux et enstatistique.

Aujourd’hui, l’interet crois-sant porte par les methodesd’apprentissage automatisepour le traitement des grandesdonnees (BigData) motivel’analyse et l’ameliorationde ces methodes dans leregime des grandes matricesaleatoires. Dans le cadredu projet ANR Jeunes Cher-cheurs (ANR RMT4GRAPH)que je dirige, nous etudionsaujourd’hui au sein de monequipe (composee d’une di-zaine de doctorants, post-doctorants et stagiaires)les algorithmes non lineaires(a base de noyaux) de clas-sification supervisee, semi-supervisee, ou non super-visee, les reseaux de neu-rones aleatoires non lineaires,recurrents ou non, profondsou non, ainsi que l’inferencesur graphes. Nos resultatsdepuis 2014 sont extremementprometteurs et permettenten particulier une meilleurecomprehension des methodes

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de classification non su-pervisee a noyaux (J ?( ?) ?17,J ?( ?) ?18),de classification semi-supervisee(J ?( ?) ?7,J ?( ?) ?5), a l’ap-prentissage par transfert(C??), des approches ap-parentees au SVM (J ?( ?) ?13),des methodes de detectionde communaute sur graphes(J ?( ?) ?14) utilisant en par-ticulier des outils puissantsde la physique statistique(C??), mais egalement per-mettent des avancees nou-velles dans l’etude des reseauxde neurones grace au developpementd’outils novateurs dans cedomaine (J ?( ?) ?12). Demaniere plus fondamentale,la plupart de ces resultatssont demontres valables pourdes modeles de donneeset algorithmes extremementrealistes, notamment gracea l’idee novatrice d’une ex-tension de la theorie desmatrices aleatoires a l’aidede l’outil de la concentra-tion de la mesure (J ?( ?) ?4,C??,C??).Ces travaux ont par ailleursdonne lieu a des idees in-novantes pratiques developpeesau cours de projets indus-triels (I ?( ?) ?1).L’objectif a moyen et longtermes de ces etudes estde developper de nouvellesmethodes d’apprentissagemieux adaptees aux grandesdonnees ainsi qu’un nou-veau paradigme d’analyseet d’amelioration des per-formances de ces methodesau moyen de la theorie desgrandes matrices aleatoires.

Une description detailleede tous les articles publiesdepuis 2007 est presenteeci-apres.

Articles aveccomite derelecture

J1. C. Louart, R. Couillet“A Concentration of Mea-

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sure Approach to LargeDimensional Robust Sta-tistics”, (submitted to)Annals of Applied Pro-bability, 2020.

Abstract. This articlestudies the robust co-variance matrix estima-tion of a data collec-tion X = (x1, ..., xn)with xi = τizi + m,where zi ∈ Rp is a concen-trated vector (e.g., anelliptical random vec-tor), m ∈ Rp a deter-ministic signal and τi ∈R a scalar perturba-tion of possibly largeamplitude, under theassumption where bothn and p are large. Thisestimator is defined asthe fixed point of a func-tion which we show iscontracting for a so-called stable semi-metric.We exploit this semi-metric along with concen-tration of measure ar-guments to prove theexistence and unique-ness of the robust es-timator as well as eva-luate its limiting spec-tral distribution.

J2. K. Elkhalil, A. Kam-moun, R. Couillet, T.Al-Naffouri, M-S. Alouini,“A Large DimensionalAnalysis of Regulari-zed Discriminant Ana-lysis Classifiers” (to ap-pear) IEEE Transac-tions on Signal Proces-sing, 2020.

Abstract. Abstract—Inthis paper, we conducta large dimensional studyof regularized discrimi-nant analysis classifierswith its two popularvariants known as re-gularized LDA and re-

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gularized QDA. The ana-lysis is based on theassumption that the datasamples are drawn froma Gaussian mixture mo-del with different meansand covariances and re-lies on tools from ran-dom matrix theory (RMT).We consider the regimein which both the datadimension and trainingsize within each classtends to infinity withfixed ratio. Under mildassumptions, we showthat the probability ofmisclassification convergesto a deterministic quan-tity that describes inclosed form the perfor-mance of these classi-fiers in terms of theclass statistics as wellas the problem dimen-sion. The result allowsfor a better understan-ding of the underlyingclassification algorithmsin terms of their per-formances in practicallarge but finite dimen-sions. Further exploi-tation of the results per-mits to optimally tunethe regularization pa-rameter with the aimof minimizing the pro-bability of misclassifi-cation. The analysis isvalidated with nume-rical results involvingsynthetic as well as realdata from the USPSdataset yielding a highaccuracy in predictingthe performances andhence making an in-teresting connection bet-ween theory and prac-tice.

J3. L. Dall’Amico, R. Couillet,N. Tremblay, “A uni-

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fied framework for spec-tral clustering in sparsegraphs”, (submitted to)Journal of Machine Lear-ning Research, 2020.

Abstract. This articleconsiders spectral com-munity detection in theregime of sparse net-works with heteroge-neous degree distribu-tions, for which we de-vise an algorithm toefficiently retrieve com-munities. Specifically,we demonstrate thata conveniently parame-trized form of regula-rized Laplacian matrixcan be used to performspectral clustering insparse networks, withoutsuffering from its de-gree heterogeneity. Be-sides, we exhibit im-portant connections bet-ween this proposed ma-trix and the now po-pular non-backtrackingmatrix, the Bethe-Hessianmatrix, as well as thestandard Laplacian ma-trix. Interestingly, asopposed to competitivemethods, our proposedimproved parametriza-tion inherently accountsfor the hardness of theclassification problem.These findings are sum-marized under the formof an algorithm capableof both estimating thenumber of communi-ties and achieving high-quality community re-construction.

J4. C. Louart, R. Couillet,“Concentration of Mea-sure and Large Ran-dom Matrices with anapplication to SampleCovariance Matrices”,

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(submitted to) RandomMatrix Theory and Ap-plications, 2019.

Abstract. The presentwork provides an ori-ginal framework for ran-dom matrix analysis ba-sed on revisiting theconcentration of mea-sure theory from a pro-babilistic point of view.By providing variousnotions of vector concen-tration (q-exponential,linear, Lipschitz, convex),a set of elementary toolsis laid out that allowsfor the immediate ex-tension of classical re-sults from random ma-trix theory involving ran-dom concentrated vec-tors in place of vectorswith independent en-tries. These findings areexemplified here in thecontext of sample co-variance matrices butfind a large range ofapplications in statis-tical learning and beyond,thanks to the broad adap-tability of our hypo-theses.

J5. X. Mai, R. Couillet,“Consistent Semi-SupervisedGraph Regularizationfor High DimensionalData”, (submitted) Jour-nal of Machine Lear-ning Research, 2019.

Abstract. Semi-supervisedLaplacian regularization,a standard graph-basedapproach for learningfrom both labelled andunlabelled data, is de-monstrated by the recentwork of (Mai and Couillet,2017) to have an insi-gnificant high dimen-sional learning efficiencywith respect to unla-

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belled data, causing itto be outperformed byits unsupervised coun-terpart, spectral clus-tering, given sufficientunlabelled data. Fol-lowing a detailed dis-cussion on the originof this inconsistency pro-blem, a novel regula-rization approach is pro-posed as solution, whichis shown both theore-tically and empiricallyto have a superior per-formance over Lapla-cian regularization.

J6. R. Couillet, M. Tio-moko, S. Zozor, E. Moi-san, “Random matrix-improved estimation ofcovariance matrix dis-tances”, Journal of Mul-tivariate Analysis, vol.174, pp. 104531, 2019.

Abstract. Given two setsx(1)1 , . . . , x

(1)n1 and x

(2)1 , . . . , x

(2)n2 ∈

Rp (or Cp) of randomvectors with zero meanand positive definite co-variance matrices C1

and C2 ∈ Rp×p (or Cp×p),respectively, this articleprovides novel estima-tors for a wide rangeof distances between C1

and C2 (along with di-vergences between somezero mean and cova-riance C1 or C2 pro-bability measures) ofthe form 1

p

∑ni=1 f(λi(C

−11 C2))

(with λi(X) the eigen-values of matrix X).These estimators arederived using recent ad-vances in the field ofrandom matrix theoryand are asymptoticallyconsistent as n1, n2, p→∞ with non trivial ra-tios p/n1 < 1 and p/n2 <1 (the case p/n2 > 1

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is also discussed). A first“generic” estimator, va-lid for a large set off functions, is provi-ded under the form ofa complex integral. Then,for a selected set of f ’sof practical interest (na-mely, f(t) = t, f(t) =log(t), f(t) = log(1 +st) and f(t) = log2(t)),a closed-form expres-sion is provided. Be-side theoretical findings,simulation results sug-gest an outstanding per-formance advantage forthe proposed estima-tors when compared tothe classical “plug-in”estimator 1

p

∑ni=1 f(λi(C

−11 C2))

(with Ca = 1na

∑na

i=1 x(a)i x

(a)Ti ),

and this even for verysmall values of n1, n2, p.

