Koszul Information Geometry & Souriau Lie Group Thermodynamics

21/09/2014
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Koszul Information Geometry & Souriau Lie Group Thermodynamics

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application/pdf Koszul Information Geometry & Souriau Lie Group Thermodynamics Frédéric Barbaresco

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www.thalesgroup.com Thales Air Systems Date Koszul Information Geometry & Souriau Lie Group Thermodynamics Frédéric BARBARESCO, THALES AIR SYSTEMS Senior Scientist & Advanced Studies Manager, Advanced Radar Concepts Dept. SEE Emeritus Member, Ampere Medal 2007 Aymé Poirson Prize 2014 (French Academy of Sciences) 2 /2 / Corpus of Geometric Science of Information GSI federates skills from Geometry, Probability and Information Theory: shape spaces (geometric statistics on manifolds and Lie groups, deformations in shape space,…), probability/optimization & algorithms on manifolds (structured matrix manifold, structured data/Information, …), relational and discrete metric spaces (graph metrics, distance geometry, relational analysis,…), computational and hessian information geometry, algebraic/infinite dimensionnal/Banach information manifolds, divergence geometry, tensor-valued morphology, optimal transport theory, manifold & topology learning Thales Air Systems 11/06/2014 3 /3 / Geometric Science of Information Corpus & applications Thales Air Systems 11/06/2014 Information Theory Hessian, Kähler & Symplectic Geometries Lie Group Theory Metric Space Geometry Probability & Statistics Theory 4 /4 / Related International Publications or Events Leon Brillouin Seminar on « Geometric Science of Information », Institut Henri Poincaré & IRCAM, launched by THALES since December 2009, with Ecole Polytechnque & INRIA/IRCAM http://repmus.ircam.fr/brillouin/past-events French/Indian Workshop on « Matrix Information Geometry », Ecole Polytechnique & Thales Research & Technology, 23-25th February 2011 (with Prof. Rajendra Bhatia) http://www.lix.polytechnique.fr/~schwander/resources/mig/slides/ SMAI’11 Congress, Mini-Symposium on « Information Geometry », 23-27th Mai 2011 http://smai.emath.fr/smai2011/programme_detaille.php Symposium on « Information Geometry & Optimal Transport », hosted at Institut Henri Poincaré, 12th February 2012 (with GDR CNRS MIA) https://www.ceremade.dauphine.fr/~peyre/mspc/mspc-thales-12/ SMF/SEE GSI’13 Conference on « Geometric Science of Information », Ecole des Mines de Paris, August 2013 http://www.see.asso.fr/gsi2013 Special Issue "Information, Entropy & their Geometric Structures", in MDPI Entropy http://www.mdpi.com/journal/entropy/special_issues/entropy-Geome SEE GSI’15 Conference « Geometric Science of Information », Ecole Polytechnique, 28-30 October 2015, Paris Thales Air Systems 11/06/2014 5 /5 / Information Geometry and Geometric Science of Information French/Indian Workshop on « Matrix Information Geometry », Ecole Polytechnique & Thales Research & Technology, 23-25th February 2011 (with Prof. Rajendra Bhatia) http://www.lix.polytechnique.fr/~schwander/resources/mig/slides/ SMAI’11 Congress, Mini-Symposium on « Information Geometry », 23-27th Mai 2011 http://smai.emath.fr/smai2011/programme_detaille.php Thales Air Systems 11/06/2014 6 /6 / Information Geometry and Geometric Science of Information Symposium on « Information Geometry & Optimal Transport », hosted at Institut Henri Poincaré, 12th February 2012 (with GDR CNRS MIA) https://www.ceremade.dauphine.fr/~peyre/mspc/mspc-thales-12/ SMF/SEE Conference on « Geometric Science of Information », Ecole des Mines de Paris, August 2013 http://www.see.asso.fr/gsi2013 Thales Air Systems 11/06/2014 Yann Ollivier & Jean-Louis Koszul 7 /7 / Information Geometry and Geometric Science of Information Leon Brillouin Seminar on « Geometric Science of Information », Institut Henri Poincaré & IRCAM, launched by THALES since December 2009, with Ecole Polytechnque & INRIA/IRCAM http://repmus.ircam.fr/brillouin/past-events Thales Air Systems 11/06/2014 8 /8 / Geometric Science of Information: Books by SPRINGER 3 Books published by SPRINGER Matrix Information Geometry http://www.springer.com/engineering/signals/b ook/978-3-642-30231-2 Geometric Science of Information http://www.springer.com/computer/image+proc essing/book/978-3-642-40019-3 Geometric Theory of Information http://www.springer.com/engineering/signals/b ook/978-3-319-05316-5 Special Issue of « Entropy » Journal Information, Entropy and their Geometric Structures http://www.mdpi.com/journal/entropy/special_is sues/entropy-Geome Organized at THALES RESEARCH & TECHNOLOGY Sponsored by THALES Thales Air Systems 11/06/2014 www.thalesgroup.com Introduction 10 /10 / Geometric Science of Information Takeshi SASAKI W. BLASCHKE Eugenio CALABI Calyampudi R. RAO Nikolai N. CHENTSOV Hirohiko SHIMA Jean-Louis KOSZUL Von Thomas FRIEDRICH Y. SHISHIDO Homogeneous Convex Cones G. E.B. VINBERG Jean-Louis KOSZUL Homogeneous Symmetric Bounded Domains G. Elie CARTAN Carl Ludwig SIEGEL Probability in Metric Space Maurice R. FRECHET Information Theory Nicolas .L. BRILLOUIN Claude. SHANNON Probability on Riemannian Manifold Michel EMERY Marc ARNAUDON Geometric Science of Information KOSZUL-VINBERG METRIC (KOSZUL-VINBERG CHARACTERISTIC FUNCTION) FISHER METRIC Probability/G. on structures Y. OLLIVIER M. GROMOV Contact G. Vladimir ARNOLD 11 /11 / Thales Air Systems Date Probability on Riemannian Manifold M. Arnaudon, L. Miclo, ”Means in complete manifolds: uniqueness and approximation”, http://arxiv.org/abs/1207.3232 M. Arnaudon, C. Dombry, A. Phan, Le Yang, « Stochastic algorithms for computing means of probability measures. », Stochastic Processes and their Applications122, pp. 1437-1455, 2012 M. Arnaudon, A. Thalmaier, “Brownian motion and negative curvature”, Boundaries and Spectra of Random Walks, Progress in Probability, Vol. 64, 145--163, Springer Basel ,2011 M. Arnaudon, F. Barbaresco, Le Yang, ”Medians and means in Riemannian geometry: existence, uniqueness and computation” Matrix Information Geometry, Nielsen, Frank; Bhatia, Rajendra (Eds.), Springer, http://arxiv.org/pdf/1111.3120v1 Le Yang, « Médianes de mesures de probabilité dans les variétés riemanniennes et applications à la détection de cibles radar », PhD with advisors M. arnaudon & F. Barbaresco http://tel.archives-ouvertes.fr/docs/00/66/41/88/PDF/Dissertation- Le_YANG.pdf , THALES PhD Award 2012 Marc Arnaudon PhD with M. Emery, Bordeaux University P-Means Computation on Riemannian Manifold Stochastic Flow on Riemannian Manifold Michel Emery IRMA Lab, Strasbourg University Probability on Riemannian Manifold Stochastc Calculus on Manifolds M. Émery, G. Mokobodzki, « Sur le barycentre d'une probabilité dans une variété », Séminaire de probabilités de Strasbourg, 25 , p. 220-233, 1991 http://archive.numdam.org/article/SPS_1991__25__220 _0.pdf M. Émery, P.A. Meyer, « Stochastic calculus in manifolds », Springer 1989 M. Émery, W. Zheng, « Fonctions convexes et semimartingales dans une variété », Séminaire de Probabilités XVIII, Lecture Notes in Mathematics 1059, Springer 1984 M. Fréchet, « L’intégrale abstraite d’une fonction abstraite et son application à la moyenne d’un élément aléatoire de nature quelconque », Revue Scientifique, 483-512, 1944 M. Fréchet, « Les éléments aléatoires de nature quelconque dans un espace distancié ». Annales IHP, 10 no. 4, p. 215-310, 1948 http://archive.numdam.org/article/AIHP_1948__10_4_2 15_0.pdf [ ] functionconvexcontinuous,)()( ϕϕϕ xEx ≤ [ ] ( )( ) 0)(exp)( 1 =⇒= ∫ − dxPxgxgEb M b 12 /12 / Thales Air Systems Date Probability on structures: Gromov/Ollivier work on Fisher Metric Y. Ollivier, « Probabilités sur les espaces de configuration d'origine géométrique », PhD with advisers M. Gromov & P. Pansu http://www.yann-ollivier.org/rech/publs/these.pdf Y. Ollivier, « Le Hasard et la Courbure (Randomness and Curvature) », habilitation: http://www.yann-ollivier.org/rech/publs/hdr_intro.pdf Y. Ollivier & Youhei Akimoto, « Objective improvement in information-geometric optimization », FOGA 2013, preprint 2012 http://www.yann-ollivier.org/rech/publs/deeptrain.pdf Yann Ollivier, Ludovic Arnold, Anne Auger, and Nikolaus Hansen, « Information-geometric optimization: A unifying picture via invariance principles », arXiv:1106.3708v1, 2011 http://www.yann-ollivier.org/rech/publs/quantile_igo.pdf Video « seminaire Brillouin »: http://archiprod- externe.ircam.fr/video/VI02023900-226.mp4 Yann Ollivier, Paris-Sud University, LRI Dept. CNRS Bronze Medal 2011 Introduction of probabilistic models on structured objects Interplay between probability and differential geometry Natural Gradient by Fisher Information Matrix (IGO) Misha Gromov, IHES Abel Prize 2009 Mathematics about "interesting structures“ Category-theoretic approach of Fisher Metric M. Gromov, « In a Search for a Structure, Part 1: On Entropy », preprint, July 2012 http://www.ihes.fr/~gromov/PDF/structre-serch- entropy-july5-2012.pdf Chap 2. « Fisher Metric and Von Neumann Entropy» Such a rescaling, being a non-trivial symmetry, is a significant structure in its own right; for example, the group of families of such "rescalings" leads the amazing orthogonal symmetry of the Fisher metric M. Gromov, « Convex sets and Kähler manifolds», in Advances in J. Differential Geom., F. Tricerri ed., World Sci., Singapore, pp. 1-38, 1990 http://www.ihes.fr/~gromov/PDF/%5B68%5D.pdf 13 /13 / Thales Air Systems Date Koszul forms, characteristic function & metric: Hessian structure H. Shima, “The Geometry of Hessian Structures”, World Scientific, 2007 http://www.worldscientific.com/worldscibooks/10.1142/6241 dedicated to Prof. Jean-Louis Koszul (« the content of the present book finds their origin in his studies ») H. Shima,Symmetric spaces with invariant locally Hessian structures, J. Math. Soc. Japan,, pp. 581-589., 1977 H. Shima, « Homogeneous Hessian manifolds », Ann. Inst. Fourier, Grenoble, pp. 91-128., 1980 H. Shima, « Vanishing theorems for compact Hessian manifolds », Ann. Inst. Fourier, Grenoble, pp.183-205., 1986 H. Shima, « Harmonicity of gradient mappings of level surfaces in a real affiffiffiffine space », Geometriae Dedicata, pp. 177-184., 1995 H. Shima, « Hessian manifolds of constant Hessian sectional curvature », J. Math. Soc. Japan, pp. 735-753., 1995 H. Shima, « Homogeneous spaces with invariant projectively flat affiffiffiffine connections », Trans. Amer. Math. Soc., pp. 4713-4726, 1999 Hirohiko Shima, Emeritus Professor of Yamaguchi Univ. PhD from Osaka University Interplay between the Geometry of Hessian Structures and Information Geometry Jean-Louis Koszul, French Sciences Academy PhD student of Henri Cartan, Bourbaki member Introduction of Koszul forms, Koszul-Vinberg characteristic function & metric J.L. Koszul, « Sur la forme hermitienne canonique des espaces homogènes », complexes, Canad. J. Math. 7, pp. 562-576., 1955 J.L. Koszul, « Domaines bornées homogènes et orbites de groupes de transformations affines », Bull. Soc. Math. France 89, pp. 515-533., 1961 J.L. Koszul, « Ouverts convexes homogènes des espaces affines », Math. Z. 79, pp. 254-259., 1962 J.L. Koszul, « Variétés localement plates et convexité », Osaka J. Maht. 2, pp. 285-290., 1965 J.L. Koszul, « Déformations des variétés localement plates », .Ann Inst Fourier, 18 , 103-114., 1968 See: M. N. Boyom, « Convexité locale dans l’espace des connexions symétriques. Critère de comparaison des modèles statistiques »,, March 2012, IHP, Paris http://www.ceremade.dauphine.fr/~peyre/mspc/mspc- thales-12/ « Les connexions symétriques est un sous-ensemble convexe contenant le sous-ensemble des connexions localement plates » ( ) ( ) ψψψ ψψ logdetlogloglog 0log,)( connectionflatcones,dualand, * * *, * dsdsd Ddgdxex Dx xx =−= >== ΩΩΩ∈ ∫ Ω − o Koszul-Vinberg Characteristic Function & metric of regular convex cone ΩΩΩΩ 14 /14 / The Geometry of Hessian Structures Hessian Geometry and J.L. Koszul Works Hirohiko Shima Book, « Geometry of Hessian Structures », world Scientific Publishing 2007, dedicated to Jean-Louis Koszul J.L. Koszul www.thalesgroup.com Thales Air Systems Date Hessian Information Geometry & Lie Group Thermodynamics 16 /16 / History: Cartan by Poincaré « la forme des mathématiques = la structure du groupe » Thales Air Systems Date 17 /17 / From Elie Cartan to Jean-Louis Koszul & Jean-Marie Souriau Thales Air Systems Date 2 2 2 2 log)(log x (x) x p E x ∂ ∂ =      ∂ ∂ − Ωψξ ξ Ω∈∀= ∫ Ω − Ω xdex x )( * , ξψ ξ ( ) ( ) )(.,, 22121 1 ZadQZZfZZf Z+=β [ ]( ) ( ) ()( ) [ ],..Im,,,,, 212121 ββ βββ =∈∀∈∀= adZZZZfZZg g * gg ∈ ∂ ∂ −=∈∈ − Q Q CfKereTemperatur CapacityHeat ,,:)( β ββ β ∫ ∈ −− = * ξ xx x deep ξξ ξξ ,, /)( GeDf toassociatedcocyclewith))(( θθ= ( ) [ ] ( ) MapMomentSouriauwith ,,.)(),( 2121,2,1 µ µξξσ ZZfZZZZ MM += ( ) ( ) InvolutionCartan,where ),(with)(,, g adadTryxByxByx yx ∈ =−= η η β ψ ∂ ∂ −= ∂ ∂ = Ω Q x (x) IFisher 2 2 log Koszul Forms Koszul Characteristic Function Koszul Hessian Metric Jean-Louis Koszul Elie Cartan [ ] [ ]( ) [ ]( )2121 ,,,,, ZZfZZg βββ ββ = (x)d Ω= ψα log Ω== ψα log2 dDg Marcel Berger Cartan Classification of Symmetric Spaces Jean-Marie Souriau Souriau Moment Map Souriau Geometric Temperature/Heat Capacity Souriau Metric from Symplectic Cocycle 18 /18 / Thales Air Systems Date Koszul Information Geometry, Souriau Lie Group Thermodynamics Koszul Information Geometry Model Souriau Lie Groups Thermodynamics Model Characteristic function Ω∈∀−=Φ ∫ Ω − xdex x log)( * , ξξ g∈∀−=Φ ∫ − βωβ ξβ log)( )(. M U de Entropy ξξξ dppx xx∫ Ω −=Φ * )(log)()( ** ∫−= M dpps ωξξ )(log)( Legendre Transform )(,)( *** xxxx Φ−=Φ )(.)( ββ Φ−= QQs Density of probability ∫ − − +− = = * ξ,x ξ,x x Φ(x)x,ξ x dξe e p ep )( )( ξ ξ ∫ − − Φ+− = = M U U U de e p ep ω ξ ξ ξβ ξβ β βξβ β )(. )(. )()(. )( )( Dual Coordinate Systems ** and Ω∈Ω∈ xx ∫ ∫ ∫ − Ω − Ω == * ξ,x ξ,x x dξe de dpx * * . )(.* ξξ ξξξ * gg ∈∈ Qandβ ∫ ∫ ∫ − − == M U M U M de deU dpUQ ω ωξ ωξξ ξβ ξβ β )(. )(. )( )().