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Entropy2016,18, 425 4. Hsu,E.P. StochasticAnalysis onManifolds;AmericanMathematicalSociety: Providence,RI,USA,2002. 5. FrĂ©chet,M. LesĂ©lĂ©mentsalĂ©atoiresdenaturequelconquedansunespacedistancie. Annalesde l’Institut HenriPoincarĂ©1948,10, 215–310. 6. Fletcher,P.; Lu,C.; Pizer, S.; Joshi, S. Principalgeodesicanalysis for the studyofnonlinear statisticsof shape. IEEETrans.Med. Imaging2004,23, 995–1005. 7. Vaillant, M.; Miller, M.; Younes, L.; TrouvĂ©, A. Statistics on diffeomorphisms via tangent space representations. NeuroImage2004,23, S161–S169. 8. Younes,L. ShapesandDiffeomorphisms; Springer: Berlin/Heidelberg,Germany,2010. 9. Pennec,X. Intrinsic statisticsonriemannianmanifolds: Basic tools forgeometricmeasurements. J.Math. ImagingVis. 2006,25, 127–154. 10. Karcher,H.Riemanniancenterofmassandmollifiersmoothing.Commun.PureAppl.Math.1977,30, 509–541. 11. Huckemann,S.;Hotz,T.;Munk,A. Intrinsic shapeanalysis: GeodesicPCAforRiemannianmanifolds modulo isometric liegroupactions. Stat. Sin. 2010,20, 1–100. 12. Jung,S.;Dryden, I.L.;Marron, J.S. Analysisofprincipalnestedspheres. Biometrika2012,99, 551–568. 13. Pennec,X.Barycentric subspacesandafïŹnespans inmanifolds. InProceedingsof theSecondInternational Conference on Geometric Science of Information, Paris, France, 28–30 October 2015; Nielsen, F., Barbaresco,F.,Eds.;LectureNotes inComputerScience;pp. 12–21. 14. Sommer,S.Horizontaldimensionalityreductionanditeratedframebundledevelopment. InGeometricScience of Information; Springer: Berlin/Heidelberg,Germany,2013;pp. 76–83. 15. Zhang,M.;Fletcher,P. Probabilisticprincipalgeodesicanalysis. InProceedingsof the26thInternational Conference on Neural Information Processing Systems, Lake Tahoe, Nevada, 5–10 December 2013; pp.1178–1186. 16. Tipping, M.E.; Bishop, C.M. Probabilistic principal component analysis. J. R. Stat. Soc. Ser. B 1999, 61, 611–622. 17. Elworthy,D. Geometric aspects ofdiffusionsonmanifolds. InÉcole d’ÉtĂ© deProbabilitĂ©s de Saint-Flour XV–XVII, 1985–1987; Hennequin, P.L., Ed.; Number 1362 in LectureNotes inMathematics; Springer: Berlin/Heidelberg,Germany,1988;pp. 277–425. 18. Mok,K.P. Onthedifferentialgeometryof framebundlesofRiemannianmanifolds. J.ReineAngew.Math. 1978,1978, 16–31. 19. Taubes,C.H.DifferentialGeometry: Bundles,Connections,Metrics andCurvature, 1sted.;OxfordUniversity Press:Oxford,UK;NewYork,NY,USA,2011. 20. KolĂĄrˇ, I.; SlovĂĄk, J.;Michor,P.W.NaturalOperations inDifferentialGeometry; Springer: Berlin/Heidelberg, Germany,1993. 21. Andersson, L.; Driver, B.K. Finite dimensional approximations towienermeasure andpath integral formulasonmanifolds. J.Funct.Anal. 1999,165, 430–498. 22. Fujita,T.;Kotani,S.i. TheOnsager-Machlupfunctionfordiffusionprocesses. J.Math. KyotoUniv. 1982, 22, 115–130. 23. Strichartz,R.S. Sub-Riemanniangeometry. J.Differ. Geom. 1986,24, 221–263. 24. Bloch,A.M. Nonholonomicmechanics and control. In InterdisciplinaryAppliedMathematics; Springer: NewYork,NY,USA,2003;Volume24, 25. Marsden, J.E.;Ratiu,T.S. Introductiontomechanicsandsymmetry.InTexts inAppliedMathematics;Springer: NewYork,NY,USA,1999;Volume17, 26. Leite, F.S.; Krakowski, K.A. Covariant Differentiation under Rolling Maps; Centro de MatemĂĄtica da UniversidadedeCoimbra:Coimbra,Portugal, 2008. 27. Hinkle, J.;Fletcher,P.T.; Joshi,S. Intrinsicpolynomials forregressiononriemannianmanifolds. J.Math. ImagingVis. 2014,50, 32–52. 28. Noakes, L.;Heinzinger,G.; Paden,B. Cubic splinesoncurvedspaces. IMAJ.Math. Control Inf. 1989, 6, 465–473. 29. Camarinha, M.; Silva Leite, F.; Crouch, P. On the geometry of Riemannian cubic polynomials. Differ.Geom.Appl. 2001,15, 107–135. 30. Team,T.T.D.;Al-Rfou,R.;Alain,G.;Almahairi,A.;Angermueller,C.;Bahdanau,D.;Ballas,N.;Bastien,F.; Bayer, J.;Belikov,A.; etal. Theano:APythonframeworkfor fastcomputationofmathematicalexpressions. arXiv2016, arXiv:1605.02688. 422
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Differential Geometrical Theory of Statistics
Title
Differential Geometrical Theory of Statistics
Authors
Frédéric Barbaresco
Frank Nielsen
Editor
MDPI
Location
Basel
Date
2017
Language
English
License
CC BY-NC-ND 4.0
ISBN
978-3-03842-425-3
Size
17.0 x 24.4 cm
Pages
476
Keywords
Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
Categories
Naturwissenschaften Physik
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