J7. X. Mai, R. Couillet,“A Random Matrix Ana-lysis and Improvementof Semi-Supervised Lear-ning for Large Dimen-sional Data”, Journalof Machine Learning Re-search, vol. 19, no. 79,pp. 1-27, 2018.

Abstract. This articleprovides an original un-derstanding of the be-havior of a class of graph-oriented semi-supervisedlearning algorithms inthe limit of large andnumerous data. It isdemonstrated that theintuition at the rootof these methods col-lapses in this limit andthat, as a result, mostof them become incon-sistent. Corrective mea-sures and a new data-driven parametrizationscheme are proposed alongwith a theoretical ana-

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lysis of the asympto-tic performances of theresulting approach. Asurprisingly close be-havior between theo-retical performances onGaussian mixture mo-dels and on real data-sets is also illustratedthroughout the article,thereby suggesting theimportance of the pro-posed analysis for dea-ling with practical data.As a result, significantperformance gains areobserved on practicaldata classification usingthe proposed parame-trization.

J8. A. Kammoun, R. Couillet,“Subspace Kernel Clus-tering of Large Dimen-sional Data” (submit-ted to) Annals of Ap-plied Probability, 2017.

Abstract. Let x1, . . . , xnbe independent obser-vations of size p, eachof them belonging toone of c distinct classes.We assume that obser-vations within class aare characterized by theirdistributionN (0, 1pCa)where here C1, . . . , Ccare some non-negativedefinite p×pmatrices.This paper studies theasymptotic behavior ofthe symmetric matrixΦkl =

√p((xTk xl)

2δk 6=l)when p and n grow toinfinity with n/p→ c0.Particularly, we provethat, if the class co-variance matrices aresufficiently close in acertain sense, the ma-trix Φ behaves as a low-rank perturbation of aWigner matrix, presen-ting possibly some iso-

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lated eigenvalues out-side the bulk of the semi-circular law. We carryout a careful analysisof some of the isolatedeigenvalues and eigen-vectors of matrix Φ, andillustrate how these re-sults can help unders-tand spectral clusteringmethods that use Φ asa kernel matrix.

J9. L. Yang, M. McKay,R. Couillet, “High-DimensionalMVDR Beamforming:Optimized Solutions ba-sed on Spiked RandomMatrix Models”, IEEETransactions on SignalProcessing, vol. 66, no.1, pp. 1933-1947, 2018.

Abstract. Minimum va-riance distortionless res-ponse (MVDR) beam-forming (or Capon beam-forming) is among themost popular adaptivearray processing stra-tegies due to its abi-lity to provide noise re-silience while nullingout interferers. A prac-tical challenge with thisbeamformer is that itinvolves the inverse co-variance matrix of thereceived signals, whichmust be estimated fromdata. Under modern high-dimensional applications,it is well-known thatclassical estimators canbe severely affected bysampling noise, whichcompromises beamfor-mer performance. Herewe propose a new ap-proach to MVDR beam-forming which is sui-ted to high-dimensionalsettings. In particular,by drawing an analogywith the MVDR pro-

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blem and the so-called“spiked models” in ran-dom matrix theory, wepropose robust beam-forming solutions whichare shown to outper-form classical approaches(e.g., matched filters andsample matrix inver-sion techniques), as wellas more robust solu-tions, such as methodsbased on diagonal loa-ding. The key to ourmethod is the designof an optimized inversecovariance estimator whichapplies eigenvalue clip-ping and shrinkage func-tions that are tailoredto the MVDR appli-cation. Our proposedMVDR solution is simple,in closed form, and easyto implement.

J10. A. Karadimitrakis, A.L. Moustakas, R. Couillet,“Gallager Bound for MIMOChannels: Large-N Asymp-totics” IEEE Transac-tions on Wireless Com-munications, vol. 17,no. 2, pp. 1323-1330,2018.

Abstract. The use ofmultiple antenna arraysin transmission and re-ception has become anintegral part of modernwireless communications.To quantify the per-formance of such sys-tems, the evaluation ofbounds on the error pro-bability of realistic fi-nite length codewordsis important. In thispaper, we analyze thestandard Gallager er-ror bound for both constraintsof maximum averagepower and maximuminstantaneous power. Ap-

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plying techniques fromrandom matrix theory,we obtain analytic ex-pressions of the errorexponent when the lengthof the codeword increasesto infinity at a fixedratio with the antennaarray dimensions. Ana-lyzing its behavior atrates close to the er-godic rate, we find thatthe Gallager error boundbecomes asymptoticallyclose to an upper er-ror bound obtained re-cently by Hoydis et al.2015. We also obtainan expression for theGallager exponent inthe case when the co-delength spans severalRayleigh fading blocks,hence taking into ac-count the situation whenthe channel varies du-ring each transmission.

J11. N. Auguin, D. Mo-rales, M. McKay, R.Couillet, “Large-dimensionalbehavior of regularizedMaronna’s M-estimatorsof covariance matrices”IEEE Transactions onSignal Processing, vol.66, no. 13, pp. 3529–3542, 2018.

Abstract. Robust esti-mators of large cova-riance matrices are consi-dered, comprising re-gularized (linear shrin-kage) modifications ofMaronna’s classical M-estimators. These es-timators provide robust-ness to outliers, whilesimultaneously being well-defined when the num-ber of samples does notexceed the number ofvariables. By applying

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tools from random ma-trix theory, we charac-terize the asymptoticperformance of such es-timators when the num-bers of samples and va-riables grow large to-gether. In particular,our results show that,when outliers are ab-sent, many estimatorsof the regularized-Maronnatype share the same asymp-totic performance, andfor these estimators wepresent a data-drivenmethod for choosing theasymptotically optimalregularization parame-ter with respect to aquadratic loss. Robust-ness in the presence ofoutliers is then studied :in the non-regularizedcase, a large-dimensionalrobustness metric is pro-posed, and explicitlycomputed for two par-ticular types of estima-tors, exhibiting inter-esting differences de-pending on the under-lying contamination mo-del. The impact of out-liers in regularized es-timators is then stu-died, with remarkabledifferences with respectto the non-regularizedcase, leading to new prac-tical insights on the choiceof particular estimators.

J12. C. Louart, Z. Liao,R. Couillet, “A Ran-dom Matrix Approachto Neural Networks”Annals of Applied Pro-bability, vol. 28, no. 2,pp. 1190–1248, 2018.

Abstract. This articlestudies the Gram ran-dom matrix modelG =

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1T ΣTΣ, Σ = σ(WX),classically found in theanalysis of random fea-ture maps and randomneural networks, whereX = [x1, . . . , xT ] ∈ Rp×Tis a (data) matrix ofbounded norm, W ∈Rn×p is a matrix of in-dependent zero-meanunit variance entries,and σ : R → R is aLipschitz continuous (ac-tivation) function — σ(WX)being understood entry-wise. We prove that,as n, p, T grow large atthe same rate, the re-solventQ = (G+γIT )−1,for γ > 0, has a si-milar behavior as thatmet in sample covariancematrix models, invol-ving notably the mo-ment Φ = T

nE[G], whichprovides in passing adeterministic equivalentfor the empirical spec-tral measure ofG. Thisresult, established bymeans of concentrationof measure arguments,enables the estimationof the asymptotic per-formance of single-layerrandom neural networks.This in turn providespractical insights intothe underlying mecha-nisms into play in ran-dom neural networks,entailing several unex-pected consequences, aswell as a fast practi-cal means to tune thenetwork hyperparame-ters.

J13. Z. Liao, R. Couillet,“A Large DimensionalAnalysis of Least SquareSupport Vector Machi-nes” IEEE Transactionson Signal Processing,

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vol. 67, no. 4, pp. 1065-1074, 2018.

Abstract. In this article,a large dimensional per-formance analysis of ker-nel least squares sup-port vector machines(LS-SVMs) is providedunder the assumptionof a two-class Gaussianmixture model for theinput data. Building uponrecent random matrixadvances, when boththe dimension of datap and their number ngrow large at the samerate, we show that theLS-SVM decision func-tion converges to a normal-distributed variable, themean and variance ofwhich depend explicitlyon a local behavior ofthe kernel function. Thistheoretical result is thenapplied to real data setswhich, despite their non-Gaussianity, exhibit asurprisingly similar be-havior. Our analysis pro-vides a deeper unders-tanding of the mecha-nism into play in SVM-type methods and inparticular of the im-pact on the choice ofthe kernel function aswell as some of theirtheoretical limits.

J14. R. Couillet, H. Tio-moko Ali, “Improvedspectral community de-tection in large hete-rogeneous networks” Jour-nal of Machine Lear-ning Resarch, vol. 18,no. 225, pp. 1–49, 2018.