( heatGeometricor MapMomentSouriauofMean: mapMomentSouriau: eTemperaturGeometricSouriau: Q U β Dual Coordinate Systems x x x ∂ Φ∂ = )(* and * ** )( x x x ∂ Φ∂ = β∂ Φ∂ =Q and Q s ∂ ∂ =β Hessian Metric )(22 xdds Φ−= ( )βΦ−= 22 dds Fisher metric 2 ,2 2 2 2 2 * log )( )(log )( x de x (x) xI x p ExI x x ∂ ∂ = ∂ Φ∂ −==       ∂ ∂ −= ∫ Ω − ξ ξ ξ ξ 2 )(.2 2 2 2 2 log )( )(log )( β ω β β β β ξ β ξβ β ξ ∂ ∂ = ∂ Φ∂ −=         ∂ ∂ −= ∫ − M U de )( I p EI CapacityGeometricSouriau: )( 2 2 β ββ β β ∂ ∂ −= ∂ ∂ −= ∂ Φ∂ −= Q K Q)( I 19 /19 / Thales Air Systems Date Koszul Information Geometry, Souriau Lie Group Thermodynamics Koszul Information Geometry Model Souriau Lie Groups Thermodynamics Model Convex Cone Ω∈x Ω convex cone Ω∈β Ω convex cone: largest open subset of g , Lie algebra of G, such that ∫ − M U de ωξβ )(. and ∫ − M U de ωξ ξβ )(. . are convergent integrals Transformation ( )Ω∈→ Autggxx with )(ββ ga→ Transformation of Potential (non invariant) ( )gxgxx detlog)()()( +Φ=Φ→Φ ΩΩΩ ( ) ( ) ( ) ( )βθβββ 1 )( − −Φ=Φ→Φ aag Transformation of Entropy (invariant) ( ) ( )***** *** )( x x gx x Ω Ω ΩΩ Φ=      ∂ Φ∂ Φ→Φ x x x ∂ Φ∂ = Ω )( with * ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )βθβββ θ β βθ β ββ ββ 1 '' )( )( ' ' ' ' .with )(.'''.'' − −Φ=Φ=Φ=Φ += ∂ +Φ∂ = ∂ Φ∂ = = =Φ−=Φ−=→ aa aQa a aa Q a QsQQQsQs g g g g g * Information Geometry Metric (invariant) ( ) ( )[ ] ( )xI x x x gx gxI = ∂ Φ∂ −= ∂ +Φ∂ −= ΩΩ 2 2 2 2 )(detlog)( ( ) ( ) ( )[ ] ( ) ( )β β β β βθβ β I a aI = ∂ Φ∂ −= ∂ −Φ∂ −= − 2 2 2 12 )(g 20 /20 / Thales Air Systems Date Koszul-Vinberg Characteristic Function François Massieu in 1869 demonstrated that some thermal properties of physical systems could be derived from “characteristic functions”. This idea was developed by Gibbs and Duhem with the notion of potentials in thermodynamics, and introduced by Poincaré in probability. We will study generalization of this concept by Jean-Louis Koszul in Mathematics Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of “Information Geometry” theory: defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF (their gradients defining mutually inverse diffeomorphisms) Fisher Information Metrics as hessian of these dual functions. Koszul proved that these metrics are invariant by all automorphisms of the convex cones. 21 /21 / Thales Air Systems Date Koszul-Vinberg Characteristic Function Jean-Marie Souriau has extended the Characteristic Function in Statistical Physics: looking for other kinds of invariances through co-adjoint action of a group on its momentum space defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector (a vector valued temperature of Planck), giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean “Moment map” by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the exclusive “Inner Product” selection using symmetric bilinear “Cartan-Killing form” (introduced by Elie Cartan in 1894). www.thalesgroup.com Legendre Duality in Information Geometry 23 /23 / Duality in Projective Geometry: Pascal’s Mystic Hexagram In projective geometry, Pascal's theorem (the Hexagrammum Mysticum Theorem) states that: if an arbitrary six points are chosen on a conic (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon The dual of Pascal’s Theorem is known as Brianchon’s Theorem: Pascal’s Theorem: If the vertices of a simple hexagon are points of a point conic, then its diagonal points are collinear. Brianchon’s Theorem: If the sides of a simple hexagon are lines of a line conic, then the diagonal lines are concurrent. 24 /24 / Duality of Information Geometry: Multivariate Gaussian Law Dual Coordinates systems & Potential functions Potential Functions are Dual and related by Legendre transformation : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    −−−+−= +Θ−Θ= ⇒     +Σ−== == −−−− −−−− −− enΗ)(ηΗηΗΦ nTrΘΨ mmmη,ΗΗ Σ)m,(Σθ,ΘΘ T T T π πθθ 2log2detlog21log2 ~~ )log(2detlog22 ~~ , ~ 2 ~ scoordinateDual 1111 1112 11        = ∂ ∂ = ∂ ∂       = ∂ ∂ = ∂ ∂ Θ Η Φ θ η Φ Η Θ Ψ η θ Ψ ~ ~ and~ ~ ( )TT ΘΗTrH,Θ ΨH,ΘΦ += −≡ θη ~~ with ~~~~ ( ) [ ]pEΗΦ log ~~ = Entropy ji * ij ji ij ΗΗ Φ g ΘΘ Ψ g ∂∂ ∂ ≡ ∂∂ ∂ = ~ and ~ 22 Hessians are convexe and define Riemannian metrics : 25 /25 / Duality of Information Geometry: Multivariate Gaussian Law Link with Kullback Divergence For each Dual Geometry, we can built a divergence that is directly related to Kulback-Leibler Divergence : ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) dx mxp mxp mxp,Σm,N,ΣmNDiv H,ΘΗΦΘΨ,Σm,N,ΣmNDiv H,ΘΗΦΘΨ,Σm,N,ΣmNDiv ∫ Σ Σ Σ= ≥−+≡ ≥−+≡ 22 11 111122 21211122 * 12121122 ,/ ,/ log,/ 0 ~~~~~~ 0 ~~~~~~ Riemannian Metric As Potential are convexe, their Hessians define Riemannian Metrics : ( ) ( )( ) ( )( ) ( )( ) ( )( )( )         −−Σ+ Σ−ΣΣ+ΣΣ− = − −−− T mmmmTr Tr ,Σm,N,ΣmNDiv 2121 1 2 1 1 1 21 1 21 1122 detlog 2 1 ( ) ( )332 2 1 2 1 iji ij ijiij dΗOdΗdΗgdΘOdΘdΘgds +=+= www.thalesgroup.com Historical Development of Characteristic Function Concept 27 /27 / Paul LEVY (general use of characteristic function in Probability) Henri POINCARE (Introduction of characteristic function Ψ in Probability) ψφφ logor == eψ 1869 François MASSIEU (introduction of characteristic function in Thermodynamic: Gibbs-Duhem Potentials) ( )TT S /1 . 1 ∂ ∂ −= φ φ « je montre, dans ce mémoire, que toutes les propriétés d’un corps peuvent se déduire d’une fonction unique, que j’appelle la fonction caractéristique de ce corps» Koszul Characteristic: Massieu/Poincaré/Levy/Balian Roger BALIAN (metric for quantum states by hessian metric from Von- Neumann Entropy) Roger Balian, 1986 DISSIPATION IN MANY-BODY SYSTEMS: A GEOMETRIC APPROACH BASED ON INFORMATION THEORY XDXFDS ˆ,ˆ)ˆ()ˆ( −= XTrXF ˆexpln)ˆ( = [ ]DdDdTrSdds ˆln.ˆ22 =−= 28 /28 / Thales Air Systems Date Influence of Massieu on Poincaré [M. Massieu showed that, if we make choice for independent variables of v and T or of p and T, there is a function, moreover unknown, of which three functions of variables, p, U and S in the first case, v, U and S in the second, can be deducted easily. M. Massieu gave to this function, the form of which depends on the choice of variables, name of characteristic function.] [Because from functions of M. Massieu, we can deduct the other functions of variables, all the equations of the Thermodynamics can be written not so as to contain more than these functions and their derivatives; it will thus result from it, in certain cases, a large simplification. We shall see soon an important application of these functions.] 2nd edition of Poincaré Lecture on « Thermodynamic » 29 /29 / Pierre Duhem Thermodynamic Potentials Duhem P., « Sur les équations générales de la Thermodynamique », Annales Scientifiques de l’Ecole Normale Supérieure, 3e série, tome VIII, p. 231, 1891 “Nous avons fait de la Dynamique un cas particulier de la Thermodynamique, une Science qui embrasse dans des principes communs tous les changements d’état des corps, aussi bien les changements de lieu que les changements de qualités physiques “ four scientists were credited by Duhem with having carried out “the most important researches on that subject”: F. Massieu had managed to derive Thermodynamics from a “characteristic function and its partial derivatives” J.W. Gibbs had shown that Massieu’s functions “could play the role of potentials in the determination of the states of equilibrium” in a given system. H. von Helmholtz had put forward “similar ideas” A. von Oettingen had given “an exposition of Thermodynamics of remarkable generality” based on general duality concept in “Die thermodynamischen Beziehungen antithetisch entwickelt“, St. Petersburg 1885 WTSEG +−=Ω )( Duhem-Massieu Potentials in Thermodynamics 30 /30 / Projective Legendre Duality and Koszul Characteristic Function (projective) LEGENDRE DUALITY LEGENDRE TRANSFORM (between Dual Space) CONTACT/SYMPLECTIC GEOMETRIES (Legendre mapping, fibration,…) (Analytic) FOURIER DUALITY FOURIER TRANSFORM (Time-/Frequency Dual Spaces) LINEAR ALGEBRA (Linear Signal Processing) 31 /31 / Projective Legendre Duality and Koszul Characteristic Function LEGENDRE TRANSFORM FOURIER/LAPLACE TRANSFORM ENTROPY= LEGENDRE(- LOG[LAPLACE]) Ψ=Φ−= 22 log ddg INFORMATION GEOMETRY METRIC Sddg 2*2* =Ψ= ∫ Ω − −=Φ−=Ψ * , log)(log)( dyexx yx )(,)( *** xxxx Ψ−=Ψ ξξξ dpp xx∫ Ω −=Ψ * )(log)(* Φ(x)x,ξξ,xξ,x x edξeep * +−−− == ∫/)(ξ ∫ Ω = * )(.* ξξξ dpx x Legendre Transform of minus logarithm of characteristic function (Laplace transform) = ENTROPY !!! ds2=d2ENTROPY ds2=-d2LOG[FOURIER] 32 /32 / Maurice René Fréchet: Probability in Metric Spaces Maurice René Fréchet 1948 (Annales de l’IHP) Les éléments aléatoires de nature quelconque dans un espace distancié Extension of Probability/Statistic in abstract/Metric space 1928 Book from Fréchet PhD Abstract Spaces Invention of Metric Space 1928 Jacques Hadamard PhD Supervisor 33 /33 / ab h h2 = a2 + b2 « Mean » of structured data: Fréchet Barycenter in Metric Space Right Triangle {a,b,h} h2=a2+b2 { } 222 1 with,, iii N iiii bahhba +== { } { } { }{ }( )HBAhbadMinHBA iii N i p geodesique HBA ,,,,,arg,, 1 ,, ∑= = a b h >90° 222 bah +> Inside the cone (bounded domain) Ambligone Triangle p=2 : Mean, p=1 Median 34 /34 / Thales Air Operations Laplace Median Definition : « valeur probable » ( )mxEMinmediandxxP median m X −=⇔=∫ 5.0).( 0 Laplace has proved in 1774 that : let F be the cumulative distribution function of an (absolute continuous) random variable; the median is defined as the value µ such that F(µ) = 0.5. Laplace proved that µ is also the value minimizing the average of the absolute deviations, where the deviation between two values is their L1 (called also Manhattan) distance i.e., the absolute value of their differences. Laplace called this value "le milieu de probabilité" or "la valeur probable". The term median has been introduced by Cournot in l’Exposition de la théorie des chances in 1883. 35 /35 / Cartan Center of Mass and Karcher Flow Cartan Center of Mass Elie Cartan has proved that the following functional : is strictly convexe and has only one minimum (center of mass of A for distribution da) for a manifold of negative curvature Karcher Flow Hermann Karcher has proved the convergence of the following flow to the Center of Mass : E. J. Cartan H. Karcher ∫∈ A daamdmf ),(: 2 aΜ ( ) )()0(avec)(.exp)(1 nnnnmnnn mfmfttm n −∇=∇−==+ γγ & )(exp 1 ∫ − −=∇ A m daaf 36 /36 / Median by Median Reflection/Verblunsky Coefficients µµµµk www.thalesgroup.com Koszul-Vinberg Characteristic Function, Entropy & Density 38 /38 / Hessian Geometry by J.L. Koszul Hirohiko Shima Book, « Geometry of Hessian Structures », world Scientific Publishing 2007, dedicated to Jean-Louis Koszul Hirohiko Shima Keynote Talk at GSI’13 http://www.see.asso.fr/file/5104/download/9914 Prof. M. Boyom tutorial : http://repmus.ircam.fr/_media/brillouin/ressources/une -source-de-nouveaux-invariants-de-la-geometrie-de-l- information.pdf Jean-Louis Koszul J.L. Koszul, « Sur la forme hermitienne canonique des espaces homogènes complexes », Canad. J. Math. 7, pp. 562-576., 1955 J.L. Koszul, « Domaines bornées homogènes et orbites de groupes de transformations affines », Bull. Soc. Math. France 89, pp. 515-533., 1961 J.L. Koszul, « Ouverts convexes homogènes des espaces affines », Math. Z. 79, pp. 254-259., 1962 J.L. Koszul, « Variétés localement plates et convexité », Osaka J. Maht. 2, pp. 285-290., 1965 J.L. Koszul, « Déformations des variétés localement plates », .Ann Inst Fourier, 18 , 103-114., 1968 Foundation of Information Geometry: Jean-Louis Koszul Works 39 /39 / Koszul-Vinberg Characteristic Function/Metric of convex cone J.L. Koszul and E. Vinberg have introduced an affinely invariant Hessian metric on a sharp convex cone through its characteristic function. is a sharp open convex cone in a vector space of finite dimension on (a convex cone is sharp if it does not contain any full straight line). is the dual cone of and is a sharp open convex cone. Let the Lebesgue measure on dual space of , the following integral: is called the Koszul-Vinberg characteristic function Ω E R * Ω Ω ξd * E E Ω∈∀= ∫ Ω − Ω xdex x )( * , ξψ ξ 40 /40 / Koszul-Vinberg Characteristic Function/Metric of convex cone is analytic function defined on the interior of and as If then is logarithmically strictly convex, the function is strictly convex Koszul 1-form αααα: The differential 1-form is invariant by . If and then and Koszul 2-form ββββ : The symmetric differential 2-form is a positive definite symmetric bilinear form on invariant under (from Schwarz inequality and ) Koszul-Vinberg Metric: defines a Riemanian structure invariant by and then the Riemanian metric Ω∈∀= ∫ Ω − Ω xdex x )( * , ξψ ξ Ωψ +∞→Ω )(xψΩ Ω∂→x ( )Ω∈ Autg ( ) )(det 1 xggx Ω − Ω = ψψ Ωψ ( ))(log)( xx ΩΩ = ψφ ΩΩΩΩ === ψψψφα /log ddd ( )Ω= AutG Ω∈x Eu ∈ ∫ Ω − −= * , .,, ξξα ξ deuu x x * Ω−∈xα Ω== ψαβ log2 dD E ( )Ω= AutG ( ) ∫ Ω − Ω = * ,2 ,,,log ξξξψ ξ devuvud u αD ( )ΩAut Ω= ψlog2 dg D: flat connection D, (D,g) is called a Hessian structure 41 /41 / Koszul-Vinberg Characteristic Function/Metric of convex cone Koszul-Vinberg Metric : We can define a diffeomorphism by: with When the cone is symmetric, the map is a bijection and an isometry with a unique fixed point (the manifold is a Riemannian Symmetric Space given by this isometry): , and is characterized by is the center of gravity of the cross section of : Ω= ψlog2 dg [ ] ( )loglog 2 1log log)(log 2 u 2 22 ∫∫ ∫∫ ∫ ∫ ∫ − +== dudv dudvdd du dud dudxd vu vuv u uu u ψψ ψψψψ ψ ψψ ψψ )(log* xdx x Ω−=−= ψα )()(),( 0 tuxf dt d xfDuxdf t u +== = Ω xx α−=* xx =** )( nxx =* , cstexx =ΩΩ )()( * *ψψ * x { }nyxyyx =Ω∈= ,,/)(minarg ** ψ * x { }nyxy =Ω∈ ,,* * Ω ∫∫ Ω − Ω − = ** ,,* /. ξξξ ξξ dedex xx 42 /42 / Koszul Metric of Convex Cone Koszul Metric: defines a Riemanian structure invariant by The Riemanian metric is given by: This result is obtained using Schwarz inequality, and where and Thales Air Systems Date αD ( )ΩAut Ω= ψlog2 dg ( ) ξξξ ξξξξξξξ ψ ψ ξξ ,)(and)(with 0)().()(.)