Seite - 310 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 442 6. Watanabe,S.;Yamazaki,K.;Aoyagi,M.Kullback InformationofNormalMixture isNotanAnalyticFunction; TechnicalReportof IEICENeurocomputing;The InstituteofElectronics, InformationandCommunication Engineers,Tokyo, Japan,2004;pp. 41–46. (In Japanese) 7. Michalowicz, J.V.;Nichols, J.M.; Bucholtz, F. Calculationofdifferential entropy for amixedGaussian distribution. Entropy2008,10, 200–206. 8. Pichler,G.;Koliander,G.;Riegler,E.;Hlawatsch,F. Entropyforsingulardistributions. InProceedingsof the IEEEInternationalSymposiumonInformationTheory(ISIT),Honolulu,HI,USA,29 June–4 July2014; pp.2484–2488. 9. Huber,M.F.; Bailey, T.; Durrant-Whyte,H.; Hanebeck,U.D. Onentropy approximation forGaussian mixturerandomvectors. InProceedingsof the IEEEInternationalConferenceonMultisensorFusionand Integration for Intelligent Systems, Seoul,Korea, 20–22August 2008; IEEE:NewYork,NY,USA,2008; pp.181–188. 10. Yamada,M.; Sugiyama,M. Direct importanceestimationwithGaussianmixturemodels. IEICETrans. Inf.Syst. 2009,92, 2159–2162. 11. Durrieu, J.L.;Thiran, J.P.;Kelly,F. Lowerandupperbounds forapproximationof theKullback-Leibler divergencebetweenGaussianMixtureModels. InProceedingsof the IEEEInternationalConferenceon Acoustics,SpeechandSignalProcessing(ICASSP),Kyoto, Japan,25–30March2012; IEEE:NewYork,NY, USA,2012;pp. 4833–4836. 12. Schwander,O.;Marchand-Maillet,S.;Nielsen,F. Comix: Jointestimationandlightspeedcomparisonof mixturemodels. InProceedingsof the2016IEEEInternationalConferenceonAcoustics,SpeechandSignal Processing, ICASSP2016,Shanghai,China,20–25March2016;pp. 2449–2453. 13. Moshksar,K.;Khandani,A.K. ArbitrarilyTightBoundsonDifferentialEntropyofGaussianMixtures. IEEETrans. Inf. Theory2016,62, 3340–3354. 14. Mezuman,E.;Weiss,Y. ATightConvexUpperBoundontheLikelihoodofaFiniteMixture. arXiv2016, arXiv:1608.05275. 15. Amari,S.-I. InformationGeometryandItsApplications; Springer: Tokyo, Japan,2016;Volume194. 16. Nielsen, F.; Sun, K. GuaranteedBounds on theKullback–LeiblerDivergence ofUnivariateMixtures. IEEESignalProcess. Lett. 2016,23, 1543–1546. 17. Nielsen,F.;Garcia,V. Statisticalexponential families:Adigestwithflashcards. arXiv2009, arXiv:0911.4863. 18. Calafiore,G.C.;ElGhaoui,L.OptimizationModels;CambridgeUniversityPress:Cambridge,UK,2014. 19. Shen,C.;Li,H. Onthedual formulationofboostingalgorithms. IEEETrans. PatternAnal.Mach. Intell. 2010,32, 2216–2231. 20. Beck,A. Introduction toNonlinearOptimization: Theory,Algorithms, andApplicationswithMATLAB; Society for IndustrialandAppliedMathematics: Philadelphia,PA,USA,2014. 21. Boyd,S.;Vandenberghe,L.ConvexOptimization;CambridgeUniversityPress:Cambridge,UK,2004. 22. De Berg, M.; vanKreveld, M.; Overmars, M.; Schwarzkopf, O.C. Computational Geometry; Springer: Heidelberg,Germany,2000. 23. Setter,O.;Sharir,M.;Halperin,D.ConstructingTwo-DimensionalVoronoiDiagramsviaDivide-and-Conquerof Envelopes inSpace; Springer:Heidelberg,Germany,2010. 24. Devillers,O.;Golin,M.J. Incremental algorithms forfinding the convexhulls of circles and the lower envelopesofparabolas. Inf. Process. Lett. 1995,56, 157–164. 25. Nielsen, F.; Yvinec,M. Anoutput-sensitive convexhull algorithm for planar objects. Int. J. Comput. Geom.Appl. 1998,8, 39–65. 26. Nielsen,F.;Nock,R. Entropiesandcross-entropiesofexponential families. InProceedingsof the17thIEEE InternationalConferenceonImageProcessing (ICIP),HongKong,China, 26–29September2010; IEEE: NewYork,NY,USA,2010;pp. 3621–3624. 27. Sharir, M.; Agarwal, P.K. Davenport-Schinzel Sequences and Their Geometric Applications; Cambridge UniversityPress:Cambridge,UK,1995. 28. Bronstein,M. Algorithms and computation inmathematics. InSymbolic Integration. I. Transcendental Functions; Springer: Berlin,Germany,2005. 29. Carreira-Perpinan,M.A.Mode-findingformixturesofGaussiandistributions. IEEETrans. PatternAnal. Mach. Intell. 2000,22, 1318–1323. 310