Abstract. In this article,we study spectral me-thods for communitydetection based on α-parametrized normali-

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zed modularity matrixhereafter called Lα inheterogeneous graph mo-dels. We show, in a re-gime where communitydetection is not asymp-totically trivial, that Lαcan be well approxi-mated by a more trac-table random matrixwhich falls in the fa-mily of spiked randommatrices. The analy-sis of this equivalentspiked random matrixallows us to improvespectral methods for com-munity detection andassess their performancesin the regime under study.In particular, we provethe existence of an op-timal value αopt of theparameter α for whichthe detection of com-munities is best ensu-red and we provide anon-line estimation of αopt

only based on the know-ledge of the graph ad-jacency matrix. Unlikeclassical spectral me-thods for communitydetection where clus-tering is performed onthe eigenvectors asso-ciated with extreme ei-genvalues, we show throughour theoretical analy-sis that a regulariza-tion should instead beperformed on those ei-genvectors prior to clus-tering in heterogeneousgraphs. Finally, througha deeper study of theregularized eigenvectorsused for clustering, weassess the performancesof our new algorithmfor community detec-tion. Numerical simu-lations in the courseof the article show that

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our methods outperformstate-of-the-art spectralmethods on dense he-terogeneous graphs.

J15. R. Couillet, M. McKay,“Optimal block-sparsePCA for high dimen-sional correlated sam-ples” (submitted to) Jour-nal of Multivariate Ana-lysis, 2016.

Abstract. A new prin-cipal component ana-lysis (PCA) method isproposed which is per-formed on a subset ofblocks of consecutiveentries of the popula-tion data vectors. Thisblock-based dimensio-nality reduction intro-duces a trade-off by whichthe accuracy of the do-minant eigenvector ofthe dimension-reducedsample covariance ma-trix is enhanced whilesome population entriesare discarded. This schemeis particularly suited(but not restricted) topopulation eigenvectorswith localized energyand rather sparse struc-tures. Unlike many sparsePCA algorithms, theoriginality of our schemelies in its providing anonline selection of thesubset of blocks which,in the large dimensio-nal regime where bothpopulation and samplesizes grow large, pro-vably ensures optimalalignment between po-pulation and sample ei-genvectors. Moreover,our method inherentlyhandles (a priori unk-nown) linear correla-tion between sample data.

J16. R. Couillet, G. Wain-

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rib, H. Sevi, H. Tio-moko Ali, “The asymp-totic performance of li-near echo state neuralnetworks” Journal ofMachine Learning Re-search, vol. 17, no. 178,pp. 1-35, 2016.

Abstract. In this article,a study of the mean-square error (MSE) per-formance of linear echo-state neural networksis performed, both fortraining and testing tasks.Considering the realis-tic setting of noise presentat the network nodes,we derive determinis-tic equivalents for theaforementioned MSE inthe limit where the num-ber of input data T andnetwork size n both growlarge. Specializing thenthe network connecti-vity matrix to specificrandom settings, we fur-ther obtain simple for-mulas that provide newinsights on the perfor-mance of such networks.

J17. R. Couillet, F. Benaych-Georges, “Kernel Spec-tral Clustering of LargeDimensional Data” Elec-tronic Journal of Sta-tistics, vol. 10, no. 1,pp. 1393-1454, 2016.

Abstract. This articleproposes a first ana-lysis of kernel spectralclustering methods inthe regime where thedimension p of the datavectors to be cluste-red and their numbern grow large at the samerate. We demonstrate,under a k-class Gaus-sian mixture model, thatthe normalized Lapla-cian matrix associated

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with the kernel matrixasymptotically behavessimilar to a so-calledspiked random matrix.Some of the isolatedeigenvalue-eigenvectorpairs in this model areshown to carry the clus-tering information upona separability conditionclassical in spiked ma-trix models. We eva-luate precisely the po-sition of these eigen-values and the contentof the eigenvectors, whichunveil important pro-perties concerning spec-tral clustering, in par-ticular in simple toymodels. Our results arethen compared to thepractical clustering ofimages from the MNISTdatabase, thereby re-vealing an importantmatch between theoryand practice.

J18. F. Benaych-Georges,R. Couillet, “SpectralAnalysis of the GramMatrix of Mixture Mo-dels” ESAIM : Proba-bility and Statistics, DOIhttp ://dx.doi.org/10.1051/ps/2016007,2016.

Abstract. This text isdevoted to the asymp-totic study of some spec-tral properties of theGram matrixWTW builtupon a collection w1, . . . , wn ∈Rp of random vectors(the columns of W ),as both the number nof observations and thedimension p of the ob-servations tend to in-finity and are of simi-lar order of magnitude.The random vectors w1, . . . , wnare independent obser-vations, each of them

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belonging to one of kclasses C1, . . . , Ck. Theobservations of each classCa (1 ≤ a ≤ k) arecharacterized by theirdistributionN (0, p−1Ca),where C1, . . . , Ck aresome non negative de-finite p × p matrices.The cardinality na ofclass Ca and the dimen-sion p of the observa-tions are such that na/n(1 ≤ a ≤ k) and p/nstay bounded away from0 and +∞. We pro-vide deterministic equi-valents to the empiri-cal spectral distribu-tion of WTW and tothe matrix entries ofits resolvent (as wellas of the resolvent ofWWT). These deter-ministic equivalents aredefined thanks to thesolutions of a fixed-pointsystem. Besides, we provethatWTW has asymp-totically no eigenvaluesoutside the bulk of itsspectrum, defined thanksto these deterministicequivalents. These re-sults are directly usedin the companion pa-per ( ?( ?) ?17), whichis devoted to the ana-lysis of the spectral clus-tering algorithm in largedimensions. They alsofind applications in va-rious other fields suchas wireless communi-cations where functio-nals of the aforemen-tioned resolvents allowone to assess the com-munication performanceacross multi-user multi-antenna channels.

J19. R. Couillet, Estima-tion robuste et matrices

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aleatoires, revue Trai-tement du Signal, vol.33, no. 2-3, pp. 273-320, 2016.

Abstract. This articleprovides a technical sur-vey of the recent ad-vances between the fieldsof robust estimation ofscatter and of large di-mensional random ma-trix theory. An expo-sition of the theoreti-cal results will be madewhich we shall applyto various contexts inthe area of statisticsand signal processingat large. The theoreti-cal results essentiallyshow that, while robustestimators of scatter areimplicitly defined andthus difficult objects tomanipulate, in the largedimensional random ma-trix regime where boththe population size andthe number of samplesare simultaneously large,these implicit robust es-timators tend to be-have similar to muchsimpler random matrixmodels, amenable to ana-lysis. This induces thatmany statistical pro-perties of these estima-tors could be unear-thed which we shall dis-cuss. In terms of ap-plications, these robustestimators of scatter arelong-standing structu-ral elements to handleboth outliers and heavy-tailed behavior in theobserved data. Theseimpulsiveness harnes-sing effects will be pre-cisely documented andshall be instrumentalto develop improved ro-bust statistics methods

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for detection and esti-mation in antenna ar-rays, portfolio optimi-zation, etc.

J20. R. Couillet, G. Wain-rib, Perspectives en ma-trices aleatoires et grandsreseaux, revue Traite-ment du Signal, vol. 33,no. 2-3, pp. 351-376,2016.

Abstract. In this article,several research pers-pectives in random ma-trix theory applied tograph theory at largeare discussed. Specificfocus will be made onthe spectrum analysisof the adjacency or La-placian matrices of largedimensional graphs forcommunity detection innetworks, of kernel ran-dom matrices for clus-tering in large datasets,along with applicationsto neural networks.

J21. M. Sadeghi, L. San-guinetti, R. Couillet,Y. Chau, “Large Sys-tem Analysis of PowerNormalization Techniquesin Massive MIMO”, IEEETransactions on Vehi-cular Technologies, vol.66, no. 10, pp. 9005-9017, 2017.

Abstract. Linear pre-coding has been widelystudied in the contextof Massive MIMO to-gether with the two com-mon power normaliza-tion techniques, namely,matrix normalization(MN) and vector nor-malization (VN). Ho-wever, the effect of bothon the system perfor-mance has not been tho-roughly studied. The

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aim of this paper is toaddress this problemusing large system ana-lysis. Considering a sys-tem model that accountsfor channel estimation,pilot contamination, ar-bitrary pathloss, andper-user channel cor-relation, we computetight approximations forthe signal-to-interference-plus-noise ratio (SINR)and the rate of eachuser equipment (UE)in the system while em-ploying maximum ra-tio transmission (MRT),zero forcing (ZF), andregularized ZF (RZF)precoding under bothMN and VN techniques.Exploiting such results,we reveal the effect ofpower normalization onthe performance of MRTand ZF, and determinehow it affects noise, in-terference, pilot conta-mination, and signal po-wers of any arbitraryUE. We show that thepower normalization canconvey a notion of fair-ness or sum rate maxi-mization for ZF. Nu-merical results are usedto validate the accu-racy of the asympto-tic analysis and to showthat in Massive MIMO,non- coherent interfe-rence and noise, ratherthan pilot contamina-tion, are often the ma-jor limiting factors ofthe considered preco-ding schemes. .