( )( 1 )()(log , 2 1 , 2 1 2 22 2 2 *** ueGeF dGFdGdF u uxd xx −− ΩΩΩ == >                 −= ∫∫∫ ψ ψ ψ d d =log 22 2 log       −= ψ ψ ψ ψ ψ dd d ( ) ∫ Ω − −= * ,)()( , ξξψ ξ dueuxd x ( ) ∫ Ω − = * 2,2 ,)()( ξξψ ξ dueuxd x 43 /43 / Koszul Entropy via Legendre Transform we can deduce “Koszul Entropy” defined as Legendre Transform of minus logarithm of Koszul-Vinberg characteristic function : with and where Demonstration: we set Using and we can write: and )(,)( *** xxxx Φ−=Φ Φ= xDx* * * Φ= x Dx )(log)( xx Ω−=Φ ψ Ω∈∀= ∫ Ω − Ω xdex x )( * , ξψ ξ ∫∫ Ω − Ω − = ** ,,* /. ξξξ ξξ dedex xx ∫∫ Ω − Ω − Ω −==− ** ,,* /,)(log, ξξξψ ξξ dedehxdhx xx h ∫∫ Ω − Ω −− =− ** ,,,* /.log, ξξ ξξξ dedeexx xxx ∫∫∫∫ ∫∫∫ Ω − Ω −− Ω − Ω − Ω − Ω − Ω −−         −        =Φ +−=Φ **** *** ,,,,,** ,,,,** /.loglog.)( log/.log)( ξξξξ ξξξ ξξξξξ ξξξξ dedeededex dededeex xxxxx xxxx 44 /44 / Koszul-Vinberg Characteristic Function Legendre Transform Thales Air Systems Date                         −=Φ =             −             =Φ           −=Φ         −         =Φ +−=Φ−=Φ ∫ ∫∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫∫∫∫ ∫∫∫ Ω Ω − − Ω − − Ω Ω − − Ω Ω − − − Ω Ω − − Ω − Ω Ω − − − Ω − Ω − Ω −− Ω − Ω − Ω − Ω − Ω −− * ** * * * * * * * * * * **** *** , , , , ** , , , , , , , ,** , , ,,** ,,,,,** ,,,,*** log.)( 1with.log.log)( .loglog)( /.loglog.)( log/.log)(,)( ξ ξξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξξξξ ξξξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξξ ξξξξξ ξξξξ d de e de e x d de e d de e ed de e dex d de e edex dedeededex dededeexxxx x x x x x x x x x x x x x x xx xxxxx xxxx 45 /45 / [ ]( ) [ ] [ ]∫ Ω Φ=Φ≥Φ⇒ Φ−≥Φ Φ≤Φ⇒Φ * * * )()()()( )(,)(:TransformLegendre )(conv.:Ineq.Jensen *** *** ** ξξξξ ξξ Edpx xxxx EE x Koszul Entropy via Legendre Transform We can then consider this Legendre transform as an entropy, that we could named “Koszul Entropy”: With and ξξξξ ξξ ξ ξ ξ ξ dppd de e de e xxx x x x ∫ ∫ ∫ ∫ Ω Ω − − Ω Ω − − −=−=Φ * * * * )(log)(log , , , , * Φ(x)x,ξ dξex,ξ ξ,xξ,x x eedξeep * ξ,x * +− ∫−− −− === − ∫ log /)(ξ ( ) ξξξξξξ dedξ.edpΦDx ξΦΦ(x)x,ξ xx * * − Ω +− Ω ∫∫∫ ==== ** .)(.* [ ] 1log)()( loglog)( * * * * * * * )()( )()(, =⇒−Φ=Φ −=−=Φ ∫∫ ∫∫ Ω Φ− Ω Φ− Ω Φ+Φ− Ω − ξξ ξξ ξξ ξξ dedexx dedex xx [ ] [ ]( )ξξ ξξξξξξ ξξξξξξ ξξ ξξ EE dpdp xdpdpp eep xx xxx xx x ** ** *** *)()(, )(or )(.)()(ifonlyandif )()()()()(log )(loglog)(log ** ** * Φ=Φ         Φ=Φ Φ=Φ=−⇒ Φ−=== ∫∫ ∫∫ ΩΩ ΩΩ Φ−Φ+− 46 /46 / Barycentre & Koszul Entropy Thales Air Systems Date [ ] [ ]( )ξξ EE * )( * Φ=Φ [ ])(,)( *** xxxSupx x Φ−=Φ ( ) ξξξξξξ dedξ.edpΦDx ξΦΦ(x)x,ξ xx * * − Ω +− Ω ∫∫∫ ==== ** .)(.*         Φ=Φ ∫∫ ΩΩ ** )(.)()( ** ξξξξξξ dpdp xx Φ(x)x,ξ dξex,ξ ξ,xξ,x x eedξeep * ξ,x * +− ∫−− −− === − ∫ log /)(ξ Barycenter of Koszul Entropy = Koszul Entropy of Barycenter 47 /47 / Koszul metric & Fisher Metric To make the link with Fisher metric given by matrix , we can observe that the second derivative of is given by: We could then deduce the close interrelation between Fisher metric and hessian of Koszul-Vinberg characteristic logarithm. [ ] 2 2 2 2 2 2 2 2 2 2 2 2 * log)(log )( ,)()(log ,)()()(log x (x) x (x) x p ExI x (x) x xx x p xxp x x x ∂ ∂ = ∂ Φ∂ −=      ∂ ∂ −=⇒ ∂ Φ∂ = ∂ −Φ∂ = ∂ ∂ −Φ=Φ−= Ωψξ ξξ ξξξ ξ 2 2 2 2 log)(log )( x (x) x p ExI x ∂ ∂ =      ∂ ∂ −= Ωψξ ξ FISHER METRIC (Information Geometry) = KOSZUL HESSIAN METRIC (Hessian Geometry) )(xI )(log ξxp 48 /48 / Koszul Metric and Fisher Metric as Variance We can also observed that the Fisher metric or hessian of KVCF logarithm is related to the variance of : Thales Air Systems Date ξ ∫ ∫ ∫ − − − −= ∂ ∂ ⇒= * * * ξ,x ξ,x ξ,x dξe dξex (x)Ψ dξe(x)Ψ . 1log loglog ξ                 +−         −= ∂ ∂ ∫∫∫ ∫ −−− − 2 2 22 2 ... 1log *** * ξ,xξ,xξ,x ξ,x dξedξedξe dξe x (x)Ψ ξξ 2 2 2 2 2 2 )(.)(... log         −=             −= ∂ ∂ ∫∫∫ ∫ ∫ ∫ − − − − *** * * * xxξ,x ξ,x ξ,x ξ,x dξpdξpdξ dξe e dξ dξe e x (x)Ψ ξξξξξξ [ ] [ ] )( log)(log )( 22 2 2 2 2 ξξξ ψξ ξξξ VarEE x (x) x p ExI x =−= ∂ ∂ =      ∂ ∂ −= Ω 49 /49 / Relation of Koszul density with Maximum Entropy Principle The density from Maximum Entropy Principle is given by: If we take such that: Then by using the fact that with equality if and only if , we find the following: Thales Air Systems Date       = =         − ∫ ∫ ∫ Ω Ω Ω *(.) * * * )(. 1)( such)(log)( xdp dp dppMax x x xx px ξξξ ξξ ξξξ ∫−− −− − == ∫ * ξ,x * dξex,ξ ξ,xξ,x x edξeeq log /)(ξ        −−== == ∫ ∫∫∫ Ω − ∫−− −− − * , log log,log)(log 1/).( ξξξ ξ ξ dexeq dξedξedξq x dξex,ξ x ξ,xξ,x x * ξ,x *** ( )1 1log − −≥ xx 1=x ξ ξ ξ ξξ ξ ξ ξ d p q pd q p p x x x x x x ∫∫ ΩΩ       −−≤− ** )( )( 1)( )( )( log)( 50 /50 / Relation of Koszul density with Maximum Entropy Principle We can then observe that: because We can then deduce that: If we develop the last inequality, using expression of : Thales Air Systems Date 0)()( )( )( 1)( *** =−=      − ∫∫∫ ΩΩΩ ξξξξξ ξ ξ ξ dqdpd p q p xx x x x 1)()( ** == ∫∫ ΩΩ ξξξξ dqdp xx ξξξξξξξ ξ ξ ξ dqpdppd q p p xxxx x x x ∫∫∫ ΩΩΩ −≤−⇒≤− *** )(log)()(log)(0 )( )( log)( )(ξxq ξξξξξξξ ξ ddexpdpp x xxx ∫ ∫∫ Ω Ω − Ω         −−−≤− * ** , log,)()(log)( ∫∫∫ Ω − ΩΩ +≤− *** , log)(.,)(log)( ξξξξξξξ ξ dedpxdpp x xxx )(,)(log)( * * xxxdpp xx Φ−≤− ∫ Ω ξξξ )()(log)( ** * xdpp xx Φ≤− ∫ Ω ξξξ 51 /51 / New Definition of Maximum Entropy We have then observed that Koszul Entropy provides density of Maximum Entropy: with and where and Thales Air Systems Date ( ) ( ) ∫ − − Θ− Θ− = * ξ, ξ, dξe e p ξ ξ ξ ξ 1 1 )( )(1 ξ− Θ=x dx xd x )( )( Φ =Θ=ξ ξξξξ ξ dp∫ Ω = * )(. ∫ Ω − −=Φ * , log)( ξξ dex x 52 /52 / Crouzeix relation on hessian of dual potentials Previously, we have used the duality between dual potential functions that is recovered by this relation: with and where If we develop relations, we can deduce that the hessian of one potential function is the inverse of the hessian of the dual potential function, then Information Geometry metric could be given in two systems of dual coordinates: Thales Air Systems Date *** ,)()( xxxx =Φ+Φ dx d x Φ =* * * dx d x Φ = )(log)( xx Ω−=Φ ψ 2* 2* *22 * 2* *21 2* *2 2 2 2 2 1 2* *2 2 2 2* *2 2 2 *2* *2 * 2 2 * * * ... 1. dx dx d dx dx d dx d dx dx d ds dx d dx d dx d dx d dx dx dx d dx dx dx d x dx d x dx d Φ −=      Φ       Φ −= Φ −=⇒       Φ = Φ ⇒= ΦΦ ⇒        = Φ = Φ ⇒       = Φ = Φ − − 53 /53 / Geodesics equation for Koszul Hessian Metric Last contribution has been given by Rothaus that has studied the construction of geodesics for this hessian metric geometry, using: with Or expressed also according the Christoffel symbol of the 1st kind: Then geodesic is given by: that could be developed with previous relation: We can then observe that: The geodesic equation can then be rewritten: and Thales Air Systems Date lkj il l jk j lk k ljili jk xxx (x) g x g x g x g g ∂∂∂ ∂ =         ∂ ∂ − ∂ ∂ + ∂ ∂ =Γ Ωψlog 2 1 2 1 3 ji ij xx (x) g ∂∂ ∂ = Ωψlog2 02 2 2 2 =Γ+=Γ+ ds dx ds dx ds xd g ds dx ds dx ds xd ji ijk k kl jik ij k 0 log 2 1log 32 2 2 = ∂∂∂ ∂ + ∂∂ ∂ ΩΩ jil ji lk k xxxds dx ds dx xxds xd ψψ lk k lji ji l xx ψ ds xd xxx ψ ds dx ds dx x ψ ds d ∂∂ ∂ + ∂∂∂ ∂ =      ∂ ∂ logloglog 2 2 23 2 2 0 loglog 2 22 2 2 =      ∂ ∂ + ∂∂ ∂ llk k x ψ ds d xx ψ ds xd lkjk ij j ki i jk ijk xxx (x) x g x g x g ∂∂∂ ∂ =         ∂ ∂ − ∂ ∂ + ∂ ∂ =Γ Ωψlog 2 1 2 1 3 0)( 2 *2 2 2 =− ds xd ds xd xI 54 /54 / Koszul-Vey Theorem J.L. Koszul and J. Vey have proved the following theorem: Koszul J.L., Variétés localement plates et convexité, Osaka J. Math. , n°2, p.285-290, 1965 Vey J., Sur les automorphismes affines des ouverts convexes saillants, Annali della Scuola Normale Superiore di Pisa, Classe di Science, 3e série, tome 24,n°4, p.641-665, 1970 Koszul-Vey Theorem: Let be a connected Hessian manifold with Hessian metric . Suppose that admits a closed 1-form such that and there exists a group of affine automorphisms of preserving : - If is quasi-compact, then the universal covering manifold of is affinely isomorphic to a convex domain real affine space not containing any full straight line. - If is compact, then is a sharp convex cone. M g α gD =α G M α GM / M ΩGM / Ω [] Koszul J.L., Variétés localement plates et convexité, Osaka J. Math. , n°2, p.285-290, 1965 [] Vey J., Sur les automorphismes affines des ouverts convexes saillants, Annali della Scuola Normale Superiore di Pisa, Classe di Science, 3e série, tome 24,n°4, p.641-665, 1970 55 /55 / Sasaki Works on Affine Geometry If we denote by the level surface of : which is a non-compact submanifold in , and by the induced metric of on , then assuming that the cone is homogeneous under , Sasaki proved that is a homogeneous hyperbolic affine hypersphere and every such hyperspheres can be obtained in this way . Sasaki also remarks that is identified with the affine metric and is a global Riemannian symmetric space when  is a self-dual cone. Let be a regular convex cone and let be the canonical Hessian metric, then each level surface of the characteristic function is a minimal surface of the Riemannian manifold . cS Ωψ { }cxSc == Ω )(ψ Ω cω Ωψlog2 d cS Ω )(ΩG cω cS Ω Ω= ψlog2 dgΩ Ωψ ),( gΩ 56 /56 / Koszul Forms/Metric for Homogenous Siegel Domains SD Koszul Forms for Homogeneous Bounded domains Koszul has developed his previously described theory for Homogenous Siegel Domains SD. He has proved that there is a subgroup G in the group of the complex affine automorphisms of these domains (Iwasawa subgroup), such that G acts on SD simply transitively. The Lie algebra of G has a structure that is an algebraic translation of the Kähler structure of SD. There is an integrable almost complex structure J on and there exists such that defines a J-invariant positive definite inner product on . Koszul has proposed as admissible form , the form : Koszul has proved that coincides, up to a positive number multiple with the real part of the Hermitian inner product obtained by the Bergman metric of SD by identifying with the tangent space of SD. The Koszul forms are then given by: Thales Air Systems Date * g∈η [ ]ηη ,,, YJXYX = * g∈η ξ ( ) ( )[ ] g∈∀−==Ψ XXadJJXadTrXX )(.,ξ ξ YX , ( )XdΨ−= 4 1 α αβ D= g g g g 57 /57 / Koszul Forms 1st Koszul Form : 2nd Koszul Form: Application for Poincaré Upper-Hal Plane: With and We can deduce that Koszul Forms/Metric for Homogenous Siegel Domains SD ( ) ( )[ ] gXXJadJXadTrX bg ∈∀−=Ψ )(/ ( )XdΨ−= 4 1 α αβ D= { }0/ >+== yiyxzV dx d yX = dy d yY = ( ) [ ] [ ]      = −= = ⇒ YJX YYX ZYZYad , ,. ( ) ( )[ ] ( ) ( )[ ]   =− =− 0 2 YJadJYadTr XJadJXadTr dydx y ∧=Ω 2 1 ( ) 2 22 2 2 2 2 1 4 1 2 y dydx ds y dydx d y dx X + =⇒ ∧ −=−=⇒= ΨαΨ g 58 /58 / Koszul form for Siegel Upper-Half Plane: Symplectic Group : Associated Lie Algebra: Jean-Louis Koszul Forms for Siegel Upper-Half Plane { }0/ >+== YiYXZV ( )       − =      =     == += 0 0 and 0 with , 1 I I J D BA S BDDBIDA DBAZSZ TTT -       + =        − =     −= =       00 0 , 0 0 baseandwith 0 jiij ij ji ij ijT T ee e e ad bb d ba βα ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         + = ∧ + =Ψ−=Ω + =+Ψ ⇒     +=Ψ =Ψ −− −− − ZddZYYTr p ds ZdYdZYTr P d dXYTr p idYdX pijij ij 112 11 1 8 13 8 13 4 1 2 13 13 0 δβ α 59 /59 / Point of View of Geometers: Cartan-Siegel Domains Metric Henri Poincaré (upper-half plane model of hyperbolic geometry) n=1 Elie Cartan (classification in 6 types of symmetric homogeneous bounded domains ) n<=3 Ernest Vinberg (link with homogeneous convexe cones, Siegel domains of 2nd kind) Fine structure of Information Geometry (Hessian Geometry, Kählerian Geometry) « Il est clair que si l’on parvenait à démontrer que tous les domaines homogènes dont la forme est définie positive sont symétriques, toute la théorie des domaines bornés homogènes serait élucidée. C’est là un problème de géométrie hermitienne certainement très intéressant » Dernière phrase de Elie Cartan, dans « Sur les domaines bornés de l'espace de n variables complexes », Abh. Math. Seminar Hamburg, 1935 ( ) ji ji ji zddz zz zzK ∑ ∂∂ ∂ = , 2 ,log Φ Jean-Louis Koszul (canonical hermitian form of complex homogeneous spaces, a complex homogeneous space with positive definite canonical hermitian form is isomorphic to a bounded domain,,Study of Affine Transform Groups of locally flat manifolds) Carl Ludwig Siegel (Siegel domains in framework of Symplectic Geometry) Lookeng Hua (Bergman, Cauchy and Poisson Kernels in Siegel domains) 60 /60 /Cartan-Siegel Symmetric Homogeneous Bounded Domains { } ( ) ( )2* ** 11111 1 1 ,,1/ zw wzKzzCzIVIIIIII , − =<∈==== Henri Poincaré (n=1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) domain.theofvolumeeuclideaniswhere ,:IVTypefor2ZZ1 1 , 1,:IIIType 1,:IIType ,:IType fordet 1 , -**t* * Ωµ ν Ωµ ν ν ν Ωµ ν ν nZWWWWZK p p qp ZWIWZK IV n III p II p I p,q =−+=       −= += += −= + −+ Lookeng Hua 02ZZ1,1ZZ :line1androwsnwithmatricescomplex:IVType porderofmatricessymmetricskewcomplex:IIIType porderofmatricessymmetriccomplex:IIType rowsqandlinespwithmatricescomplex:IType ):( MatrixrRectangulaComplex: 2tt >−+< −< + ++ ZZ conjugatetransposedIZZ Z IV n III p II p I p,q Carl Ludwig Siegel Elie Cartan (n<=3) 61 /61 / Hyperbolic Poincaré Space Action of SLAction of SLAction of SLAction of SL2222R Group on hyperbolic Poincaré PlanR Group on hyperbolic Poincaré PlanR Group on hyperbolic Poincaré PlanR Group on hyperbolic Poincaré Plan Möbius Transform is a transitive action that transforms upper half- plane to itself (homogeneous space) : M and –M have same action then we consider the quotient Group : Complex unit disk is link to upper half plane by Cayley transform : { }0)Im(:and)(,2 >∈=∈ + + =∈      = zCzHz dcz baz zMRSL dc ba M 222 / IRSLRPSL ±= { } 1 1et 1: z z iz HD iz iz z DH zCzD − + → + − → <∈= aa 1- 1 i - i 2 2 2 22 2 y dz y dydx ds = + = x y H D ( )22 2 2 1 4 z dz ds − = Carl Ludwig Siegel Contribution in his seminal book « Symplectic Geometry » : a generalization of Poincaré Space 1=−bcad 62 /62 / Siegel Metric for the Siegel Upper-Half Plane: Upper-Half Plane : Isometries of are given by the quotient group: with the Symplectic Group: Unique Metric invariante by : Invariance by Automorphisms: Siegel Metric of SHn nSH { }nIRnSpRnPSp 2/),(),( ±≡ ),( FnSp ( )( ) 1 )( − ++=⇒      = DCZBAZZM DC BA M     =− ⇔∈      = n TT TT IBCDA DBCA FnSp DC BA M symmetricet ),( { } ),2( 0 0 ,/),2(),( RnSL I I JJJMMFnGLMFnSp n nT ∈      − ==∈≡ )(ZM ( ) ( )( )ZdYdZYTrdsSiegel 112 −− =    = = nRY X 0 ( )( )( )212 nn dRRTrds − = iYXZ += { }0Im/),( >=∈+== Y(Z)CnSymiYXZSHn 63 /63 / Siegel Uppel-Half Plane Geometry { }0Im/),( >=∈+== Y(Z)CnSymiYXZSHn Siegel Upper Half Space X 0>Y ( ) ( )( )ZdYdZYTrds 112 −− = ( ) ( ) ( )( )∑= −+= n i iiZZd 1 2 21 2 1/1log, λλ kkk YiXZ .+= 1=k 2=k ( )[ ]212 dRRTraceds − = ( )kkkk RNWRiZ ,0if. ≡= 1=k 2=k ( ) ( )∑= = n k kRRd 1 2 21 2 log, λ ( ) ( )( ) ( )( ) 1 2121 1 212121, −− −−−−= ZZZZZZZZZZR ( ) 0...det 2/1 12 2/1 1 =−−− IRRR λ ( )( ) 0.,det 21 =− IZZR λ 64 /64 / Air Systems Division6 4 « At one point Siegel thought that too many unnecessary things were being published, so he decided not to publish anything at all » George Polya The Polya Picture Album, Encounters of a Mathematician, Birkäuser Carl Ludwig Siegel With George Polya ( ) ( )( ) iYXZZdYdZYTrdsSiegel +== −− with112 Carl Ludwig siegel 65 /65 / QUANTIZATION IN COMPLEX SYMMETRIC SPACES F. Berezin { } ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )BBItraceBBIgFABg ZFZgFZBAtraceZFZgF ZZItraceZZIzF AZBBAZZg BAAB IBBAA I I JJJgg BB BA g IZZZSD t t t n ++− ++ − + + + +=+=⇒= ∂∂=∂∂⇒++= −−=−−= ++=     =− =−       − ==      = <= logdetlog))0(()0( )()(logRe2)())(( logdetlog)(:potentialKähler )(with 0 where 0 0 withandwith / 1* **** 1** * * ** Z gZ ZgjZZKZgjzgjgZgZKZZd K ZZK hc ZZd K ZZK ZgZfhcgf h h ∂ ∂ ==      =       = ∫ ∫ − − − ),(,),(),(),(),(with),( )0,0( ),( )( ),( )0,0( ),( )()()(, **** /1* 1 * /1* µ µ ( ) ( ) ( ) ( ) ( )ν βααβ βα βα βα π µ µ −+ += ∂∂ ∂ −== = ∑ WWIWWF WW WWF gdWdWgds WWd WWFWWd n L det),(where ,log with , ,, * * *2 , * , 2 * ** * A 66 /66 / Information Geometry for Multivariate Gaussian Law of zero Mean and intrinsec Geometry of Hermitian Positive Definite Matrices (particular case of Siegel Upper-Half Plane) provide the same metric Information Geometry: Geometry of Siegel Upper-Half Plane: ( )( )( )212 nn dRRTrds − = [ ] ( )( ) [ ] nn nnnnn RRTr n n nn RRE mZmZR eRRZp nn = −−= = + −−− − ˆand .ˆwith ..)()/( 1 .ˆ1 π         −= * 2 . )/(ln )( ji nn ij Zp Eg ∂θ∂θ θ∂ θ 0=nm with Siegel HPD Matrix Metric = IG Metric of Multiv. Gaussian Law { }0Im/),( >=∈+== Y(Z)CnSymiYXZSHn ( ) ( )( ) iYXZZdYdZYTrdsSiegel +== −− with112    = = nRY X 0 ( )( )( )212 nn dRRTrds − = 67 /67 / Siegel Distance: Particular Case (X=0) and General Case: Particular Case (pure imaginary axis) : General Case of Siegel Upper-Half Plane Distance: Distance between HPD Matrices: particular case of Siegel ( ) ( ) ( )∑= −− == n k kRRRRRd 1 222/1 12 2/1 121 2 log..log, λ ( ) 0det 12 =− RR λwith 0with ≠∈+= XSHiYXZ n ( ) n n k k k Siegel SHZZZZd ∈                 − + = ∑= 21 1 2 21 2 ,with 1 1 log, λ λ with ( ) 0.),(det 21 =− IZZR λ ( ) ( )( ) ( )( ) 1 2121 1 212121, −− −−−−= ZZZZZZZZZZR 0avec >= RiRZ 68 /68 / Siegel has deduced an other distance from : Another distance deduced from Siegel Work ( )[ ]212 .ds − ΣΣ= dTr ( )( ) 1212 2 12 11 2 1 1 1 2 1 112 .T=Rand. TTT +−−−−− =Σ+ΣΣ−Σ= R is hermitian positive definite matrix with eigen-values : )..(and)(rwith 1 1 2/1)1(1)2(2/1)1( kk 2 nnn k k k RRRRr − ==      + − = σλσ λ λ We deduce that : ( ) ∑=       − + =            − + =ΣΣ n k k k n n r r RI RI Trd 1 2/1 2/1 2 2/1 2/1 2 21 2 1 1 lnln, because : [ ] [ ] ∑∑ = ∞ = − =      + ==      − + n k j k k k n n ra k R RR RI RI 1 j 2 0 22/11 2/1 2/1 2 RTrnd 1.2 ..4tanh.4ln 69 /69 / HPD Matrices Geometry Geometry of Hermitian Positive Definite Matrices given by: Geodesic : Properties of this space Symetric Space as studied by Elie Cartan : Existence of bijective geodesic isometry Bruhat-Tits Space : semi-parallelogram inequality Cartan-Hadamard Space (Complete, simply connected with negative sectional curvature Manifold) [ ]10with),(.))(,( ,tRRdttRd YXX ∈=γ ( ) ( ) YXYX X t XYXXX RRRt X RRRR RRRRRReRt XYX o=== == −−−− )2/1(and)1(,)0( )( 2/12/12/12/12/1log2/1 2/12/1 γγγ γ ( ) ( ) 2/12/12/12/12/11- ),( avec)(X ABAAABABABAXG BA −− == ooo Xx)d(x,x)d(x,xd(x,z)),xd(x xzxx ∈∀+≤+ ∀∃∀ 224 quetel, 2 2 2 1 22 21 21 70 /70 / x 1x 2x z U VU + VU − V Cartan’s Symmetric Space Symetric Space as studied by Elie Cartan : Existence bijective geodesic isometry Bruhat-Tits Space : semi-parallelogram inequality 2222 22 VUVUVU +=++− ),( AGB BA = BGA BA ),( = ( ) )(X-1 ),( BABAXG BA oo= X ( ) 2/1/212/12/12/1 ABAAABA −− =o ( ) [ ]1,0 )( 2/12/12/12/1 ∈ = −− t ABAAAt t γ A=)0(γ B=)1(γ 2 2 2 1 22 21 224 )d(x,x)d(x,xd(x,z)),xd(x +≤+ E. J. Cartan M. Berger J. Tits 71 /71 / Air Systems Division Symmetric space classification • Premier Miracle : La théorie des espaces symétriques peut être considérée comme le premier miracle de la géométrie riemannienne, en fait comme un nœud de forte densité dans l’arbre de toutes les mathématiques . … On doit à Elie Cartan dans les années 1926 d’avoir découvert que ces géométries sont , dans une dimension donnée, en nombre fini, et en outre toutes classées. • Second Miracle : Entre les variétés localement symétriques et les variétés riemanniennes générales, il existe une catégorie intermédiaire, celle des variétés kählériennes. … On a alors affaire pour décrire le panorama des métriques kählériennes sur notre variété, non pas à un espace de formes différentielles quadratiques, très lourd, mais à un espace vectoriel de fonctions numériques [le potentiel de Kähler]. … La richesse Kählérienne fait dire à certains que la géométrie kählérienne est plus importante que la géométrie riemannienne. • Pas d’espoir d’autre miracle : Ne cherchez pas d’autres miracles du genre des espaces (localement) symétriques et des variétés kählériennes. En effet, c’est un fait depuis 1953 que les seules variétés riemanniennes irréductibles qui admettent un invariant par transport parallèle autre que g elle-même (et sa forme volume) sont les espaces localement symétriques, les variétés kählériennes, les variétés kählérienne de Calabi-Yau, et les variétés hyperkählériennes. Marcel Berger (IHES), « 150 ans de Géométrie Riemannienne », Géométrie au 20ième siècle, Histoire et horizons, Hermann Éditeur, 2005 Read More : Marcel Berger, « A Panoramic View of Riemannian Geometry », Springer 2003 M. Berger 72 /72 / This isometry for metric space : Is an extension of this one : To be compared with Euclidean « symmetric » space Symmetric Space ( ) ( ) ( ) ( )    = =• ••= −− −− 22/12/12 2/12/12/12/12/1 1- ),( log, with)(X BAABA ABAAABA BABAXG BA δ ( ) ( ) ( ) ( )    = = = − 212 1 log, with abba abba baxbaxG - (a,b) δ o oo ( )     −= + =       + +      + = 22 , 2with 22 baba ba baba -x ba xG(a,b) δ o 73 /73 / is the only fixed point because : due to trace property of : Unicity of fixed point ( ) 2/12/12/12/12/1 ABAAA −− ( )( ) CXICXX ICXXCXXXCCX =⇒=⇒ =⇒= −− −−−−− 2/12/1 rootsquareofunicity 2/12/12/12/11 ( )( ) ( )( ) ( ))(,2,)(X),(X 1-1-1 BAXdIXBABAdXBABAd •=••=•• − { } 1 1 1 2 )(ofseigenvaluewith log)( − = = •       =• ∑ XBA )BAd(X, n ii n k i λ λ 74 /74 / For space of Symmetric Positive Definite matrices, the analogue of is : It constitutes a natural framework for a generalized theory of convexity, where the role of arithmetic mean is played by a midpoint pairing : is called -convex if For space of symmetric positive definite matrices, an affine function is given by : and is closely related to entropy : Convexity ba )1( λλ −+ ( ) 2/12/12/12/1 ABAAABA λ λ −− =• BA λ• BAA 0•= BAA 1•= f ( )21,•• ( ) ( ) ( )YfXfYXf 21 •≤• detlog=f ( ) csteREntropy +−= detlog 75 /75 / Bruhat-Tits Space & Semi-parallelogram Law Bruhat-Tits Space : Space (Sym++,δ) is called Bruhat-Tits Space, where, according to δ metric, point at equal distance of 2 points is unique and given by geometric mean: ( ) ( ) ),(),(, ),(2),(2),( 0detwithlog),( definitepositive),(, 11 1 1 2 −− − = == == =−      = ∈∀ ∑ BABABCCACC BABBAABA IABBA RnSymBA TT n i i δδδ δδδ λλδ oo Bruhat-Tits Space is a complete metric space that verifies semi-parallelogram law : Xx)δ(x,x)δ(x,xδ(x,z)),xδ(x zxx ∈+≤+ ∃∈∀ allfor224 :such thatX, 2 2 2 1 22 21 21 Bruhat-Tits Space are particular cases of Cartan-Hadamard Manifold [1] F. Bruhat & J. Tits, « Groupes réductifs sur un corps local », IHES, n°41, pp.5-251, 1972 « Bruhat-Tits Space» (Metric Space) 76 /76 / Bruhat-Tits Space & Semi-Parallelogram Law Bruhat-Tits Space : Semi-parallelogram Law : Xx)δ(x,x)δ(x,xδ(x,z)),xδ(x zxx ∈+≤+ ∃∈∀ allfor224 :such thatX, 2 2 2 1 22 21 21 x 1x 2x z U VU + VU − 2222 22 VUVUVU +=++− Deduced from parallelogram law : V 2 22 2 12 2 32 2 212 22 )(x,xd)(x,xd)(x,xd),x(xd +=+ 2 22 2 12 2 2 2 212322 224.2 )(x,xd)(x,xd(x,z)d),x(xd)(x,xd(x,z)d +=+⇒= 3x 77 /77 / Geodesic between matrix P and Geodesic between Q & R : The geodesic projection is a contraction : Geodesic Projection Q R P P Q ( ) ( )[ ] 2/12/12/12/12/12/12/12/1 ,)( PPQRQQQPPtst ts s −−−− == σσ ( ) ( )QPdistQPdist MM ,)(),( ≤ΠΠ M : geodesically convex closed submanifold of Hadamard Manifold ( )SPdistP MS M ,minarg)( ∈ =Π ΠΠΠΠM(P) ΠΠΠΠM(Q) P Q 78 /78 / Kähler Potential & Bergman Kernel We can find Bergman Kernel of Kähler geometry for Siegel Upper- half space by mean of Siegel Unit Disk : Analogy with Bergman Kernel for Poincaré unit disk : ),(ln ZZKΦ B−= ( ) { }0/on),( >−= ZI-ZZWZIWZKB { } ( ) ( ) ( ) ( ) ( )22 22 2 2222 22 2 2 2 1 1 1 1 )(1 1 1ln 1/on1),(with),(ln z dz zdzd zz zgdzdds zz zzz zz g z z z z zzwzwzKzzK BB − = ∂∂ Φ∂ == − = − −−− = ∂∂ Φ∂ =⇒ − = ∂ Φ∂ ⇒−−=Φ <−=−=Φ ( )( ) 1− +− →Ψ nn nn iIZiIZZ SD:SH a { } ZZZIZSym(n,C)/ZSDn =<∈= 22 with With Bergman Kernel ( ) ( )( )ZdZZIdZZZITrZdZd ZZ ds 11 2 2 −− −−= ∂∂ Φ∂ = 79 /79 / Upper-Half Planes & Disks of H. Poincaré and C.L. Siegel ( ) ( ) ( ) *1*1*2 22 2 2 11 1 dwwwdwwwds w dw ds −− −−= − = iz iz w + − = ( )( ) 1− +−= iIZiIZW Upper-Half Plane of H. Poincaré Unit Disk of Poincaré Siegel Upper-Half Plane Unit Siegel Disk ( ) ),(and),( with 112 CnHPDYCnHermX iYXZ ZddZYYTrds ∈∈ += = −− 0andwith *112 2 2 2 22 2 >+= = = + = −− yiyxz dzdzyyds y dz y dydx ds ( ) ( )[ ]+−+−+ −−= dWWWIdWWWITrds 112 www.thalesgroup.com Koszul-Vinberg Characteristic Function, Entropy & Density for Symmetric/Hermitian Positive Definite Matrices 81 /81 / Koszul-Vinberg Characteristic Function/Metric of convex cone Let v be the volume element of g. We define a closed 1-form αααα and ββββ a symmetric bilinear form by: and The forms αααα and ββββ are called the first Koszul form and the second Koszul form for a Hessian structure (D; g) : A pair (D; g) of a flat connection D and a Hessian metric g is called a Hessian structure. J.L. Koszul studied a flat manifold endowed with a closed 1-form such that is positive definite, whereupon is a Hessian metric. This is the ultimate origin of the notion of Hessian structures. A Hessian structure (D; g) is said to be of Koszul type, if there exists a closed 1-form such that . vXvDX )(α= αβ D= [ ]( ) [ ]( ) [ ]       ∂∂ ∂ = ∂ ∂ = ∂ ∂ = ⇒∧= ji kl j i ij ijii n/ ij xx g x vg x α ..dxdxgv detlog 2 1 detlog det 2 2 1 121 α β α αD αD α αDg = 82 /82 / Koszul-Vinberg Characteristic Function/Metric of convex cone We can apply this Koszul theory for Symmetric Positive Definite Matrices. Let the inner product be the set of symmetric positive definite matrices is an open convex cone and is self-dual : Let be the regular convex cone consisting of all positive definite symmetric matrices of degree n. Then is a Hessian structure on , and each level surface of is a minimal surface of the Riemannian manifold ( ) )(,,, RSymyxxyTryx n∈∀= Ω Ω=Ω* )(det)( 2 1 dual-self )(, , * * n n xyTryx x Ixdex ψξψ ξ + − Ω=Ω = Ω − Ω == ∫ xd n dg detlog 2 1 log 22 + −== Ωψ 1* 2 1 detlog 2 1 log − Ω + = + =−= x n xd n dx ψ )detlog,( xDdD Ω xdet )detlog,( xDdg −=Ω ( ) 1 .det det with − = ∂ ∂ xx x x 83 /83 / Koszul Density: Application for SPD matrices We can then named this new density as “Koszul Density”: With Φ(x)x,ξ dξex,ξ ξ,xξ,x x eedξeep * ξ,x * +− ∫−− −− === − ∫ log /)(ξ ( ) ξξξξξξ dedξ.edpΦDx ξΦΦ(x)x,ξ xx * * − Ω +− Ω ∫∫∫ ==== ** .)(.* ( ) ( )[ ] ( ) ∫ Ω −− + +− === − * 1 ).(.withdet)( 1 detlog 2 1 ξξξξξαξ ξξααξ dpeep x Tr x n xTr x ( )           + = + =−== == ∈∀= − Ω + − Ω=Ω = Ω − Ω ∫ 1* 2 1 dual-self )(, , 2 1 detlog 2 1 log )(det)( )(,,, * * x n xd n dx Ixdex RSymyxxyTryx n n xyTryx x n ψξ ψξψ ξ www.thalesgroup.com Koszul Metric for Toeplitz Hermitian Positive Definite Matrices 85 /85 /Trench/Verblunsky Theorem & Partial Iwasawa Parameterization All Toeplitz Hermitian Positive Definite Matrix can be parameterized by Reflection/Verblunsky Coefficients: Block structure of covariance matrix & Verblunsky Parameterization: Verblunsky/Trench Theorem: Exitence of diffeomorphism ϕ       + = + −−− − −−− + −−−− 111 1 111 1111 ... . nnnnnn nnn n AARA A R αα αα       − −+ = −−− − + −−− + − − − 111 11111 1 1 . ... nnn nnnnnn n RAR RAARA R α [ ] *)( )( 11 1- 00 1 1 21 . 001 00 100 where 1 . 0 and.1with VV AA A P n n n n nnn           =      +      = =−= − − −− − − − Nµ ααµα ( ) { }1/with ,...,, : 110 1* <∈= ×→ − − + zCzD PR DRTHPD nn n n µµ ϕ a S. Verblunsky (PhD student of Littlewood) 86 /86 / Non-Symmetric Square Root of Siegel Group If we consider Cholesky decomposition of covariance matrix : Cholesky decomposition (Goldberg inversion algorithm) : All distribution of n-dimensionnal variable is associated with Affine Group. It is the element such that its action on vector Is transformed to random vector : This representation of Affine Group elements could be considered as non symmetric square root of Siegel Group element : ( ) ( ) + −−− −− + −−−− + −+− ΩΩ=Ω      Ω −=       +Ω −===Ω 2/1 1 2/1 112/1 11 2 n 1111 121 .and 01 1with . 1 .1.. nnn nn n nnnn n nnnnnn A W AAA A WWR µ µα ),0(~ nn INZ ),(~ nnn ANX Ω       =      +Ω =            Ω −−−− XAZZA nnnn 1 . 11 . 01 1 2/1 1 2/1 11       +Ω + −−−− + − 1111 1 . 