J22. M. Sadeghi, L. San-guinetti, R. Couillet,Y. Chau, “Reducing theComputational Com-plexity of Multicasting

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in Large-Scale AntennaSystems”, IEEE Tran-sactions on Wireless Com-munications, vol. 16,no. 5, pp. 2963-2975,2017.

Abstract. In this pa-per, we study the phy-sical layer multicastingto multiple co-channelgroups in large-scale an-tenna systems. The userswithin each group areinterested in a commonmessage and differentgroups have distinct mes-sages. In particular, weaim at designing theprecoding vectors sol-ving the so-called qua-lity of service (QoS) andweighted max-min fair-ness (MMF) problems,assuming that the chan-nel state informationis available at the basestation (BS). To solveboth problems, the ba-seline approach exploitsthe semidefinite relaxa-tion (SDR) technique.Considering a BS withN antennas, the SDRcomplexity is more thanO(N6), which preventsits application in large-scale antenna systems.To overcome this issue,we present two new classesof algorithms that, notonly have significantlylower computational com-plexity than existing so-lutions, but also lar-gely outperform the SDRbased methods. Moreo-ver, we present a no-vel duality between trans-formed versions of theQoS and the weightedMMF problems. Theduality explicitly de-termines the solutionto the weighted MMF

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problem given the so-lution to the QoS pro-blem, and vice versa.Numerical results areused to validate the ef-fectiveness of the pro-posed solutions and tomake comparisons withexisting alternatives un-der different operatingconditions.

J23. L. Sanguinetti, R. Couillet,M. Debbah, “Large Sys-tem Analysis of BaseStation Cooperation forPower Minimization”IEEE Transactions onWireless Communica-tions, vol. 15, no. 8,pp. 5480-5496, 2016.

Abstract. Abstract Thiswork focuses on a large-scale multi-cell multi-user MIMO system inwhich L base stations(BSs) of N antennaseach communicate withK single-antenna userequipments. We consi-der the design of thelinear precoder that mi-nimizes the total po-wer consumption whileensuring target user rates.Three configurations withdifferent degrees of co-operation among BSsare considered : the singlecell processing scheme(no cooperation betweenBSs), the coordinatedbeamforming scheme (onlychannel state informa-tion is shared betweenBSs) and the coordi-nated multipoint MIMOprocessing technology(channel state and datacooperation). The ana-lysis is conducted as-suming that N and Kgrow large with a nontrivial ratio K/N and

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imperfect channel stateinformation is availableat the BSs. Tools ofrandom matrix theoryare used to compute,in explicit form, deter-ministic approximationsfor : (i) the parame-ters of the optimal pre-coder ; (ii) the powersneeded to ensure tar-get rates ; and (iii) thetotal transmit power.These results are ins-trumental to get fur-ther insight into the struc-ture of the optimal pre-coders and also to re-duce the complexity ofits implementation inlarge-scale networks. Nu-merical results are usedto validate the asymp-totic analysis in the fi-nite system regime andto make comparisonsamong the different confi-gurations.

J24. A. Abboud, F. Iut-zeler, R. Couillet, H.Siguerdidjane, M. Deb-bah, “Distributed Production-Sharing Optimizationand Application to Po-wer Grid Networks,”IEEE Transactions onSignal and InformationProcessing over Networks,vol. 2, no. 1, pp. 1628,2016.

Abstract. Based on recentworks on asynchronousversions of the distri-buted Alternating Di-rection Method of Mul-tipliers (ADMM) algo-rithm, we develop andprove the convergenceof a distributed asyn-chronous method forProduction-Sharing Pro-blems over networks.The asynchronous na-

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ture of the algorithmallows both for the re-laxation of the synchro-nization constraint of-ten inherent to distri-buted ADMM-based me-thods and distributedoptimization methodsat large, but also al-lows for random localfailures to occur in fullycentralized methods. Thesetwo considerations mo-tivate the applicationof the method to theDirect-Current OptimalPower Flow (DC-OPF)problem in power trans-mission networks. Ap-plied to the DC-OPF,this method leads toan overall network op-timal production ob-tained through a se-quence of local com-putations in subareasof the network (eacharea waking up randomlywhile the rest of thenetwork is non-operational)and neighboring dataexchanges. In anotherscenario, the DC-OPFis performed via itera-tions of a centralizednetwork-wide ADMMmethod which may containdisconnected nodes (ingeneral with low pro-bability and for a shortduration). In both cases,this method still convergesand thus provides ad-ditional flexibility to clas-sical DC-OPF algorithms.The proposed algorithm,inherently designed fornetworks of overlappingsubareas, is then ex-tended to networks ofnon-overlapping areas.Simulations are carriedout on the IEEE-30 andIEEE-118 bus test sys-

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tems which illustratethe convergence, sca-lability and effective-ness of the proposedalgorithms.

J25. A. Kammoun, R. Couillet,F. Pascal, M.-S. Alouini,“Optimal Design of theAdaptive NormalizedMatched Filter Detec-tor using RegularizedTyler Estimator” IEEETransactions on Aeros-pace and Electronic Sys-tems, vol. 54, no. 2,pp. 755–769, 2018.

Abstract. This articleaddresses improvementson the design of theadaptive normalized mat-ched filter (ANMF) forradar detection. It iswell-acknowledged thatthe estimation of thenoise-clutter covariancematrix is a fundamen-tal step in adaptive ra-dar detection. In thispaper, we consider re-gularized estimation me-thods which force byconstruction the eigen-values of the scatter es-timates to be greaterthan a positive regu-larization parameter ρ.This makes them moresuitable for high dimen-sional problems witha limited number of se-condary data samplesthan traditional samplecovariance estimates. Whilean increase of ρ seemsto improve the condi-tioning of the estimate,it might however causeit to significantly de-viate from the true co-variance matrix. Thesetting of the optimalregularization parame-ter is a difficult ques-

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tion for which no convin-cing answers have thusfar been provided. Thisconstitutes the majormotivation behind ourwork. More specifically,we consider the designof the ANMF detec-tor for two kinds of re-gularized estimators, na-mely the regularized samplecovariance matrix (RSCM),appropriate when theclutter follows a Gaus-sian distribution andthe regularized Tylerestimator (RTE) for non-Gaussian spherically in-variant distributed clut-ters. The rationale be-hind this choice is thatthe RTE is efficient inmitigating the degra-dation caused by thepresence of impulsivenoises while inducinglittle loss when the noiseis Gaussian. Based onrecent random matrixtheory results studyingthe asymptotic fluctua-tions of the statisticsof the ANMF detec-tor when the numberof samples and theirdimension grow toge-ther to infinity, we pro-pose a design for theregularization parame-ter that maximizes thedetection probability un-der constant false alarmrates. Simulation resultswhich support the ef-ficiency of the propo-sed method are provi-ded in order to illus-trate the gain of theproposed optimal de-sign over conventionalsettings of the regula-rization parameter.

J26. A. Kammoun, R. Couillet,

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F. Pascal, M.-S. Alouini,“Convergence and Fluc-tuations of Regulari-zed Tyler Estimators”IEEE Transactions onSignal Processing, vol.64, no. 4, pp. 1048-1060,2016.

Abstract. This articlestudies the behavior ofregularized Tyler esti-mators (RTEs) of scat-ter matrices. The keyadvantages of these es-timators are twofold.First, they guaranteeby construction a goodconditioning of the es-timate and second, beinga derivative of robustTyler estimators, theyinherit their robustnessproperties, notably theirresilience to the pre-sence of outliers. Ne-vertheless, one majorproblem that poses theuse of RTEs in prac-tice is represented bythe question of settingthe regularization pa-rameter ρ. While a highvalue of ρ is likely topush all the eigenva-lues away from zero,it comes at the costof a larger bias withrespect to the popu-lation covariance ma-trix. A deep unders-tanding of the statis-tics of RTEs is essen-tial to come up withappropriate choices forthe regularization pa-rameter. This is not aneasy task and mightbe out of reach, unlessone considers asymp-totic regimes whereinthe number of obser-vations n and/or theirsize N increase toge-ther. First asymptotic

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results have recently beenobtained under the as-sumption that N andn are large and com-mensurable. Interestin-gly, no results concer-ning the regime of ngoing to infinity withN fixed exist, even thoughthe investigation of thisassumption has usuallypredated the analysisof the most difficultNand n large case. Thismotivates our work. Inparticular, we prove inthe present paper thatthe RTEs converge toa deterministic matrixwhen n → ∞ with Nfixed, which is expres-sed as a function of thetheoretical covariancematrix. We also derivethe fluctuations of theRTEs around this de-terministic matrix andestablish that these fluc-tuations converge in dis-tribution to a multi-variate Gaussian dis-tribution with zero meanand a covariance de-pending on the popu-lation covariance andthe parameter ρ.

J27. D. Morales-Jimenez,R. Couillet, M. McKay,“Large Dimensional Ana-lysis of Robust M-Estimatorsof Covariance with Out-liers” IEEE Transac-tions on Signal Proces-sing, vol. 63, no. 21,pp. 5784-5797, 2015.