1 nnnn n AAA A 87 /87 / Thales Air Operations Partial Iwasawa Decomposition Partial Iwasawa decomposition : the components of a positive definite or semi-definite matrix in the Iwasawa coordinates : Iwasawa, K., « On some types of topological groups », Ann. Math. Vol. 50, n°3, pp. 507–558, 1949 Every n×n positive definite matrix G can be uniquely expressed using its Iwasawa components as follows. where W & V are HPD matrices of size k×k and m×m respectively By computing the matrix multiplications in previous equation, we derive the following parametrization of positive semidefinite matrices : ( )[ ] ABBBA I XI V W G m k + =            = where 00 0       + = ++ VWXXWX WXW G Iwasawa (Lie Group Theory) 88 /88 / Preservation of Toeplitz Structure by Verblunsky Coefficients Conformal Information Geometry metric (metric = Hessian of Entropy): Entropy ΦΦΦΦ as Kähler potential: Conformal metric on Verblunsky parameterization: [ ]T n n P 110 )( −= µµθ L )*()( 2 ~ n j n i ijg θθ ∂∂ Φ∂ ≡ ( ) ( ) [ ] [ ]..ln.1ln).(.logdetlog, ~ 0 1 1 21 0 PenkneR)P(R n k knn πµπ −−−−=−=Φ ∑ − = − with [ ] ( )∑ − = + − −+      == 1 1 22 22 0 0)()(2 1 )(. n i i in ij n n d in P dP ndgdds µ µ θθ E. Kähler 1t withcoefficienVerblunsky: ⇔>Ω >      + − =Ω Hadamard Compactification { } DiskUnitPoincaré: 1/ 0 1 *1 . * D zzD h iba Rh h <=∈ + = ∈ >      =Ω + µ µ µ { } { }1/ )log( with),log( <=∈ ∈ + = zzD Rh h iba h µ µµ Scale Parameter Shape Parameter a b h >90° Ambligone Triangle h2>a2+b2 ( )( ) 222 22 2222 0 02 0det bah bah bahh hiba ibah +>⇔> +±=⇒ =+−+−⇒ =      −+ −− λ λ λλ λ λ www.thalesgroup.com Hua-Cartan & Iwasawa-Cartan Coordinates 103 /103 / Thales Air Operations Cartan Decomposition on Poincaré Unit Disk Lemma of Cartan for radial coordinates in Poincaré Disk : { } ( ) ( ) ( ) ( ) )(ln21ln)( . )( 1. )2coth( . .)2(8 )( )( )( )()( )()( and 0 0 with :ionDecompositCartan 1where)(andwith)1,1( 1/ 2 2 2 22 2 2222 21* )( 22 **** τ Φττ τ τ ∆θττ τ τ τ Φ ΨΦ ΨΦ τΦ Φ Φ ΨτΦ chzzF sh dshdds ethabz sheb chea tchtsh tshtch d e e u udug ba azb baz zg ab ba gSUg zzD LB i i i i i =−−= ∂ ∂ + ∂ ∂ + ∂ ∂ =⇒+= ==⇒     = = ⇒       =        = = =− + + =      =∈ <= − − + − 104 /104 / Thales Air Operations Iwasawa Decomposition on Poincaré Unit Disk Iwasawa coordinates in Poincaré Disk : { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) τ τ θ τ θ ξτθ ξτθ ξ τ τ ξ θθ θθ eu e u isheb e u ichea N tchtsh tshtch DK i i CCgCghNDKhg ba azb baz zg ab ba gSUg zzD i i =                −=         += ⇒       =      =      − =       − === =− + + =      =∈ <= − − − with 2 2/ 2 2/ 10 1 and )()( )()( , 2/cos2/sin 2/sin2/cos 1 1 2 1 ,)(with:Dec.Iwasawa 1where)(andwith)1,1( 1/ 22/ 22/ 1 22 **** 105 /105 / Thales Air Operations Hua-Cartan Decomposition on Siegel Unit Disk Lemma of Hua for radial coordinates in Siegel Disk (Hua-Cartan) : [ ] [ ] [ ] [ ] ( ) ( ) ( )[ ] [ ])()()( diag,Let )(with 0)( )(0 exp )()( )()( )( )( norderofmatricescomplexunitaryandexistthere 0 0 0 0 ,)( )()()()( )()()()( 0with 21 21 0 2 0 21* 212 2 2 00 00 0 * 0 * 00 00 ** 210 210 1121 n t n t t t n n nnn thththdiagP ZZeigenABPPUUABZ diagZ Z Z AB BA VBUB VAUA VU V V AB BA U U gnSp AB BA g shshshdiagB chchchdiagA τττ ττττ τ τ ττ ττ τ τ ττττ ττττ τττττττ L L L L LL = ==== =       =      ⇒     = =                   =∈      = = = ≤≤≤≤= +−− + − 106 /106 / Thales Air Operations Iwasawa Decomposition on Siegel Unit Disk Iwasawa coordinates in Siegel Disk : { } ( )( ) ( ) ( ) [ ] [ ] ( ) ( ) ( )           +−=     ++=       =⇒      − −+ =      =             ==                 =      − + ==       =        +− −−+ =⇒             ==       − ==      = ++=<= −− − − −+ SBAiBUB SBAiAUA AB BA KANh SiISi SiSiI Nh AB BA Ah S I SI NN eediag eediag BA BA A C U U C UUUUi UUiUU gU U U ghg iII iII CCgCgh AB BA g AZBBAZg(Z)IZZZSD n n n 2 1 2 1 with .2/.2/ .2/.2/ )(,)( norderofmatrixreal, 0 / 0 0 0 0 0 0 2 1 norderunitary, 0 0 )(/ 2 1 with)(, and/ 0001 0001 * 1 * 1 11 00 00 00 00 * 1 ** ** * 1 ** 1** 1 1 Ν Α Κ ττ ττ L L 107 /107 / Thales Air Operations Iwasawa/Cartan Coordinates on Siegel Unit Disk Iwasawa/Cartan coordinates relation in Siegel Disk : ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 000000 11 1* 0001 0001 0 * 0 ** 0000 00 00 ~ 00 00 ~ 2 ~ 2 with 2 1 2 1 :Iwasawa :Cartan ~ with , .2/.2/ .2/.2/ − −       −+      −+= ===                        +−=       ++=     = = ⇒      = −=+       =            − −+ = S i BAAS i BABH PUUHUUABZ SBAiBUB SBAiAUA VBUB VAUA AB BA g BASSBA AB BA MM AB BA SiISi SiSiI M tt t t SSS 108 /108 / Thales Air Operations Special Berezin Coordinates For every symmetric Riemannian space, there exist a dual space being compact. The isometry groups of all the compact symmetric spaces are described by block matrices (the action of the group in terms of special coordinates is described by the same formula as the action of the group of motions of the dual domain). Berezin coordinates for Siegel domain :       ===         =      = + + + − 0 0 with,:lyequivalentor , 1 ** I I LLLI AB BA AB BA t t t ΓΓΓΓ ΓΓ ( ) ( ) ( )++− +=+=⇒= BBItraceBBIgFABg lndetln))0(()0( 1* ( )( )       == ++=⇒      = − − IiI iII CCC AWAAWAW AA AA 2 1 with:Isometry )( 1 1 22211211 2221 1211 ΓΓ ΓΓ www.thalesgroup.com Souriau Lie Group Thermodynamic: Souriau Geometric Temperature and covariant definition of thermodynamic equilibriums 110 /110 / Thales Air Systems Date Covariant Definition of Thermodynamic Equilibriums Jean-Marie Souriau , student of Elie Cartan at ENS Ulm in 1946, has given a covariant definition of thermodynamic equilibriums formulated statistical mechanics and thermodynamics in the framework of Symplectic Geometry by use of symplectic moments and distribution-tensor concepts, giving a geometric status for: Temperature Heat Entropy This work has been extended by C. Vallée & G. de Saxcé, P. Iglésias and F. Dubois. 111 /111 / Thales Air Systems Date Covariant Definition of Thermodynamic Equilibriums The first general definition of the “moment map” (constant of the motion for dynamical systems) was introduced by Jean-Marie Souriau during 1970’s with geometric generalization such earlier notions as the Hamiltonian and the invariant theorem of Emmy Noether describing the connection between symmetries and invariants (it is the moment map for a one-dimensional Lie group of symmetries). In symplectic geometry the analog of Noether’s theorem is the statement that the moment map of a Hamiltonian action which preserves a given time evolution is itself conserved by this time evolution. The conservation of the moment of a Hamilotnian action was called by Souriau the “Symplectic or Geometric Noether theorem” considering phases space as symplectic manifold, cotangent fiber of configuration space with canonical symplectic form, if Hamiltonian has Lie algebra, moment map is constant along system integral curves. Noether theorem is obtained by considering independently each component of moment map 112 /112 / Thales Air Systems Date Souriau Covariant Model Let be a differentiable manifold with a continuous positive density and let E a finite vector space and a continuous function defined on with values in E. A continuous positive function solution of this problem with respect to calculus of variations: is given by: and and Entropy can be stationary only if there exist a scalar and an element belonging to the dual of E. Entropy appears naturally as Legendre transform of : M ωd )(ξU M )(ξp       = =       −= ∫ ∫ ∫ QdpU dp dppsArgMin M M Mp ωξξ ωξ ωξξ ξ )()( 1)( such that)(log)( )( )(.)( )( ξββ ξ U ep −Φ = ∫ − −=Φ M U de ωβ ξβ )(. log)( ∫ ∫ − − = M U M U de deU Q ω ωξ ξβ ξβ )(. )(. )( ∫−= M dpps ωξξ )(log)( Φ β Φ )(.)( ββ Φ−= QQs 113 /113 / Thales Air Systems Date Souriau Covariant Model This value is a strict minimum of s, and the equation: has a maximum of one solution for each value of Q. The function is differentiable and we can write and identifying E with its dual: Uniform convergence of proves that and that is convex. Then, and are mutually inverse and differentiable, where . Identifying E with its bidual: )(.)( ββ Φ−= QQs ∫ ∫ − − = M U M U de deU Q ω ωξ ξβ ξβ )(. )(. )( )(βΦ Qdd .β=Φ β∂ Φ∂ =Q ∫ − ⊗ M U deUU ωξξ ξβ )(. )()( 02 2 > ∂ Φ∂ − β )(βΦ− )(βQ )(Qβ dQds .β= Q s ∂ ∂ =β 114 /114 / Thales Air Systems Date Liouville Theorem Souriau has applied this approach for classical statistical mechanic system. Considering a mechanical system with n parameters , its movement could be defined by its phase at arbitrary time t on a manifold of dimension 2n: Liouville theorem shows that coordinate changes have a Jacobian equal to unity, and a Liouville density could be defined on manifold : that will not depend on choice to t. A system state is one point on 2n-Manifold and a statistical state is a law of probability defined on such that and its time evolution is driven by: where H is the Hamiltonian nqq ,,1 L nn ppqq ,,,,, 11 LL M nn dpdpdqdqd LL 11=ω M M 1)()( =∫M dp ξωξ jjjj p H q p q H p p t p ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ = ∂ ∂ ∑ 115 /115 / Thales Air Systems Date Thermodynamic Equilibrium A thermodynamic equilibrium is a statistical state that maximizes the entropy: among all states giving the mean value of energy Q: Apply for free particles, for ideal gas, equilibrium is given for (with k Boltzmann constant) and if we set previous relation provides: and are identified with Massieu-Duhem Potential. We recover also the Maxwell Speed law: ∫−= M dpps ωξξ )(log)( QdpH M =∫ ωξξ )().( kT 1 =β kT 1 =β skS .= dQds .β= T dQ dS = ∫= T dQ S ∫= T dQ S )(βΦ kT H ecstep − = .)(ξ 116 /116 / Thales Air Systems Date Covariance of Thermodynamic Equilibrium Discovery of Jean-Marie Souriau: previous thermodynamic equilibrium is not covariant on a relativity point of view ! J.M. Souriau has proposed a covariant definition of thermodynamic equilibrium where the previous definition is a particular case. In previous formalization, manifold was solution of the calculus of variations problem: with We can then consider the time variable t as other variables through an arbitrary parameter , and defined the new calculus of variations problem by: with and , where .Variables are not changed. we have the relation: M 0,, 1 0 =      ∫ dt dt dq qtld t t j j j j q l p ∂ ∂ = jq τ ( ) 0, 1 0 =∫ τdqqLd t t JJ & 1+= nqt τd dq q J J =& 1,...,2,1 += nJ ( ) t t q qtlqqL j jJJ & & & &       = ,,, jp ∑−=+ j j jn dt dq plp .1 117 /117 / Thales Air Systems Date Covariance of Thermodynamic Equilibrium If we compare with classical mechanic, we have: with (H is Legendre transform of l) H is energy of system that is conservative if the Lagragian doesn’t depend explicitly of time t. It is a particular case of Noether Theorem: If Lagrangian L is invariant by an infinitesimal transform , then: is first integral of variations equations. As Energy is not the conjugate variable of time t, or the value provided by Noether theorem by system invariance to time translation, the thermodynamic equilibrium is not covariant. Then, Jean-Marie Souriau proposes a new covariant definition of thermodynamic equilibrium ∑−=+ j j jn dt dq plp .1 Hpn −=+1 l dt dq pH j j j −= ∑ . )( KJJ QFdQ = ∑= J JJ dQpu 118 /118 / Thales Air Systems Date Covariance of Thermodynamic Equilibrium Souriau Definition of Thermodynamic Equilibrium: Let a mechanical system with a Lagragian invariant by a Lie Group G. Equilibrium states by Group G are statistical states that maximizes the Entropy, while providing given mean values to all variables associated by Noether theorem to infinitesimal transforms of group G. Noether theorem allows associating to all system movement , a value belonging to the vector space dual of Lie Algebra of group G. is called the moment of the group. For each derivation δδδδ of this Lie algebra , we take: With previous development, as is dual of , value belongs to this Lie algebra , geometric generalization of thermodynamic temperature. Value Q is a geometric generalization of heat and belongs to , the dual of . ξ )(ξU g ∑= J JJ QpU δδξ .))(( * g g β g * g g )(ξU 119 /119 / Thales Air Systems Date Souriau Theorem about Covariant Equilibrium A statistical state is invariant by δδδδ if for all (then is invariant by finite transform of G generated by δδδδ). Jean-Marie Souriau gave the following theorem: Souriau Theorem 1: An equilibrium state allowed by a group G is invariant by an element δδδδ of Lie Algebra , if and only if (with [.], the Lie Bracket), with the generalized equilibrium temperature. For classical thermodynamic, where G is an Abelian group of translation with respect to time t, all equilibrium states are invariant under G. )(ξp [ ] 0)( =ξδ p ξ )(ξp g [ ] 0, =βδ β )(.)( )( ξββ ξ U ep −Φ = ∫ − −=Φ M U de ωβ ξβ )(. log)( ∫ ∫ − − = M U M U de deU Q ω ωξ ξβ ξβ )(. )(. )( )(.)( ββ Φ−= QQs Qdd .β=Φ dQds .β= 120 /120 / Thales Air Systems Date Souriau Geometric Temperature For Group of transformation of Space-Time, elements of Lie Algebra of G could be defined as vector fields in Space-Time. The generalized temperature previously defined, would be also defined as a vector field. For each point of manifold , we could then define: Temperature Vector: with Unitary Mean Speed: with Eigen Absolute Temperature: Classical formula of thermodynamics are thus generalized, but variables are defined with a geometrical status: the geometrical temperature an element of the Lie algebra of the Galileo or Poincaré groups, interpreted as the field of space-time vectors. Souriau proved that in relativistic version is a time like vector with an orientation that characterizes the arrow of time. The temperature vector and entropy flux are in duality. β M kT V M =β M M V β β = 1=V Mk T β. 1 = β β www.thalesgroup.com Souriau-Gibbs Canonical Ensemble of Dynamical Group and Lie Group Thermodynamics 122 /122 / Thales Air Systems Date Souriau-Gibbs Canonical Ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system that is being maintained in thermodynamic equilibrium. Souriau has defined this Gibbs canonical ensemble on Symplectic manifold M for a Lie group action on M The seminal idea of Lagrange was to consider that a statistical state is simply a probability measure on the manifold of motions In Jean-Marie Souriau approach, one movement of a dynamical system (classical state) is a point on manifold of movements. For statistical mechanics, the movement variable is replaced by a random variable where a statistical state is probability law on this manifold. 