Abstract. A large di-mensional characteri-zation of robust M-estimatorsof covariance (or scat-ter) is provided underthe assumption that thedataset comprises in-dependent (essentially

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Gaussian) legitimate samplesas well as arbitrary de-terministic samples, re-ferred to as outliers.Building upon recentrandom matrix advancesin the area of robuststatistics, we specificallyshow that the so-calledMaronna M-estimatorof scatter asymptoti-cally behaves similarto well-known randommatrices when the po-pulation and sample sizesgrow together to infi-nity. The introductionof outliers leads the ro-bust estimator to be-have asymptotically asthe weighted sum ofthe sample outer pro-ducts, with a constantweight for all legitimatesamples and differentweights for the outliers.A fine analysis of thisstructure reveals impor-tantly that the propen-sity of the M-estimatorto attenuate (or enhance)the impact of outliersis mostly dictated bythe alignment of theoutliers with the inversepopulation covariancematrix of the legitimatesamples. Thus, robustM-estimators can bringsubstantial benefits overmore simplistic estima-tors such as the per-sample normalized ver-sion of the sample co-variance matrix, whichis not capable of diffe-rentiating the outlyingsamples. The analysisshows that, within theclass of Maronna’s es-timators of scatter, theHuber estimator is mostfavorable for rejectingoutliers. On the contrary,

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estimators more simi-lar to Tyler’s scale in-variant estimator (of-ten preferred in the li-terature) run the riskof inadvertently enhan-cing some outliers.

J28. L. Yang, R. Couillet,M. McKay, “A RobustStatistics Approach toMinimum Variance Port-folio Optimization” IEEETransactions on SignalProcessing, vol. 63, no.24, pp. 6684–6697, 2015.

Abstract. We study thedesign of portfolios un-der a minimum risk cri-terion. The performanceof the optimized port-folio relies on the ac-curacy of the estima-ted covariance matrixof the portfolio assetreturns. For large port-folios, the number ofavailable market returnsis often of similar or-der to the number ofassets, so that the samplecovariance matrix per-forms poorly as a co-variance estimator. Ad-ditionally, financial mar-ket data often containoutliers which, if notcorrectly handled, mayfurther corrupt the co-variance estimation. Weaddress these shortco-mings by studying theperformance of a hy-brid covariance matrixestimator based on Ty-ler’s robust M-estimatorand on Ledoit-Wolf’sshrinkage estimator whileassuming samples withheavy-tailed distribu-tion. Employing recentresults from random ma-trix theory, we deve-lop a consistent esti-

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mator of (a scaled ver-sion of) the realized port-folio risk, which is mi-nimized by optimizingonline the shrinkage in-tensity. Our portfoliooptimization method isshown via simulationsto outperform existingmethods both for syn-thetic and real marketdata.

J29. R. Couillet, A. Kam-moun, F. Pascal, “Se-cond order statistics ofrobust estimators of scat-ter. Application to GLRTdetection for ellipticalsignals” Elsevier Jour-nal of Multivariate Ana-lysis, vol. 143, pp. 249-274, 2015.

Abstract. A central li-mit theorem for bili-near forms of the typea∗CN (ρ)−1b, where a, b ∈CN are unit norm de-terministic vectors andCN (ρ) a robust-shrinkageestimator of scatter pa-rametrized by ρ andbuilt upon n independentelliptical vector obser-vations, is presented.The fluctuations of a∗CN (ρ)−1bare found to be of or-derN−

12 and to be the

same as those of a∗SN (ρ)−1bfor SN (ρ) a matrix ofa theoretical tractableform. This result is ex-ploited in a classicalsignal detection problemto provide an impro-ved detector which isboth robust to ellipti-cal data observations(e.g., impulsive noise)and optimized acrossthe shrinkage parame-ter ρ.

J30. A. Muller, R. Couillet,E. Bjørnson, S. Wag-

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ner, M. Debbah, “Interference-Aware RZF Precodingfor Multi-Cell Down-link Systems” IEEE Tran-sactions on Signal Pro-cessing, vol. 63, no. 15,pp. 3959-3973 2015.

Abstract. Recently, thestructure of the opti-mal linear precoder formulti-cell downlink sys-tems has been descri-bed. Other referenceshave used simplified ver-sions of the precoderto obtain promising per-formance gains. Thesegains have been hypo-thesized to stem fromproviding additional de-grees of freedom thatallow for interferencemitigation through in-terference relegation toorthogonal subspaces.However, no conclusiveor rigorous understan-ding has yet been pro-posed. In this paper,we take an interference-aware adaption of thegenerally optimal pre-coding structure andanalyze the rate per-formance in multi-cellscenarios. A special em-phasis is placed on in-duced interference mi-tigation. For example,we will verify the in-tuitive expectation thatthe precoder structurecan either completelyremove induced inter-cell or intra-cell inter-ference. We state newresults from large-scalerandom matrix theory,that make it possibleto give more intuitiveand insightful explana-tions of the precoderbehavior, also for casesinvolving imperfect chan-

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nel state information(CSI). We remark es-pecially that the interference-aware precoder makesuse of all available in-formation about inter-fering channels to im-prove performance. Evenextremely bad CSI canbe used to enhance thesum rate. Our obtai-ned insights are thenused to propose heu-ristic precoder parame-ters for arbitrary sys-tems, whose effective-ness is shown in moreinvolved system scena-rios. Furthermore, de-termining these para-meters does not requireexplicit inter base sta-tion cooperation. Usinga simple heuristic ver-sion of the interferenceaware precoder, one findsthat a sum rate per-formance, close to theoptimally parameteri-zed precoder one, canbe achieved.

J31. R. Couillet, “Robustspiked random matricesand a robust G-MUSICestimator” Elsevier Jour-nal of Multivariate Ana-lysis, vol. 140, pp. 139-161, 2015.

Abstract. A class of ro-bust estimators of scat-ter applied to information-plus-impulsive noise samplesis studied, where thesample information ma-trix is assumed of lowrank ; this generalizesthe study ( ?( ?) ?34) (res-tricted to a noise onlysetting) to spiked ran-dom matrix models. Itis precisely shown that,as opposed to samplecovariance matrices which

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may have asymptoti-cally unbounded (eigen-)spectrum due to thesample impulsiveness,the robust estimator ofscatter has bounded spec-trum and may containisolated eigenvalues whichwe fully characterize.We show that, if foundbeyond a certain de-tectability threshold, theseeigenvalues allow oneto perform statisticalinference on the eigen-values and eigenvectorsof the information ma-trix. We use this re-sult to derive new ei-genvalue and eigenvec-tor estimation proce-dures, which we applyin practice to the po-pular array processingproblem of angle of ar-rival estimation. Thisgives birth to an im-proved algorithm ba-sed on the MUSIC me-thod, which we referto as robust G-MUSIC.

J32. R. Couillet, M. McKay,“Large Dimensional Ana-lysis and Optimizationof Robust Shrinkage Co-variance Matrix Esti-mators” Elsevier Jour-nal of Multivariate Ana-lysis, vol. 131, pp. 99-120, 2014.

Abstract. This articlestudies two regularizedrobust estimators of scat-ter matrices proposed(and proved to be welldefined) in parallel in(Chen et al., 2011) and(Pascal et al., 2013),based on Tyler’s robustM-estimator (Tyler, 1987)and on Ledoit and Wolf’sshrinkage covariance ma-

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trix estimator (Ledoitand Wolf, 2004). Thesehybrid estimators havethe advantage of conveying(i) robustness to out-liers or impulsive samplesand (ii) small samplesize adequacy to theclassical sample cova-riance matrix estima-tor. We consider herethe case of i.i.d. ellip-tical zero mean samplesin the regime where bothsample and populationsizes are large. We de-monstrate that, underthis setting, the esti-mators under study asymp-totically behave simi-lar to well-understoodrandom matrix models.This characterizationallows us to derive op-timal shrinkage stra-tegies to estimate thepopulation scatter ma-trix, improving signi-ficantly upon the em-pirical shrinkage me-thod proposed in (Chenet al., 2011).

J33. Y. Chitour, R. Couillet,F. Pascal “On the conver-gence of Maronna’s M-estimators of scatter”IEEE Signal ProcessingLetters, vol. 22, no. 6,pp. 709-712, 2014.

Abstract. In this pa-per, we propose an al-ternative proof for theuniqueness of Maron-na’s M-estimator of scat-ter for N vector ob-servations y1, . . . , yN ∈Rm under a mild constraintof linear independenceof any subset of m ofthese vectors. This en-tails in particular al-most sure uniquenessfor random vectors yi

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with a density as longas N > m. This ap-proach allows to esta-blish further relationsthat demonstrate thata properly normalizedTyler’s M-estimator ofscatter can be consi-dered as a limit of Ma-ronna’s M-estimator. Moreprecisely, the contribu-tion is to show that eachM-estimator, verifyingsome mild conditions,converges towards a par-ticular Tyler’s M-estimator.These results find im-portant implications inrecent works on the largedimensional (random ma-trix) regime of robustM-estimation.