123 /123 / Thales Air Systems Date Souriau-Gibbs Canonical Ensemble Symplectic manifolds have a completely continuous measure, invariant by diffeomorphisms: the Liouville measure All statistical states will be the product of Liouville measure by the scalar function given by the generalized partition function defined by the generalized energy (the moment that is defined in dual of Lie Algebra of this dynamical group) and the geometric temperature , where is a normalizing constant such the mass of probability is equal to 1, Jean-Marie Souriau generalizes the Gibbs equilibrium state to all Symplectic manifolds that have a dynamical group. To ensure that all integrals could converge, the canonical Gibbs ensemble is the largest open proper subset (in Lie algebra) where these integrals are convergent. This canonical Gibbs ensemble is convex. the mean value of the energy a generalization of heat capacity Entropy by Legendre transform λ U e .β−Φ U β Φ ∫ − −=Φ M U de ωβ. log β∂ Φ∂ =Q β∂ ∂ −= Q K Φ−= Qs .β 124 /124 / Notation Lie Algebra of Lie Group Let a Lie Group and tangent space of at its neutral element Adjoint representation of with Tangent application of at neutral element of For with Curve from tangent to : and transform by : Thales Air Systems Date G GTe G e ( ) geg e iTAdGg GTGLGAd =∈ → a : 1 : − ghghig a G Ad Ad ad [ ]YXYadGTYXGTEndGTAdTad Xeeee ,)(,)(: =∈→= a e G )(KGLG n= CRK or= )(KMGT ne = 1 )(),( − =∈∈ gXgXAdGgKMX gn [ ]YXYXXYYAdTYadKMYX XeXn ,)()()()(, =−==∈ )0(cIe d == )1(cX = )exp()( tXAdt =γ )exp()( tXtc = Ad YXXYtXYtX dt d Yt dt d YAdTYad tt XeX −==== = − = 0 1 0 )exp()exp()()()()( γ 125 /125 / Action of a Lie Group on a Symplectic Manifold Let be an action of Lie Group G on differentiable manifold M, the fundamental field associated to an element of Lie algebra of group G is the vectors field on M: is hamiltonian on a Symplectic Manifold , if is symplectic and if for all , the fundamental field is globally hamiltonian There exist linear application from to differential function on We can then associate a differentiable application ,called moment of action : Thales Air Systems Date g MMG →×Φ : X MX ( )( ) 0 ,exp)( = −Φ= t M xtX dt d xX ( ) ),(),(, 2121 xggxgg Φ=ΦΦ xxe =Φ ),(with and Φ M Φ g∈X MX XJX RMC → → ∞ ),(g XJ g M Φ J g g* ∈= → XXxJxJxJx MJ X ,),()(such that)( : a 126 /126 / Action of a Lie Group on a Symplectic Manifold We associate a bilinear and anti-symmetric form , Symplectic Cocycle of Lie algebra : If with constant , then: [ ] { }YXYX JJJYX ,),( , −=Θ Θ { } BracketPoisson:.,.with [ ] [ ] [ ] 0),,(),,(),,( =Θ+Θ+Θ YXZXZYZYXwith g µ+= JJ' * g∈µ [ ]YXYXYX ,,),(),(' µ+Θ=Θ With cobord of[ ]YXYX ,,),( µµ =∂ g 127 /127 / Action of a Lie Group on a Symplectic Manifold Equivariance of moment There exist a unique affine action such that linear part is coadjoint representation and that induce equivariance of moment is called Cocycle associated to The differential of 1-cocyle associated to at neutral element : If then : a )(),( : * gAdga Ga g θξξ += →× ** gg with XAdXAd gg 1 * ,, −= ξξ J ( ) ( ) )()())(,(),( * gxJAdxJgaxgJ g θ+==Φ J* g→G:θ e θeT θ J [ ] { }YXYXe JJJYXYXT ,),(),( , −=Θ=θ µ+= JJ' [ ]YXYXYX ,,),(),(' µ+Θ=Θ µµθθ * )()(' gAdgg −+= Where is cobord of Gµµ * gAd− 128 /128 / Thales Air Systems Date Souriau Lie Group Thermodynamic For the group of time translation, this is the classical thermodynamic Souriau has observed that if we apply this theory for non- commutative group (Galileo or Poincaré groups): the symmetry has been broken Classical Gibbs equilibrium states are no longer invariant by this group This symmetry breaking provides new equations, discovered by Jean-Marie Souriau. For each temperature , Jean-Marie Souriau has introduced a tensor , equal to the sum of cocycle and Heat coboundary (with [.,.] Lie bracket): This tensor has the following properties: is a symplectic cocycle The following symmetric tensor , defined on all values of is positive definite: β βf f ( ) ( ) [ ]21222121 ,)(with)(.,, 11 ZZZadZadQZZfZZf ZZ =+=β βf βf ββ fKer∈ βg (.)βad [ ][ ]( ) [ ]( )2121 ,,,,, ZZfZZg βββ ββ = 129 /129 / Thales Air Systems Date Souriau Lie Group Thermodynamic Souriau equations are universal, because they are not dependent of the symplectic manifold but only of: the dynamical group G its symplectic cocycle the temperature the heat Souriau called this model “Lie Groups Thermodynamics”: “Peut-être cette thermodynamique des groups de Lie a-t-elle un intérêt mathématique”. For dynamic Galileo group (rotation and translation) with only one axe of rotation: this thermodynamic theory is the theory of centrifuge where the temperature vector dimension is equal to 2 (sub-group of invariance of size 2) these 2 dimensions for vector-valued temperature are “thermic conduction” and “viscosity”, unifying “heat conduction” and “viscosity”. ( ) ( ) [ ]21222121 ,)(with)(.,, 11 ZZZadZadQZZfZZf ZZ =+=β ββ fKer∈ [ ][ ]( ) [ ]( )2121 ,,,,, ZZfZZg βββ ββ = β f Q 130 /130 / Thales Air Systems Date Fundamental Souriau Theorem Let be the largest open proper subset of , Lie algebra of G, such that and are convergent integrals this set is convex and is invariant under every transformation , where is the adjoint representation of G, with: where is the cocycle associated with the group G and the moment, and is the image under of the probability measure . Rmq: is changed but with linear dependence to , then Fisher metric is unchanged by dynamical group: Ω g ∫ − M U de ωξβ )(. ∫ − M U de ωξ ξβ )(. . Ω ga gaa a )(ββ ga→ ( ) ( ) )(.1 βθβθ gaaa +Φ=−Φ→Φ − ss → ( ) )()( QaaQaQ ** gg θ θ =+→ )(ςς + → Ma θ )(ς+ Ma Ma ς Φ β ( ) ( )[ ] ( )β ββ βθ β I a aI = ∂ Φ∂ −= ∂ −Φ∂ −= − 2 2 2 12 )(g 131 /131 / Fundamental Souriau Theorem Thales Air Systems Date Gibbs canonicalGibbs canonicalGibbs canonicalGibbs canonical ensembleensembleensembleensemble Ω * Ω g * g R R ( )βΦ ( ) ( ) )(. βθβ gaa+Φ ( ) ( )ββ Φ−= QQs . Q )(Qa * gθ β )(βga ς )(ς+ Ma e a G 132 /132 / Moment of the G action Thales Air Systems Date G g Z e V x )(xZV * g µ O R ( ))(. xZZ Vωµ ≡ If is a dynamical group of a symplectic manifold , torsor is called a moment of the G-action, if there is a differential map from to such that: To every torsor , there corresponds a field of 1-forms (Maurer-Cartan forms) on which is invariant under right translation and which takes the value when is the identity element: G V µ µax V * g ( ) [ ]ZdxZV .)( µσ −≡ [ ] 0and,for)()( ==== xZaeaxaxZ VV δδδ µ [ ]ωax G µ x ωσ dV = [ ] xZgradxZ Z V . .)( µ δµ ≡≡ 133 /133 / The Cohomology of a dynamical group Thales Air Systems Date G g e a Ga V * g * g a )(aθ µ Va x ψ ψ Theorem: Let be a connected symplectic manifold and let be a dynamical group of possessing a moment . Finally, let denote the map from to the space of torsor of G: There exists a differential map : The derivative is a 2-form on the Lie algebra of : Identity hold: V G V µ ψ µax V * g * g→G:θ ( ) ( ))()()( xaxaa V ψψθ * g −≡ ))(( eDf θ= g G [ ]( ) [ ]( ) [ ]( ) 0',)''(,'')'('',')( ≡++ ZZZfZZZfZZZf ( )( ) [ ] )')((',.)()( ' ZZfZZxZxZ VV +≡ µσ ( ) ( ) ( ) )().()()( ZfZadxxZxD V +≡ψψ 134 /134 / Thales Air Systems Date Fundamental Souriau Theorem Proof the adjoint representation of G can be written: defines an action of G on its Lie algebra ,with is called the adjoint representation, that is a linear representation of G on its Lie algebra . This Souriau theorem if based on invariance property of Liouville measure: the Liouville measure on a symplectic manifold is invariant under every symplecto- morphisms of this manifold (a symplecto-morphism is an isomorphism in the category of symplectic manifolds, a diffeomorphism between two symplectic manifolds). Let be an arbitrary element of G and action of on the manifold . Since is a symplectomorphism, the image under of the Liouville measure is equal to . The integral is equal with invariance property of Liouville measure to the integral : gaa a [ ] 0and,with)( 1 ===××= − aZbebabaZa δδδg g ga g a Ma a 1− Ma 1− Ma λ λ ∫ − M U de ωξβ .)(. ( ) ∫∫ − −− = M aU M U dede M ωω ξβξβ .. )(.)(. 1 135 /135 / Thales Air Systems Date Fundamental Souriau Theorem Proof We can then use the following relation: with a symplectic cocycle of G. This cocycle is defined for: there exist then a differential map defined by: This differential map satisfy the condition and its derivative where is the identity element of G, is a 2-form on the Lie algebra of G which satisfies: ( ) ( ) ( )111 )()( −−− += aUaaU M θξξ * g θ µξ a : * g→MU θ ( ) ( ))()( : ξξ θ UaaUa G M * g * g − → a ( ) ( ) ( )( )baaba θθθ * g +=× θ ( ) )(eDf θ= e g [ ]( ) [ ]( ) [ ]( ) g∈∀=++ 321213132321 ,,,0,,,,,, ZZZZZZfZZZfZZZf 136 /136 / Thales Air Systems Date Fundamental Souriau Theorem Proof and satisfy the following identities where is the fundamental vector field on the manifold associated to : with the Lagrange form. If we use previous relation , and the property that , by defining: The integral is then defined by: ( ) ),((.)).())(,( ZZfadUZUD ZM += ξξξ )(ξMZ M g∈Z ( )[ ] 0and,for)( ==== δξδξδξ ZaeaaZ MM ( ) [ ] ( )2121,2,1 ,,.)(),( ZZfZZZZ MM += µξξσ σ ( ) ( ) ( )111 )()( −−− += aUaaU M θξξ * g ( ) 1 ).()( − = gg* aUUa ξξ ( )ββ ga=' ( ) ( ) ( ) ( ) ( )[ ] ( ) ∫∫∫∫ −+−−− − −−− === M Ua M aUaa M aUa M U deededede M ωωωω ξββθθξβξβξβ .... )(..)(.)(.)('. 1 111 *ggg 137 /137 / Thales Air Systems Date Fundamental Souriau Theorem Proof We can then deduce the equation of Souriau theorem on : The equation of Souriau theorem on uses the relation Finally, using , we can prove that the Entropy is invariant: Φ ( ) ( )( ) ( ) ( ) ( ) ( )βθβωβ ωββ ξββθ ξβ 1)(.. )('. .log'' .log'' 1 −− − −Φ=        −=Φ=Φ −=Φ=Φ=Φ ∫ ∫ − adee dea M Ua M U g Q ( ) 1 . − = gg* aQQa ( )( ) ( ) ( ) ( ) ( ) )()(.)( ' ' ' )( )( ' ' ' 1 1 aQaaaQa a Q a aa aa Q θθθ β β ββ θ ββ βθ β +=+=+      ∂ ∂ ∂ Φ∂ = ∂ Φ∂ = + ∂ Φ∂ = ∂ +Φ∂ = ∂ Φ∂ = − − * gg g gg g ( ) 1 . − = gg* aQQa ( ) ( )( ) ( )( ) ( ) ( ) ( ) sQQaaQaas aaaQaaQs =Φ−=Φ−=Φ−= +Φ−+=Φ−= ...' )()(.'''.' βββ βθθββ g 1- ggg ggg * * 138 /138 / Thales Air Systems Date Souriau Lie Group definition of Fisher Metric Let be the derivative of (symplectic cocycle of G) at the identity element and let us define: Then is a symplectic cocycle of ,that is independent of the moment of G There exists a symmetric tensor defined on the image of such that: and that gives the structure of a positive Euclidean space f θ ( ) ( ) [ ]21222121 ,)(with)(.,,, 11 ZZZadZadQZZfZZf ZZ =+=Ω∈∀ ββ βf g ( ) Ω∈∀= ββββ ,0,f βg [ ]ββ .,(.) =ad [ ]( ) ( ) ()( ).Im,,,,, 212121 βββ β adZZZZfZZg ∈∀∈∀= g ( ) ()( ).Im,,0, 2121 ββ adZZZZg ∈∀≥ 139 /139 / Thales Air Systems Date Souriau Lie Group definition of Fisher Metric: Proof could be deduced by differentiating: and taking , and . As and , we have: If we differentiate , the following relation appears. Then, writing , we have: ( ) 0, =βββf ( ) ( ) )(.)( βθβ gg aaa +Φ=Φ e=a 2Za =δ 01 =Zδ ( )[ ]ξδξ MM aZ =)( (.)ZadZ −=g [ ] ( )2121 ,, ZZfZZQ −= ( ) ( )aQaaQ θβ +)())( * gg [ ]( ) ( ) ( )11111 ,(.).,, 1 ZZfadQZZfZ Q Z ββ β =+=− ∂ ∂ [ ] ZZ, 21 == βδβ ( ) 0,0. 21 ≥⇒≥ ZZfQ βδβδ 140 /140 / Poincaré’s Lemma & Cartan’s Theorem Poincaré’s Lemma: If is a differential field of p-forms, then: . Conversely, if is a differentialble field of [p+1]-forms defined on a cell and if , then there exists a differential field of p-forms on this cell such that . Cartan’s Theorem: If is a differential field of p-forms (p≥≥≥≥1) and if is a differentiable vector field, then: with notation: Combining Poincaré’s Lemma and Cartan’s Theorem, we verify: Characteristic Foliation: Integrability conditions is automatically satisfied, is a Foliation, Characteristic Foliation of the formThales Air Systems Date ϕax [ ] 0=ϕdd θax 0≡θd ϕax ϕθ d≡ ϕax δxx a [ ]( ) ( )[ ]xdxdδL δϕδϕϕ +≡ [ ] ϕδϕ ≡≡≡ )(and)(if)(, xgxxfxgfδL [ ] [ ] )pa p-form,(ddδ LL 0≥= ϕϕδϕ [ ]( ) [ ][ ]( ) [ ] ',' 0,'LemmasPoincaré' 0,'TheoremsCartan' )()('', Ex xd x dKerKerExx ∈⇒    ≡⇒ ≡⇒ +=∈ δδ δδϕ δδϕ ϕϕδδ 'Ex a ϕ www.thalesgroup.com Synthesis of analogies between Koszul Information Geometry Model and Souriau Statistical Physics model 142 /142 / Thales Air Systems Date Koszul Information Geometry, Souriau Lie Group Thermodynamics Koszul Information Geometry Model Souriau Lie Groups Thermodynamics Model Characteristic function Ω∈∀−=Φ ∫ Ω − xdex x log)( * , ξξ g∈∀−=Φ ∫ − βωβ ξβ log)( )(. M U de Entropy ξξξ dppx xx∫ Ω −=Φ * )(log)()( ** ∫−= M dpps ωξξ )(log)( Legendre Transform )(,)( *** xxxx Φ−=Φ )(.)( ββ Φ−= QQs Density of probability ∫ − − +− = = * ξ,x ξ,x x Φ(x)x,ξ x dξe e p ep )( )( ξ ξ ∫ − − Φ+− = = M U U U de e p ep ω ξ ξ ξβ ξβ β βξβ β )(. )(. )()(. )( )( Dual Coordinate Systems ** and Ω∈Ω∈ xx ∫ ∫ ∫ − Ω − Ω == * ξ,x ξ,x x dξe de dpx * * . )(.* ξξ ξξξ * gg ∈∈ Qandβ ∫ ∫ ∫ − − == M U M U M de deU dpUQ ω ωξ ωξξ ξβ ξβ β )(. )(. )( )().( heatGeometricor MapMomentSouriauofMean: mapMomentSouriau: eTemperaturGeometricSouriau: Q U β Dual Coordinate Systems x x x ∂ Φ∂ = )(* and * ** )( x x x ∂ Φ∂ = β∂ Φ∂ =Q and Q s ∂ ∂ =β Hessian Metric )(22 xdds Φ−= ( )βΦ−= 22 dds Fisher metric 2 ,2 2 2 2 2 * log )( )(log )( x de x (x) xI x p ExI x x ∂ ∂ = ∂ Φ∂ −==       ∂ ∂ −= ∫ Ω − ξ ξ ξ ξ 2 )(.2 2 2 2 2 log )( )(log )( β ω β β β β ξ β ξβ β ξ ∂ ∂ = ∂ Φ∂ −=         ∂ ∂ −= ∫ − M U de )( I p EI CapacityGeometricSouriau: )( 2 2 β ββ β β ∂ ∂ −= ∂ ∂ −= ∂ Φ∂ −= Q K Q)( I 143 /143 / Thales Air Systems Date Geometric heat Capacity / Specific heat We observe that the Information Geometry metric could be considered as a generalization of “Heat Capacity”. Souriau called it the “Geometric Capacity”. This geometric capacity is related to calorific capacity. is related to the mean, and is related to the variance of β∂ Φ∂ =Q ββ β β ∂ ∂ −= ∂ Φ∂ −= Q)( I 2 2 )( T Q kTT kT T QQ K ∂ ∂ =             ∂ ∂ ∂ ∂ −= ∂ ∂ −= 2 1 1 βkT 1 =β Q K U [ ] [ ] 2 222 )().()(.)()(         −=−= ∂ ∂ −= ∫∫ MM dpUdpUUEUE Q I ωξξωξξ β β ββξξ [ ]UEdpU β Φ Q M ξβ ωξξ == ∂ ∂ = ∫ )().