J34. R. Couillet, F. Pas-cal, J. W. Silverstein,“The Random MatrixRegime of Maronna’sM-estimator with el-liptically distributed sam-ples”, vol. 139, pp. 56-78, Elsevier Journal ofMultivariate Analysis,2015.

Abstract. This articledemonstrates that therobust scatter matrixestimator CN ∈ CN×Nof a multivariate ellip-tical population x1, . . . , xn ∈CN originally propo-sed by Maronna in 1976,and defined as the so-lution (when existent)of an implicit equation,behaves similar to a well-known random matrixmodel in the limitingregime where the po-pulationN and samplen sizes grow at the samespeed. We show preci-sely that CN ∈ CN×Nis defined for all n largewith probability one and

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that, under some lighthypotheses, ‖CN−SN‖ →0 almost surely in spec-tral norm, where SNfollows a classical ran-dom matrix model. Asa corollary, the limi-ting eigenvalue distri-bution of CN is deri-ved. This analysis findsapplications in the fieldsof statistical inferenceand signal processing.

J35. J. Vinogradova, R. Couillet,W. Hachem, “Estima-tion of Toeplitz cova-riance matrices in largedimensional regime withapplication to sourcedetection large”, IEEETransactions on SignalProcessing, vol. 63, no.18, pp. 4903-4913, 2015.

Abstract. In this article,we derive concentrationinequalities for the spec-tral norm of two clas-sical sample estimatorsof large dimensional Toe-plitz covariance matrices,demonstrating in par-ticular their asympto-tic almost sure consis-tence. The consistencyis then extended to thecase where the aggre-gated matrix of timesamples is corrupted bya rank one (or moregenerally, low rank) ma-trix. As an applicationof the latter, the pro-blem of source detec-tion in the context oflarge dimensional sen-sor networks within atemporally correlatednoise environment is stu-died. As opposed to stan-dard procedures, thisapplication is perfor-med online, i.e., without

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the need to possess alearning set of pure noisesamples.

J36. R. Couillet, W. Ha-chem, “Analysis of thelimiting spectral mea-sure of large randommatrices of the sepa-rable covariance type”,Random Matrix Theoryand Applications, vol.3, pp. 1-23, 2014.

Abstract. Consider therandom matrix Σ =D1/2XD1/2 where Dand D are determinis-tic Hermitian nonne-gative matrices with res-pective dimensionsN×N and n×n, and whereX is a random matrixwith independent andidentically distributedcentered elements withvariance 1/n. Assumethat the dimensionsNand n grow to infinityat the same pace, andthat the spectral mea-sures ofD and D convergeas N,n→∞ towardstwo probability mea-sures. Then it is knownthat the spectral mea-sure of ΣΣ∗ convergestowards a probabilitymeasure µ characteri-zed by its Stieltjes Trans-form. In this paper, itis shown that µ has adensity away from zero,this density is analyti-cal wherever it is posi-tive, and it behaves inmost cases as

√|x− a|

near an edge a of itssupport. In addition,a complete characte-rization of the supportof µ is provided. Asidefrom its mathematicalinterest, the analysis un-derlying these results

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finds important appli-cations in a certain classof statistical estimationproblems.

J37. J. Hoydis, R. Couillet,P. Piantanida, “The Second-Order Coding Rate ofthe MIMO Rayleigh Block-Fading Channel,” IEEETransactions on Infor-mation Theory, vol. 61,no. 12, pp. 6591-6622,2015.

Abstract. The second-order coding rate of themultiple-input multiple-output (MIMO) quasi-static Rayleigh fadingchannel is studied. Wetackle this problem viaan information-spectrumapproach and statisti-cal bounds based onrecent random matrixtheory techniques. Weprecisely derive a cen-tral limit theorem (CLT)to analyze the infor-mation density in theregime where the block-length n and the num-ber of transmit and re-ceive antennas K andN , respectively, growsimultaneously large. Thisresult leads to the cha-racterization of closed-form upper and lowerbounds on the optimalaverage error probabi-lity when the codingrate is withinO(1/

√nK)

of the asymptotic ca-pacity.

J38. J. Vinogradova, R. Couillet,W. Hachem, “Statis-tical Inference in LargeAntenna Arrays underUnknown Noise Pat-tern,” IEEE Transac-tions on Signal Proces-sing, vol. 61, no. 22,pp. 5633-5645, 2013.

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Abstract. In this article,a general information-plus-noise transmissionmodel is assumed, thereceiver end of whichis composed of a largenumber of sensors andis unaware of the noisepattern. For this mo-del, and under reaso-nable assumptions, aset of results is pro-vided for the receiverto perform statisticaleigen-inference on theinformation part. In par-ticular, we introducenew methods for thedetection, counting, andthe power and subspaceestimation of multiplesources composing theinformation part of thetransmission. The theo-retical performance ofsome of these techniquesis also discussed. Anexemplary applicationof these methods to ar-ray processing is thenstudied in greater de-tail, leading in parti-cular to a novel MUSIC-like algorithm assumingunknown noise covariance.

J39. F. Chapon, R. Couillet,W. Hachem, X. Mestre,“The outliers among thesingular values of largerectangular random ma-trices with additive fixedrank deformation,” Mar-kov Processes and Re-lated Fields, vol. 20,pp. 183-228, 2014.

Abstract. Consider thematrix Σn = n−1/2XnD

1/2n +

Pn where the matrixXn ∈ CN×n has Gaus-sian standard independentelements, Dn is a de-terministic diagonal non-

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negative matrix, andPn is a deterministicmatrix with fixed rank.Under some known condi-tions, the spectral mea-sures of ΣnΣ∗n and n−1XnDnX

∗n

both converge towardsa compactly supportedprobability measure µasN,n→∞ withN/n→c. In this paper, it isproved that finitely manyeigenvalues of ΣnΣ∗n maystay away from the sup-port of µ in the largedimensional regime. Theexistence and locationsof these outliers in anyconnected componentof R\supp(µ) are stu-died. The fluctuationsof the largest outliersof ΣnΣ∗n are also ana-lyzed. The results findapplications in the fieldsof signal processing andradio communications.

J40. R. Couillet, F. Pas-cal, J. W. Silverstein,“Robust Estimates ofCovariance Matrices inthe Large DimensionalRegime,” IEEE Tran-sactions on InformationTheory, vol. 60, no. 11,2014.

Abstract. This articlestudies the limiting be-havior of a class of ro-bust population cova-riance matrix estima-tors, originally due toMaronna in 1976, inthe regime where boththe number of availablesamples and the po-pulation size grow large.Using tools from ran-dom matrix theory, weprove that, for samplevectors made of inde-pendent entries having

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some moment condi-tions, the difference bet-ween the sample cova-riance matrix and (ascaled version of) suchrobust estimator tendsto zero in spectral norm,almost surely. This re-sult can be applied tovarious statistical me-thods arising from ran-dom matrix theory thatcan be made robust wi-thout altering their firstorder behavior.

J41. G. Geraci, R. Couillet,J. Yuan, M. Debbah,I. B. Collings, “LargeSystem Analysis of Li-near Precoding in MISOBroadcast Channels withConfidential Messages,”IEEE Journal on Se-lected Area in Com-munications, vol. 31,no. 9, pp. 1660-1671,2013. Second prizeof the 2012-2013 IEEEAustralia Council StudentPaper Contest.

Abstract. In this pa-per, we study the per-formance of regulari-zed channel inversion(RCI) precoding in largeMISO broadcast chan-nels with confidentialmessages (BCC). Weobtain a deterministicapproximation for theachievable secrecy sum-rate which is almostsurely exact as the num-ber of transmit anten-nas M and the num-ber of usersK grow toinfinity in a fixed ra-tio β = K/M . We de-rive the optimal regu-larization parameter ξand the optimal net-work load β that maxi-mize the per-antenna

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secrecy sum-rate. Wethen propose a linearprecoder based on RCIand power reduction (RCI-PR) that significantlyincreases the high-SNRsecrecy sum-rate for 1 <β < 2. Our proposedprecoder achieves a per-user secrecy rate whichhas the same high-SNRscaling factor as boththe following upper bounds :(i) the rate of the op-timum RCI precoderwithout secrecy requi-rements, and (ii) thesecrecy capacity of asingle-user system wi-thout interference. Fur-thermore, we obtain adeterministic approxi-mation for the secrecysum-rate achievable byRCI precoding in thepresence of channel stateinformation (CSI) er-ror. We also analyzethe performance of ourproposed RCI-PR pre-coder with CSI error,and we determine howthe error must scale withthe SNR in order tomaintain a given rategap to the case withperfect CSI.

J42. J. Hoydis, R. Couillet,M. Debbah, “IterativeDeterministic Equiva-lents for the CapacityAnalysis of Communi-cation Systems,” Tech-nical Report.