( 144 /144 / Thales Air Systems Date Koszul Information Geometry, Souriau Lie Group Thermodynamics Koszul Information Geometry Model Souriau Lie Groups Thermodynamics Model Convex Cone Ω∈x Ω convex cone Ω∈β Ω convex cone: largest open subset of g , Lie algebra of G, such that ∫ − M U de ωξβ )(. and ∫ − M U de ωξ ξβ )(. . are convergent integrals Transformation ( )Ω∈→ Autggxx with )(ββ ga→ Transformation of Potential (non invariant) ( )gxgxx detlog)()()( +Φ=Φ→Φ ΩΩΩ ( ) ( ) ( ) ( )βθβββ 1 )( − −Φ=Φ→Φ aag Transformation of Entropy (invariant) ( ) ( )***** *** )( x x gx x Ω Ω ΩΩ Φ=      ∂ Φ∂ Φ→Φ x x x ∂ Φ∂ = Ω )( with * ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( )βθβββ θ β βθ β ββ ββ 1 '' )( )( ' ' ' ' .with )(.'''.'' − −Φ=Φ=Φ=Φ += ∂ +Φ∂ = ∂ Φ∂ = = =Φ−=Φ−=→ aa aQa a aa Q a QsQQQsQs g g g g g * Information Geometry Metric (invariant) ( ) ( )[ ] ( )xI x x x gx gxI = ∂ Φ∂ −= ∂ +Φ∂ −= ΩΩ 2 2 2 2 )(detlog)( ( ) ( ) ( )[ ] ( ) ( )β β β β βθβ β I a aI = ∂ Φ∂ −= ∂ −Φ∂ −= − 2 2 2 12 )(g 145 /145 / Thales Air Systems Date Invariance of Fisher Metric In both Koszul and Souriau models, the Information Geometry Metric and the Entropy are invariant respectively to: the automosphisms of the convex cone to adjoint representation of Dynamical group G acting on , the convex cone considered as largest open subset of , Lie algebra of G, such that and are convergent integrals. g Ω ga Ω g ∫ − M U de ωξβ )(. ∫ − M U de ωξ ξβ )(. . ( ) ( )[ ] ( )xI x x x gx gxI = ∂ Φ∂ −= ∂ +Φ∂ −= ΩΩ 2 2 2 2 )(detlog)( ( ) ( ) ( )[ ] ( ) ( )β β β β βθβ β I a aI = ∂ Φ∂ −= ∂ −Φ∂ −= − 2 2 2 12 )(g ( )Ω∈→ Autggxx with )(ββ ga→ www.thalesgroup.com From Characteristic Function to Generating Inner Product 147 /147 / Thales Air Systems Date Elie Cartan Influence on Koszul and Souriau Works Elie Cartan works have greatly influenced: Jean-Louis Koszul (Koszul Phd thesis has extended previous work of Elie Cartan) Jean-Marie Souriau (Souriau was student of Elie Cartan at ENS, the year after his aggregation). We have shown that “Information Geometry” could: be considered as a particular application domain of Hessian Geometry through Jean-Louis Koszul work (Koszul-Vinberg metric deduced from the associated Characteristic function having the main property to be invariant to all automorphisms of the convex cone) be extended in the framework of Jean-Marie Souriau theory, as an extension towards “Lie Group Thermodynamics” with vector-valued geometric temperature (providing a geometric extension of Emmy Noether theorem). Should we deduce that the “essence” of Information Geometry is limited to “Koszul Characteristic Function”? This notion seems not to be the more general one The notion of Generating Inner Products: Reduction of Koszul and Souriau definitions to exclusive “Inner Product” selection using symmetric bilinear “Cartan-Killing form” (Elie Cartan in 1894) 148 /148 / Thales Air Systems Date Cornerstone of Inner Product Selection In Koszul Geometry, we have 2 convex dual functions: and with dual system of coordinates: and defined on dual cones: and If we can define an Inner Product , we will be able to build convex function and its dual by Legendre transform because both are only dependent of the Inner product , and dual coordinate is also defined by where is also the center of gravity of the cross section of x * x Ω * Ω Ω∈∀−=Φ ∫ Ω − xdex x log)( * , ξξ )(,)( *** xxxx Φ−=Φ .,. )(log)( xx Ω−=Φ ψ { } ∫∫ Ω − Ω − Ω ==Ω∈= ** ,,** /.,,/)(minarg ξξξψ ξξ dedenyxyyx xx * x { }nyxy =Ω∈ ,,* * Ω 149 /149 / Thales Air Systems Date Cartan-Killing Form and Invariant Inner Product It is not possible to define an ad(g)-invariant inner product for any two elements of a Lie Algebra, but a symmetric bilinear form, called “Cartan-Killing form”, could be introduced (Elie Cartan PhD 1894) This form is defined according to the adjoint endomorphism of that is defined for every element of with the help of the Lie bracket: The trace of the composition of two such endomorphisms defines a bilinear form, the Cartan-Killing form: The Cartan-Killing form is symmetric: and has the associativity property: given by: xad g x g [ ]yxyadx ,)( = ( )yxadadTryxB =),( ),(),( xyByxB = [ ]( ) [ ]( )zyxBzyxB ,,,, = [ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]( )zyxBadadadTrzyxB adadadTradadTrzyxB zyx zyxzyx ,,,,, ,,, , == == 150 /150 / Thales Air Systems Date Cartan-Killing Form and Invariant Inner Product Elie Cartan has proved that if is a simple Lie algebra (the Killing form is non-degenerate) then any invariant symmetric bilinear form on is a scalar multiple of the Cartan-Killing form. The Cartan-Killing form is invariant under automorphisms of the algebra : To prove this invariance, we have to consider: Then g g )(gAut∈σ g ( ) ( )yxByxB ,)(),( =σσ [ ] [ ] [ ] [ ] 1 )( 1 rewritten ),()(, )( )(),(, − − = =⇒    = = σσ σσσ σ σσσ σ oo xx adad zxzx yz yxyx ( ) ( ) ( ) ( ) ( ) ),()(),( )(),( 1 )()( yxBadadTryxB adadTradadTryxB yx yxyx == == − σσ σσσσ σσ oo 151 /151 / Thales Air Systems Date Cartan-Killing Form and Invariant Inner Product A natural G-invariant inner product could be introduced by Cartan- Killing form: Cartan Generating Inner Product: The following Inner product defined by Cartan-Killing form is invariant by automorphisms of the algebra where is a Cartan involution (An involution on is a Lie algebra automorphism of whose square is equal to the identity). ( ))(,, yxByx θ−= g∈θ g θ g 152 /152 / Thales Air Systems Date From Cartan-Killing Form to Koszul Information Metric ( ) ( ) InvolutionCartan,with )(,, FormKillingCartan ),( g yxByx adadTryxB yx ∈ −= − = θ θ Ω∈∀−=Φ ∫ Ω − xdex x log)( FunctionsticCharacteriKoszul * , ξξ ∫ ∫ ∫ − − Ω Ω = = −=Φ Φ−=Φ * ξ,x ξ,x x x xx dξe e p dpx dppx xxxx )( DensityKoszul )(.with )(log)()( )(,)( EntropyKoszul * * * ** *** ξ ξξξ ξξξ 2 ,2 2 2 2 2 * log )( )(log )( MetricKoszul x de x (x) xI x p ExI x x ∂ ∂ = ∂ Φ∂ −==       ∂ ∂ −= ∫ Ω − ξ ξ ξ ξ www.thalesgroup.com General Definition of Entropy 154 /154 / Thales Air Systems Date General Definition of Entropy The Legendre transform is closely related to the idempotent analogue of the Fourier transform. If we consider the semiring with the operations: and In the idempotent analogues of integration on RN is given by the formula: Then, the Legendre transform is equivalent to the Fourier transform in algebra: The Legendre transform generates an idempotent version of harmonic analysis for the space of convex functions. We can then give a general definition of Entropy: [ ][ ]LaplaceLegendreEntropy log−= LaplLogLegEnt oo−= { }∞+∪= RRmin Min=⊕ +=• { }∞+∪= RRmin )()()( xfInfdxxffI N N RxR ∈ ⊕ == ∫ ( ) ( )+=•⊕ ,, Min [ ] ( ) ( )[ ])()(,)(,)( , * xFourdxxxxxSup Min x Φ=Φ•−−=Φ−=Φ + ⊕ ΩΩ∈ ∫ ξξξ ),(),( ×++−= LaplLogFourEnt Min oo www.thalesgroup.com Legendre Transform and Minimal Surface 156 /156 / Thales Air Systems Date Legendre Transform and Minimal Surface In 1787, Adrien-Marie Legendre has introduced “Legendre Transform” to solve Minimal Surface Problem equation introduced by Lagrange and partially solved by Gaspard Monge in 1784. In 1768, J.L. Lagrange, has considered the variational problem of least area surface stretched across a given closed contour. Based on Euler-Lagrange equation, Lagrange has introduced the equation of Minimal Surface : Lagrange has observed that affine functions are solutions of this equation and minimal surfaces are plans. Jean-Baptiste Marie Meusnier de La Place, student of Monge, has observed that for this surface, with mean curvature : ( )yxz , ( ) ( ) q dy dz p dx dz dy zd p dxdy zd pq dx zd q ===++−+ andwith0121 2 2 2 2 2 2 2 ( ) cybxayxz ++= .., ( ) ( ) 2/322 2 2 2 2 2 2 2 2222 1 121 11 2               +      + ++−+ =                       +      + +                       +      + = dy dz dx dz dy zd p dxdy zd pq dx zd q dy dz dx dz dy dz dy d dy dz dx dz dx dz dx d H z zH 157 /157 / Thales Air Systems Date Legendre Transform and Minimal Surface Gaspard Monge integrated this equation but with a non-rigorous approach and has requested Legendre to find a more classical solution. For this task, Legendre has introduced a change of variable that is the nowadays well-known “Legendre transform”. Adrien-Marie Legendre said “J’y suis parvenu simplement par un changement de variables qui peut être utile dans d’autres occasions”. Legendre reduced the problem to solve to determine p and q as functions of x and y such that: are exact differentials. If we set , then these others expressions are complete differentials: Legendre considered x and y as functions of p and q: 22 1 and.. qp q.dxp.dy dyqdxp ++ − + 222 1 uqp =++       +      + u q dx u p dydqydpx ..and.. dq d y dp d xddqydpx ωω ω ===+ andwith.. 158 /158 / Thales Air Systems Date Legendre Transform and Minimal Surface If we then develop , we have: That should be an exact differential. By replacing x and y, we have a new equation: This new equation is very similar to the previous one, but simpler because it depends on p and q and not their partial differentials of first order. When the function ωωωω will be known, then functions x, y and z will be also defined according to p and q thanks to “Legendre transform”:       +      u q dx u p dy .. ( )[ ] ( )[ ] 3 2 3 2 .1.1 u dq ypqxp u dp xpqyq ++−++ ( ) ( ) 01.21 2 2 2 2 2 2 2 =++++ dp d p dpdq d pq dq d q ωωω dq d y dp d x qpyqxpyxz ωω ω == −+= andwith ),(..),( 159 /159 / Thales Air Systems Date Legendre Transform and Minimal Surface About this Legendre transform, Gaston Darboux gave interpretation of this transform by Chasles “Ce qui revient suivant une remarque de M. Chasles, à substituer à la surface sa polaire réciproque par rapport à un paraboloïde”. Classical “Legendre transform” with our previous notations: ( ) ( ) ( ) ( ) ( )             =             = Φ =             Φ Φ =              =      =       =      =    = =Φ Φ−=Φ−= dQ ds dQ ds dQ ds d d d d d d Q y x q p Q Q Q qpQs yxz QQQs 2 1 2 1 2 1 2 1 and, ),( ),( with ,. β β β β β β β ω β ββββ 160 /160 / Thales Air Systems Date Legendre Transform and Minimal Surface In the following relation, we recover the definition of Entropy : The equation of the minimal surface is given by: Or by this 2nd equation:      = = ⇒      == =+ dQ ds dsdQ dq d y dp d x ddqydpx β β ωω ω . and .. 0.2 1 1 22 ==                 Φ + Φ =           + ΦH d d d d d d Q Q d d β β ββ ( ) ( ) 0 )( 1 )( .2 )( 1 2 1 2 2 1 21 2 2 2 2 2 2 =++++ dQ Qsd Q dQdQ Qsd pq dQ Qsd Q 161 /161 / Information Metric and Curvature Thales Air Systems Date        Φ =         −−= ⇒ − =        ⇒>> −=⇒<< =               Φ + Φ =           + Φ Φ Φ β β β β β β β ββ d d Q Q Q Q HI Q Q Q QQ d dQ Q Q d d Q HIQ H d d d d d d Q Q d d with 1 .2)(.. 1 2)(1 .2 1 1 2 2 2 2 22 www.thalesgroup.com Gromov Inner Product 163 /163 / Thales Air Systems Date Gromov Inner Product As other generalization of inner product, we can consider for specific case CAT(-1)-space (generalization of simply connected Riemannian manifold of negative curvature lower than unity) or for an Homogeneous Symmetric Bounded domains, a “generating” Gromov Inner Product between three points x, y and z (relatively to x) that is defined by the distance : with d(.,.) the distance in CAT(-1). Intuitively, this inner product measures the distance of x to the geodesics between y to z. This Inner product could be also defined for points on the Shilov Boundary of the domain through Busemann distance: Independent of p, where is horospheric distance, from x to y relatively to , with geodesic ray. We have the property that: ( )),(),(),( 2 1 , zydzxdyxdzy x −+= ( )),(),( 2 1 ', ' pxBpxBx ξξξξ += [ ])()(),( trytrxLimyxB t −−−= +∞→ ξ ξ )(tr x y yx yyLim ',', '' ξ ξ ξξ → → = 164 /164 / Thales Air Systems Date Gromov Inner Product ( )),(),(),( 2 1 , zydzxdyxdzy x −+= x y yx yyLim ',', '' ξ ξ ξξ → → = 165 /165 / Thales Air Systems Date Gromov Inner Product We can then define a visual metric on the Shilov boundary by; We can then define the characteristic function according to the origin: with the center of gravity All these relations are also true on the Shilov Boundary: where is the functional of Busemann barycenter on the Shilov Boundary (existence and unicity of this barycenter have been proved by Elie Cartan for Cartan-Hadamard Spaces). ( ) ( ) otherwise0', 'if', ', = ≠= − ξξ ξξξξ ξξ x x d ed x ∫ Ω − −=Φ * 0 , log)( γ γ dex x ( ) ∫ Ω −+− Ω −=Φ * ),(),0(),0( 2 1 log)( γ γγ dex xddxd ( ) )(),(),0(),0( 2 1 )(,)( ** 0 *** xxxdxdxdxxxx Φ−−+=Φ−=Φ ( ) ( ))(2),0()(2),0(),( **** xxdxxdxxd Φ−+Φ−= ∫∫ Ω − Ω − = * 0 * 0 ,,* /. γγγ γγ dedex xx ( )∫∫ Ω∂Ω∂ − −=−=Φ ** 0 '.',log'log)( 0 ', ξξξξξ ξξ ddde ( )∫ Ω∂ * '.',0 ξξξ dd 166 /166 / Thank you for your attention Thales Air Systems Date Nous avouerons qu’une des prérogatives de la géométrie est de contribuer à rendre l’esprit capable d’attention: mais on nous accordera qu’il appartient aux lettres de l’étendre en lui multipliant ses idées, de l’orner, de le polir, de lui communiquer la douceur qu’elles respirent, et de faire servir les trésors dont elles l’enrichissent, à l’agrément de la société. Joseph de Maistre Si on ajoute que la critique qui accoutume l’esprit, surtout en matière de faits, à recevoir de simples probabilités pour des preuves, est, par cet endroit, moins propre à le former, que ne le doit être la géométrie qui lui fait contracter l’habitude de n’acquiescer qu’à l’évidence; nous répliquerons qu’à la rigueur on pourrait conclure de cette différence même, que la critique donne, au contraire, plus d’exercice à l’esprit que la géométrie: parce que l’évidence, qui est une et absolue, le fixe au premier aspect sans lui laisser ni la liberté de douter, ni le mérite de choisir; au lieu que les probabilités étant susceptibles du plus et du moins, il faut, pour se mettre en état de prendre un parti, les comparer ensemble, les discuter et les peser.. Un genre d’étude qui rompt, pour ainsi dire, l’esprit à cette opération, est certainement d’un usage plus étendu que celui où tout est soumis à l’évidence; parce que les occasions de se déterminer sur des vraisemblances ou probabilités, sont plus fréquentes que celles qui exigent qu’on procède par démonstrations: pourquoi ne dirions –nous pas que souvent elles tiennent aussi à des objets beaucoup plus importants ? Joseph de Maistre www.thalesgroup.com Legendre Duality in Mechanics (Poincaré-Cartan Integral Invariant) 168 /168 / Legendre Duality in Mechanics In Mechanics, Legendre Duality gives the relation between: the variational Euler-Lagrange the symplectic Hamilton-Jacobi formulations of the equations of motion As described by Vladimir Arnold, in the general case, we can define the Hamiltonian H as the fiberwise Legendre transformation of the Lagrangian L: Due to strict convexity, supremum is reached in a unique point such that : and Young-Fenchel inequality. For all ,the following holds : with equality if and only if ( )),,(.),,( tqqLqpSuptqpH q && & −= ),,(.),,( tqqLqptqpH && −= q& ),,( tqqLp q &&∂= ),,( tqpHq p∂=& pqtq ,,, & ),,(),,(. tqpHtqqLqp +≤ && ),,( tqqLp q &&∂= 169 /169 / Legendre Duality in Mechanics If we consider total differential of Hamiltonian: Euler-Lagrange equation with and provides the 2nd Hamilton equation with in Darboux coordinates.     ∂+∂+∂= ∂−∂−=∂−∂−∂−+= HdtHdqHdpdH LdtLdqdpqLdtqLdLdqqpddpqdH tqp tqtqq &&&& &     ∂=∂− ∂= ⇒ HL Hq qq p & 0=∂−∂∂ LL qqt & Lp q&∂= HL qq ∂=∂− Hp q−∂=& Hq p∂=& 170 /170 / Legendre Duality in Mechanics ( )),,(.),,( tqqLqpSuptqpH q && & −= ),,( tqqLp q &&∂= ),,( tqpHq p∂=& 171 /171 / Pfaffian Form and Poincaré-Cartan Integral Invariant Considering Pfaffian form related to Poincaré-Cartan integral invariant, based on: and we can deduce: with P. Dedecker has observed, that the property that among all forms the form is the only one satisfying , is a particular case of more general T. Lepage congruence related to transversality condition. dtHdqp .. −=ω ( ) ϖω LdtLdtLqLdqL qqq &&& & ∂+=−∂−∂= .... dtqdq .&−=ϖ Lp q&∂= LqpH −= &. ϖθ mod.dtL≡ dtHdqp .. −=ω ϖθ mod0≡d [] Cartan E., Leçons sur les invariants intégraux, Hermann, Paris, 1922 [] Dedecker P., A property of differential forms in the calculus of variations, Pacific J. Math. Volume 7, Number 4,p. 1545-1549, 1957 [] Lepage T., Sur les champs géodésiques du calcul des variations, Bull. Acad. Roy. Belg., CL. Sci.27, p.716-729, pp. 1036-1046, 1936 172 /172 / Pfaffian Form and Poincaré-Cartan Integral Invariant www.thalesgroup.com Legendre Transform and Contact Geometry 174 /174 / Legendre Transform and Contact Geometry Legendre transform and contact geometry where used in Mechanic and in Thermodynamic. Integral submanifolds of dimension n in 2n+1 dimensional contact manifold are called Legendre submanifolds. A smooth fibration of a contact manifold, all of whose are Legendre, is called a Legendre Fibration. In the neighbourhood of each point of the total space of a Legendre Fibration there exist contact Darboux coordinates (z, q, p) in which the fibration is given by the projection (z, q, p) =>(z, q). Indeed, the fibres (z, q) = cst are Legendre subspaces of the standard contact space. A Legendre mapping is a diagram consisting of an embedding of a smooth manifold as a Legendre submanifold in the total space of a Legendre fibration, and the projection of the total space of the Legendre fibration onto the base. 175 /175 / Legendre Transform and Contact Geometry Let us consider the two Legendre fibrations of the standard contact space of 1 -jets of functions on : and the projection of the 1-graph of a function onto the base of the second fibration gives a Legendre mapping: If S is convex, the front of this mapping is the graph of a convex function, the Legendre transform of the function S: 12 +n R n R ),(),,( quqpu a ),.(),,( puqpqpu −a )(qSu =       ∂ ∂ − ∂ ∂ q S qS q S qq ),(a ( )ppS ),(* 176 /176 / Legendre Duality & Contact Geometry Symplectic geometry of even-dimensional phase spaces has an odd- dimensional twin: contact geometry. The relation between contact geometry and symplectic geometry is similar to the relation between linear algebra and projective geometry. Any fact in symplectic geometry can be formulated as a contact geometry fact and vice versa. The calculations are simpler in the symplectic setting, but their geometric content is better seen in the contact version. The functions and vector fields of symplectic geometry are replaced by hypersurfaces and line fields in contact geometry. Each contact manifold has a symplectization, which is a symplectic manifold whose dimension exceeds that of the contact manifold by one. Symplectic manifolds have contactizations whose dimensions exceed their own dimensions by one. If a manifold has a serious reason to be odd dimensional it usually carries a natural contact structure. one might now say ‘‘symplectic geometry is all geometry,’’ but I prefer to formulate it in a more geometrical form: contact geometry is all geometry. [] Arnold, V.I., Givental, A.G., Symplectic geometry, Encyclopedia of mathematical science, vol. 4., Springer Verlag (translated from Russian) 177 /177 / Legendre Duality & Contact Geometry Contact structures and Legendre submanifolds: A contact structure on an odd-dimensional manifold M2n+1 is a field of hyperplanes(of linear subspaces of codimension 1) in the tangent spaces to M at all its points. All the generic fields of hyperplanes of a manifold of a fixed dimension are locally equivalent. They define the (local) contact structures. Example: A 1-jet of a function at point x of manifold Vn is defined by the point where The natural contact structure of this space is defined by the following condition: the 1-graphs of all the functions on V should be the tangent structure hyperplane at every point. In coordinates, this conditions means that the 1-form should vanish on the hyperplanes of the contact field. ),...,,( 21 nxxxfy = 12 ),,( + ∈ n Rpyx ii xfp ∂∂= / { } ),(/),(, 1 RVJxfpxfyx n ⊂∂∂== dxpdy .− pdVTdSdE −= Gibbs contact structure In Thermodynamics 178 /178 / Projective duality and Legendre transformation A contact structure on a manifold is a nondegenerate field of tangent hyperplanes The manifold of contact elements in projective space coincides with the manifold of contact elements of the dual projective space A contact element in projective space is a pair, consisting of a point of the space and of a hyperplane containing this point. The hyperplane is a point of the dual projective space and the point of the original space defines a contact element of the dual space. The manifold of contact elements of the projective space has two natural contact structures: The first is the natural contact structure of the manifold of contact elements of the original projective space. The second is the natural contact structure of the manifold of contact elements of the dual projective space The dual of the dual hypersurface is the initial hypersurface (at least if both are smooth for instance for the boundaries of convex bodies) The affine or coordinate version of the projective duality is called the Legendre transformation. Thus contact geometry is the geometrical base of the theory of Legendre transformation. 179 /179 / History of Contact Geometry Huygens Principle: the wave front Fqo(to + t) is the envelope of the fronts Fq(t) 1872, Sophus Lie: the notion of contact transformation (Beriihrungstransformation) as a geometric tool to study systems of differential equations. 19th century and the beginning of the 20th century, F. Engel, H. Poincaré, E. Goursat and E. Cartan www.thalesgroup.com References 181 /181 / Thales Air Systems Date Tutorial Speaker References [1] F. Barbaresco, “Algorithme de burg régularisé FSDS. Comparaison avec l’algorithme de burg MFE,” XVème Colloque GRETSI 1, pp. 29–32, 1995 [2] F. Barbaresco, “Super resolution spectrum analysis regularization: Burg, capon and ago antagonistic algorithms,” in Proc. EUSIPCO’96, Trieste, pp. 2005–2008, Sept. 1996 [3] F. Barbaresco, “Information Intrinsic Geometric Flows”, Bayesian Inference and Maximum Entropy Methods In Science and Engineering, AIP Conf. Proc. 872, pp. 211-218, 2006 [4] F. Barbaresco, “Innovative tools for radar signal processing based on Cartan’s geometry of SPD matrices and information geometry,” in Proc. IEEE Int. Radar Conf., pp. 1–6, 2008 [5] N. Charon, F. Barbaresco, “A new approach for target detection in radar images based on geometric properties of covariance matrices'spaces”, Revue TS, Traitement du Signal, vol. 26, n°4, pp. 269-278, 2009 182 /182 / Thales Air Systems Date Tutorial Speaker References [6] F. Barbaresco, “Interactions between symmetric cone and information geometries,” in Proc. ETVC’08, pp. 124–163, 2009 [7] F. Barbaresco and G. Bouyt, “Espace riemannien symétrique et géométrie des espaces de matrices de covariance : équations de diffusion et calculs de médianes,” in Proc. GRETSI’09 Conf., Dijon, France, Sep. 2009 [8] F. Barbaresco, “New foundation of radar Doppler signal processing based on advanced differential geometry of symmetric spaces: Doppler matrix CFAR and radar application,” in Proc. Radar’09 Conf., Bordeaux, France, Oct. 2009 [9] L. Yang, “Riemannian median and its estimation,” LMS J. Comput. Math., vol. 13, pp. 461–479, 2010 [10] L. Yang, M. Arnaudon, and F. Barbaresco, “Riemannian median, geometry of covariance matrices and radar target detection,” in Proc. Eur. Radar Conf., pp. 415–418, 2010 183 /183 / Thales Air Systems Date Tutorial Speaker References [11] F. Barbaresco, “Robust Median-Based STAP in Inhomogeneous Secondary Data : Frechet Information Geometry of Covariance Matrices”, 2nd French-Singaporian SONDRA Workshop on “EM Modeling, New Concepts and Signal Processing For Radar Detection and Remote Sensing, Cargese France, 25-28 May 2010 [12] F. Barbaresco, “Geometric Science of Information: Modern Geometric Foundation of Radar Signal Processing”, 8th International IEEE Conference on Communications, Bucharest, Romania, 10th-12th June, 2010 [13] C. Chaure & F. Barbaresco, "New Generation Doppler Radar Processing : Ultra-fast Robust Doppler Spectrum Barycentre Computation Scheme in Poincaré’s Unit Disk", EURAD'10 conference, Paris, Sept. 2010 [14] L. Yang, M. Arnaudon, and F. Barbaresco, “Geometry of covariance matrices and computation of median,” in Proc. AIP Conf., vol. 1305, pp. 479–486, 2011 184 /184 / Thales Air Systems Date Tutorial Speaker References [15] L. Yang, “Medians of probability measures in Riemannian manifolds and applications to radar target detection,” Ph.D. dissertation supervised by F. Barbaresco & M. Arnaudon, Univ. de Poitiers, Poitiers, France, 2011 [16] F. Barbaresco, “Science géométrique de l’information: géométrie des matrices de covariance, espace métrique de Fréchet et domaines bornés homogènes de siegel,” in Proc. Conf. GRETSI’11, Bordeaux, France, Sep. 2011. [17] F. Barbaresco, “Robust statistical radar processing in Fréchet metric space: OS-HDR-CFAR and OS-STAP processing in siegel homogeneous bounded domains, proceedings of IRS’11,” in Proc. Int. Radar Conf., Leipzig, Germany, Sep. 2011. [18] F. Barbaresco, “Geometric radar processing based on Fréchet distance: information geometry versus optimal transport theory,” in Proc. IRS’11, Int. Radar Conf., Leipzig, Germany, Sep. 2011 185 /185 / Thales Air Systems Date Tutorial Speaker References [19] F. Barbaresco, Information Geometry of Covariance Matrix: Cartan-Siegel Homogeneous Bounded Domains, Mostow/Berger Fibration and Fréchet Median, Matrix Information Geometry, R. Bhatia and F. Nielsen, Eds. New York, NY, USA: Springer, pp. 199– 256, 2012 [20] F. Barbaresco, “RRP 3.0: 3rd generation robust radar processing based on matrix information geometry (MIG),” in Proc. EuRad Conf. EuRad’12, Amsterdam, The Netherlands, Nov. 2012 [21] M. Arnaudon, F. Barbaresco, and L. Yang, Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation, Matrix Information Geometry, R. Bhatia andF. Nielsen, Eds. NewYork, NY, USA: Springer, pp. 169–198, 2012 [22] B. Balaji and F. Barbaresco, “Application of Riemannian mean of covariance matrices to space-time adaptive processing,” in Proc. EuRad Conf., EuRad’12, Amsterdam, The Netherlands, Nov. 2012 186 /186 / Thales Air Systems Date Tutorial Speaker References [23] F. Barbaresco, “Information geometry manifold of Toeplitz Hermitian positive definite covariance matrices: Mostow/Berger fibration and Berezin quantization of Cartan-Siegel domains,” Int. J. Emerging Trends in Signal Process. (IJETSP), vol. 1, no. 3, Mar. 2013 [24] F. Barbaresco, “Radar detection for non-stationary time- Doppler signal based on Fréchet distance of geodesic curves on covariance matrix information geometry manifold,” in Proc. Int. Radar Symp. IRS’13, Dresden, Germany, Jun. 2013 [25] M. Arnaudon, F. Barbaresco & Y. Le, “ Riemannian Medians and Means With Applications to Radar Signal Processing”, IEEE Journal of Selected Topics in Signal Processing, vol. 7, n°. 4, August 2013 [26] B. Balaji and F. Barbaresco, “Riemannian mean and space- time adaptive processing using projection and inversion algorithms,” in Proc. SPIE Defense, Security, Sens. Conf., May 2013 187 /187 / Thales Air Systems Date Tutorial Speaker References [27] F. Barbaresco, Information/Contact Geometries and Koszul Entropy, Geometric Science of Information, Lecture Notes in Computer Science, Vol. 8085, F. Nielsen, & Barbaresco, Frederic (Eds.), pp. 604-611, 2013 [28] Decurninge, A. ; Barbaresco, F., “Robust L1/geometric covariance matrix estimator: Comparison with huber-type M- estimator”, International Radar Symposium, Dresde, June 2013 [29] F. Barbaresco, “Eidetic Reduction of Information Geometry through Legendre Duality of Koszul Characteristic Function and Entropy: from Massieu-Duhem Potentials to Geometric Souriau Temperature and Balian Quantum Fisher Metric”, Special Issue ‘Geometric Theory of Information (GTI)’, SPRINGER, 2014 [30] Barbaresco F., Koszul Information Geometry and Souriau Geometric Temperature/Capacity of Lie Group Thermodynamics, Entropy, n°16, special issue on Information Geometry, 2014 [31] Decurninge A., Barbaresco F., Burg Estimation of Radar Covariance Matrix for Mixtures of Gaussian Stationary Distributions, Radar’14, Lille, Octo. 2014