Abstract. In this report,we introduce the no-tion of nested deter-ministic equivalents offunctional of randommatrices. Nested deter-ministic equivalents ex-tend classical determi-nistic equivalents in or-

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der to study stochas-tic problems with mul-tiple independent ran-dom variables. In par-ticular, we discuss theirapplications to wirelesscommunications, andespecially to the capa-city analysis of doubly-scattering multiple in-put multiple output (MIMO)channels and of multi-hop relay channels forwhich we derive novelexpressions.

J43. R. Couillet, S. Me-dina Perlaza, H. Tem-bine, M. Debbah, “Elec-trical Vehicles in theSmart Grid: A MeanField Game Analysis,”IEEE Journal on Se-lected Areas in Com-munications : Smart GridCommunications Series,vol. 30, no. 6, pp. 1086-1096, 2012.

Abstract. In this article,we investigate the com-petitive interaction bet-ween electrical vehiclesor hybrid oil-electricityvehicles in a Cournotmarket consisting of elec-tricity transactions toor from an underlyingelectricity distributionnetwork. We providea mean field game for-mulation for this com-petition, and introducethe set of fundamen-tal differential equationsruling the behavior ofthe vehicles at the feed-back Nash equilibrium,referred here to as themean field equilibrium.This framework allowsfor a consistent analy-sis of the evolution ofthe price of electricityas well as of the ins-

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tantaneous electricitydemand in the powergrid. Simulations pre-cisely quantify those pa-rameters and suggestthat significant reduc-tion of the daily elec-tricity peak demand canbe achieved by appro-priate electricity pri-cing.

J44. J. Yao, R. Couillet,J. Najim, M. Debbah,“Fluctuations of an Im-proved Population Ei-genvalue Estimator inSample Covariance Ma-trix Models,” IEEE Tran-sactions on InformationTheory, vol. 59, no. 2,pp. 1149-1163, 2013.

Abstract. In this article,the joint fluctuationsof the extreme eigen-values and eigenvectorsof a large dimensionalsample covariance ma-trix are analyzed whenthe associated popula-tion covariance matrixis a finite-rank pertur-bation of the identitymatrix, correspondingto the so-called spikedmodel in random ma-trix theory. The asymp-totic fluctuations, as thematrix size grows large,are shown to be inti-mately linked with ma-trices from the Gaus-sian unitary ensemble(GUE). When the spi-ked population eigen-values have unit mul-tiplicity, the fluctuationsfollow a central limittheorem. This result isused to develop an ori-ginal framework for thedetection and diagno-sis of local failures inlarge sensor networks,

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for known or unknownfailure magnitude.

J45. R. Couillet, M. Deb-bah,“Signal Processingin Large Systems: a NewParadigm,” IEEE Si-gnal Processing Maga-zine, vol. 30, no. 1, pp.24-39, 2013.

Abstract. For a long time,detection and parame-ter estimation methodsfor signal processing haverelied on asymptotic sta-tistics as the numbern of observations of apopulation grows largecomparatively to thepopulation sizeN , i.e.,n/N →∞. Modern tech-nological and societaladvances now demandthe study of sometimesextremely large popu-lations and simultaneouslyrequire fast signal pro-cessing due to accele-rated system dynamics.This results in not-so-large practical ratios n/N ,sometimes even smal-ler than one. A disrup-tive change in classi-cal signal processing me-thods has therefore beeninitiated in the past tenyears, mostly spurredby the field of large di-mensional random ma-trix theory. The earlyworks in random ma-trix theory for signalprocessing applicationsare however scarce andhighly technical. Thistutorial provides an ac-cessible methodologi-cal introduction to themodern tools of ran-dom matrix theory andto the signal proces-sing methods derivedfrom them, with an em-

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phasis on simple illus-trative examples.

J46. R. Couillet, W. Ha-chem,“Fluctuations ofspiked random matrixmodels and failure diag-nosis in sensor networks,”IEEE Transactions onInformation Theory, vol.59, no. 1, pp. 509-525,2013.

Abstract. In this article,the joint fluctuationsof the extreme eigen-values and eigenvectorsof a large dimensionalsample covariance ma-trix are analyzed whenthe associated popula-tion covariance matrixis a finite-rank pertur-bation of the identitymatrix, correspondingto the so-called spikedmodel in random ma-trix theory. The asymp-totic fluctuations, as thematrix size grows large,are shown to be inti-mately linked with ma-trices from the Gaus-sian unitary ensemble(GUE). When the spi-ked population eigen-values have unit mul-tiplicity, the fluctuationsfollow a central limittheorem. This result isused to develop an ori-ginal framework for thedetection and diagno-sis of local failures inlarge sensor networks,for known or unknownfailure magnitude.

J47. A. Kammoun, R. Couillet,J. Najim, M. Debbah,“Performanceof capacity inference me-thods under colored in-terference,” IEEE Tran-sactions on InformationTheory, vol. 59, no. 2,pp. 1129-1148, 2013.

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Abstract. The problemof fast point-to-pointMIMO channel mutualinformation estimationis addressed, in the si-tuation where the re-ceiver undergoes unk-nown colored interfe-rence, whereas the chan-nel with the transmit-ter is perfectly known.The considered scena-rio assumes that theestimation is based ona few channel use ob-servations during a shortsensing period. Usinglarge dimensional ran-dom matrix theory, anestimator referred toas G-estimator is de-rived. This estimatoris proved to be consistentas the number of an-tennas and observationsgrow large and its asymp-totic performance is ana-lyzed. In particular, theG-estimator satisfies acentral limit theoremwith asymptotic Gaus-sian fluctuations. Simu-lations are provided whichstrongly support the theo-retical results, even forsmall system dimensions.

J48. R. Couillet, J. Hoy-dis, M. Debbah, “Ran-dom beamforming overquasi-static and fadingchannels: A determi-nistic equivalent approach,”IEEE Transactions onInformation Theory, vol.58, no. 10, pp. 6392-6425, 2012.

Abstract. In this work,we study the perfor-mance of random iso-metric precoders overquasi-static and corre-lated fading channels.

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We derive determinis-tic approximations ofthe mutual informationand the signal-to-interference-plus-noise ratio (SINR)at the output of theminimum-mean-square-error (MMSE) receiverand provide simple pro-vably converging fixed-point algorithms for theircomputation. Althoughthese approximationsare only proven exactin the asymptotic re-gime with infinitely manyantennas at the trans-mitters and receivers,simulations suggest thatthey closely match theperformance of small-dimensional systems. Weexemplarily apply ourresults to the perfor-mance analysis of multi-cellular communicationsystems, multiple-inputmultiple-output multiple-access channels (MIMO-MAC), and MIMO in-terference channels. Themathematical analysisis based on the Stieltjestransform method. Thisenables the derivationof deterministic equi-valents of functionalsof large-dimensional ran-dom matrices. In contrastto previous works, ouranalysis does not relyon arguments from freeprobability theory whichenables the considera-tion of random matrixmodels for which asymp-totic freeness does nothold. Thus, the resultsof this work are also anovel contribution tothe field of random ma-trix theory and appli-cable to a wide spec-trum of practical sys-

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tems.

J49. S. Wagner, R. Couillet,M. Debbah, D. T. M.Slock, “Large SystemAnalysis of Linear Pre-coding in MISO Broad-cast Channels with Li-mited Feedback”, IEEETransactions on Infor-mation Theory, vol. 58,no. 7, pp. 4509-4537,2012.

Abstract. In this pa-per, we study the sumrate performance of zero-forcing (ZF) and regu-larized ZF (RZF) pre-coding in large MISObroadcast systems un-der the assumptions ofimperfect channel stateinformation at the trans-mitter and per-user chan-nel transmit correlation.Our analysis assumesthat the number of trans-mit antennas M andthe number of single-antenna users K arelarge while their ratioremains bounded. Wederive deterministic ap-proximations of the em-pirical signal-to-interferenceplus noise ratio (SINR)at the receivers, whichare tight as M,K →∞. In the course of thisderivation, the per-userchannel correlation mo-del requires the deve-lopment of a novel de-terministic equivalentof the empirical Stieltjestransform of large di-mensional random ma-trices with generalizedvariance profile. The de-terministic SINR ap-proximations enable usto solve various prac-tical optimization pro-blems. Under sum rate

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maximization, we de-rive (i) for RZF the op-timal regularization pa-rameter, (ii) for ZF theoptimal number of users,(iii) for ZF and RZFthe optimal power al-location scheme and (iv)the optimal amount offeedback in large FDD/TDDmulti-user systems. Nu-merical simulations sug-gest that the determi-nistic approximationsare accurate even forsmall M,K.

J50. R. Couillet, J. W. Sil-verstein, Z. Bai, M. Deb-bah, “Eigen-Inferencefor Energy Estimationof Multiple Sources”,IEEE Transactions onInformation Theory, vol.57, no. 4, pp. 2420-2439,2011.

Abstract. In this pa-per, a new method isintroduced to blindlyestimate the transmitpower of multiple si-gnal sources in multi-antenna fading chan-nels, when the num-ber of sensing devicesand the number of avai-lable samples are suf-ficiently large compa-red to the number ofsources. Recent advancesin the field of large di-mensional random ma-trix theory are used thatresult in a simple andcomputationally efficientconsistent estimator ofthe power of each source.A criterion to deter-mine the minimum num-ber of sensors and theminimum number of samplesrequired to achieve sourceseparation is then in-troduced. Simulations

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are performed that cor-roborate the theoreti-cal claims and show thatthe proposed power es-timator largely outper-forms alternative po-wer inference techniques.

J51. R. Couillet, M. Deb-bah, J. W. Silverstein,“A Deterministic Equi-valent for the Analy-sis of Correlated MIMOMultiple Access Chan-nels”, IEEE Transac-tions on Information Theory,vol. 57, no. 6, pp. 3493-3514, 2011.

Abstract. In this article,novel deterministic equi-valents for the Stieltjestransform and the Shan-non transform of a classof large dimensional ran-dom matrices are pro-vided. These results areused to characterise theergodic rate region ofmultiple antenna mul-tiple access channels,when each point-to-pointpropagation channel ismodelled according tothe Kronecker model.Specifically, an approxi-mation of all rates achie-ved within the ergo-dic rate region is de-rived and an approxi-mation of the linear pre-coders that achieve theboundary of the rateregion as well as an ite-rative water-filling al-gorithm to obtain theseprecoders are provided.An original feature ofthis work is that theproposed deterministicequivalents are provedvalid even for strongcorrelation patterns atboth communication sides.

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The above results arevalidated by Monte Carlosimulations.

J52. R. Couillet, M. Deb-bah, “A Bayesian Fra-mework for Collabora-tive Multi-Source Si-gnal Sensing”, IEEE Tran-sactions on Signal Pro-cessing, vol. 58, no. 10,pp. 5186-5195, 2010.

Abstract. This paper in-troduces a Bayesian fra-mework to detect mul-tiple signals embeddedin noisy observationsfrom a sensor array. Forvarious states of know-ledge on the commu-nication channel andthe noise at the recei-ving sensors, a margi-nalization procedure ba-sed on recent tools offinite random matrixtheory, in conjunctionwith the maximum en-tropy principle, is usedto compute the hypo-thesis selection crite-rion. Quite remarkably,explicit expressions forthe Bayesian detectorare derived which en-able to decide on thepresence of signal sourcesin a noisy wireless en-vironment. The propo-sed Bayesian detectoris shown to outperformthe classical power de-tector when the noisepower is known and pro-vides very good per-formance for limited know-ledge on the noise po-wer. Simulations cor-roborate the theoreti-cal results and quan-tify the gain achievedusing the proposed Baye-sian framework.

J53. R. Couillet, A. An-

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cora, M. Debbah, “Baye-sian Foundations of Chan-nel Estimation for Cog-nitive Radios”, Advancesin Electronics and Te-lecommunications, vol.1, no. 1, pp. 41-49, 2010.

Abstract. In this pa-per, we revisit the phi-losophical foundationsof the field of channelestimation. Our mainintention is to come upwith a partial answerto the question : “gi-ven some available sen-sed signals, how shouldcognitive radios ideallyperform channel esti-mation ?”. We specifi-cally introduce a ge-neral framework to pro-vide optimal channelestimates under any priorknowledge at the sen-sing device. Our dis-cussion is articulatedas a top-down approach,introducing successively(i) a discussion on thephilosophical founda-tions of channel esti-mation as a simplifi-cation means for thegeneral problem of wi-reless detection, (ii) aninformation theoreticallyoptimal approach to chan-nel detection assumingthe sensing device hasinfinite memory, and(iii) a derived optimalapproach when limitedmemory size is accoun-ted for. The key ma-thematical tools usedin this discussion emergefrom Bayesian proba-bility theory and areknown as the maximumentropy principle andthe minimum updateprinciple. Derivationsare carried out for the

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particular case of chan-nel estimation in or-thogonal frequency di-vision multiplexing (OFDM)systems. While sometheoretical results willbe proven to match al-ready known techniques,such as Kalman filters,another set of novel re-sults will be shown bysimulations to performbetter than known chan-nel estimation schemes.

J54. R. Couillet, M. Deb-bah,“Le telephone dufutur : plus intelligentpour une exploitationoptimale des frequences”Revue de l’Electriciteet de l’Electronique, no.6, pp. 71-83, 2010.

Resume. Au jour oules communications mo-biles demandent de plusen plus de vitesse detransfert de larges vo-lumes de donnees a des-tination d’utilisateursde plus en plus nom-breux, il apparaıt queles limites physiques desprotocoles de commu-nication sont bientotatteintes. Une revolutiontechnologique est ainsinecessaire et sur le pointd’eclore : celle-ci passepar la mise en placede systemes de com-munications opportu-nistes, cooperatifs, au-tonomes et idealementsuffisamment intelligentspour servir au mieuxles requetes de l’utili-sateur. Ces differentsaspects, certains d’oreset deja d’actualite, d’autresa l’etat embryonnairesont discutes successi-vement dans cette etude.

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J55. R. Couillet, M. Deb-bah, “Mathematical foun-dations of cognitive ra-dios”, Journal of Tele-communications and In-formation Technologies,no. 4, 2009.

Abstract. Recently, muchinterest has been di-rected towards softwaredefined radios and em-bedded intelligence intelecommunication de-vices. However, no theo-retical framework forcognitive radios has everbeen proposed. In thispaper, we introduce aninformation theoreticpoint of view on cog-nitive radios. Specifi-cally, our motivationin this work is to em-bed human-like intel-ligence in mobile wi-reless devices, followingthe three century-oldwork on Bayesian pro-bability theory, the maxi-mum entropy principleand minimal probabi-lity update. This al-lows us to partially ans-wer such questions as“what are the signaldetection capabilities ofa wireless device ?” or“when facing a situa-tion in which most pa-rameters are missing,how to react ?”. As anintroduction, we willpresent two examplesfrom the same authorsusing the cognitive fra-mework namely multi-antenna channel mo-delling and signal sen-sing.

J56. R. Couillet, M. Deb-bah, “Outage perfor-mance of flexible OFDM

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schemes in packet-switchedtransmissions”, EurasipJournal on Advanceson Signal Processing,Volume 2009, ArticleID 698417, 2009.

Abstract. In this pa-per, α-OFDM, a ge-neralization of the OFDMmodulation, is propo-sed. This new modu-lation enhances the ou-tage capacity perfor-mance of bursty com-munications. The α-OFDMscheme is easily imple-mentable as it only re-quires an additional timesymbol rotation afterthe IDFT stage and asubsequent phase ro-tation of the cyclic pre-fix. The physical effectof the induced rotationis to slide the DFT win-dow over the frequencyspectrum. When suc-cessively used with dif-ferent angles α at thesymbol rate, α-OFDMprovides frequency di-versity in block fadingchannels. Interestingly,simulation results showa substantial gain interms of outage capa-city and outage BERin comparison with clas-sical OFDM modula-tion schemes. The fra-mework is extended tomulti-antenna and multi-cellular OFDM basedstandards. Practical si-mulations, in the contextof 3GPP-LTE, calledhereafter α-LTE, sus-tain our theoretical claims.

• R. Couillet, M. Debbah,Mathematical Founda-tions for Signal Pro-cessing, Communications

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and Networking, CRCPress, Taylor & Francis Group,2011 [chapitre de livre]

Abstract. From basic trans-forms to Monte Carlosimulation to linear pro-gramming, the text co-vers a broad range ofmathematical techniquesessential to understan-ding the concepts andresults in signal pro-cessing, telecommuni-cations, and networking.Along with discussingmathematical theory,each self-contained chap-ter presents examplesthat illustrate the useof various mathemati-cal concepts to solvedifferent applications.Each chapter also in-cludes a set of home-work exercises and rea-dings for additional study.

• R. Couillet, M. Deb-bah, Orthogonal Fre-quency Division Mul-tiple Access Funda-mentals and Appli-cations, Auerbach Pu-blications, CRC Press,Taylor & Francis Group,2010 [chapitre de livre]

Abstract. Supportedby the expert-leveladvice of pioneeringresearchers, Ortho-gonal Frequency Di-vision Multiple Ac-cess Fundamentalsand Applications pro-vides a comprehen-sive and accessibleintroduction to thefoundations and ap-plications of one ofthe most promisingaccess technologiesfor current and fu-ture wireless networks.It includes autho-ritative coverage of

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the history, funda-mental principles,key techniques, andcritical design issuesof OFDM systems.

• R. Couillet, M. Deb-bah, No. 08368023.1“Method for short-time OFDM trans-mission and appa-ratus for performingflexible OFDM mo-dulation”

• R. Couillet, S. Wag-ner, No. 09368025.4“Precoding processfor a transmitter ofa MU-MIMO com-munication system”

• R. Couillet, No.09368030.4 “Pro-cess for estimatingthe channel in anOFDM communi-cation system, andreceiver for doingthe same”

II1. R. Couillet, IdeeInnovante “UserSubspace Cluste-ring”

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