Author: Charles-Michel Marle
Institution: Professeur honoraire à l'Université Pierre et Marie Curie, Institut Mathématique de Jussieu, Correspondant de l’Académie des Sciences, Paris, France.
Website: http://charles-michel.marle.pagesperso-orange.fr/
Video: http://www.youtube.com/watch?v=qdySpxlolrM
Slides: Marle_Actions of Lie Groups algebras.pdf
Presentation: https://www.see.asso.fr/node/13649
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Abstract:
I will present some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. Lie group and Lie algebra actions on symplectic and Poisson manifolds, momentum maps and their equivariance properties, first integrals associated to symmetries of Hamiltonian systems will be discussed. Reduction methods taking advantage of symmetries will be discussed.

References :

• Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Addison-Wesley, Reading (1978).
• Arnold, V.I.: Mathematical methods of Classical Mechanics, 2nd edn. Springer, New York (1978).
• Ganghoffer, J.-F., Maldenov, E. (editors): Similarity and Symmetry Methods; Applications in Elasticity and mechanics of Materials. Lecture Notes in Applied and Computational Mechanics 73, Springer, Heidelberg (2014).
• Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984).
• Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson structures. Springer, Berlin (2013).
• Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987)

Bio:
Charles-Michel Marle was born in 1934; He studied at Ecole Polytechnique (1953--1955), Ecole Nationale Supérieure des Mines de Paris (1957--1958) and Ecole Nationale Supérieure du Pétrole et des Moteurs (1957--1958). He obtained a doctor's degree in Mathematics at the University of Paris in 1968. From 1959 to 1969 he worked as a research engineer at the Institut Français du Pétrole. He joined the Université de Besançon as Associate Professor in 1969, and the Université Pierre et Marie Curie, first as Associate Professor (1975) and then as full Professor (1981). His resarch works were first about fluid flows through porous media, then about Differential Geometry, Hamiltonian systems and applications in Mechanics and Mathematical Physics.

Author(s): Amor Keziou, Philippe Regnault
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_49
Video: http://www.youtube.com/watch?v=VGucQ4FTMR4
Slides: Regnault_Semi-parametric estimation.pdf
Presentation: https://www.see.asso.fr/node/14348
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Abstract:
We derive independence tests by means of dependence measures thresholding in a semiparametric context. Precisely, estimates of mutual information associated to ϕ-divergences are derived through the dual representations of ϕ-divergences. The asymptotic properties of the estimates are established, including consistency, asymptotic distribution and large deviations principle. The related tests of independence are compared through their relative asymptotic Bahadur efficiency and numerical simulations.

Author(s): Amor Keziou
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_48
Video: No autorisation
Slide: No autorisation
Presentation: https://www.see.asso.fr/node/14347
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Abstract:
We introduce what we will call multivariate divergences between K, K_³ 1, signed finite measures (Q1, . . . , Q K ) and a given reference probability measure P on a _-field (X,B), extending the well known divergences between two measures, a signed finite measure Q1 and a given probability distribution P. We investigate the Fenchel duality theory for the introduced multivariate divergences viewed as convex functionals on well chosen topological vector spaces of signed finite measures. We obtain new dual representations of these criteria, which we will use to define new family of estimates and test statistics with multiple samples under multiple semiparametric density ratio models. This family contains the estimate and test statistic obtained through empirical likelihood. Moreover, the present approach allows obtaining the asymptotic properties of the estimates and test statistics both under the model and under misspecification. This leads to accurate approximations of the power function for any used criterion, including the empirical likelihood one, which is of its own interest. Moreover, the proposed multivariate divergences can be used, in the context of multiple samples in density ratio models, to define new criteria for model selection and multi-group classification.

Author(s): Imre Csiszár, Michel Broniatowski, Thomas Breuer
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_47
Video: not available
Slides: Breuer_Information geometry finance.pdf
Presentation: https://www.see.asso.fr/node/14346
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Abstract:
Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family _ of plausible risk factor distributions IP. We consider _ determined by a bound on a convex integral functional of the density of IP, thus _ may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution IPo. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing EIP(X) subject to IP__), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When _ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.

Author(s): Ben Anthonis, Jan Naudts, Michel Broniatowski
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_46
Video: http://www.youtube.com/watch?v=wnhAE9sVYZE
Slides: Naudts_Extension of information geometry.pdf
Presentation: https://www.see.asso.fr/node/14345
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Abstract:
Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.

Author: Diaa Al Mohamad, Michel Broniatowski
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_45
Video: http://www.youtube.com/watch?v=eVrLiyB0M9A
Slides: ALMohamad_generalized EM algo.pdf
Presentation: https://www.see.asso.fr/node/14344
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Abstract:
Minimum divergence estimators are derived through the dual form of the divergence in parametric models. These estimators generalize the classical maximum likelihood ones. Models with unobserved data, as mixture models, can be estimated with EM algorithms, which are proved to converge to stationary points of the likelihood function under general assumptions. This paper presents an extension of the EM algorithm based on minimization of the dual approximation of the divergence between the empirical measure and the model using a proximaltype algorithm. The algorithm converges to the stationary points of the empirical criterion under general conditions pertaining to the divergence and the model. Robustness properties of this algorithm are also presented. We provide another proof of convergence of the EM algorithm in a two-component gaussian mixture. Simulations on Gaussian andWeibull mixtures are performed to compare the results with the MLE.

Author: Eric Justh, P. S. Krishnaprasad
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_60
Video: http://www.youtube.com/watch?v=leK1MggION0
Slides: Justh_Enlargement geodesics collectives.pdf
Presentation: https://www.see.asso.fr/node/14343
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Abstract:
We investigate optimal control of systems of particles on matrix Lie groups coupled through graphs of interaction, and characterize the limit of strong coupling. Following Brockett, we use an enlargement approach to obtain a convenient form of the optimal controls. In the setting of drift-free particle dynamics, the coupling terms in the cost functionals lead to a novel class of problems in subriemannian geometry of product Lie groups.

Author: Fátima Leite, Krzysztof Krakowski, Luís Machado
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_59
Video: http://www.youtube.com/watch?v=Dm1ht6tNJbo
Slides: Machado_Rolling Symmetric Spaces.pdf
Presentation: https://www.see.asso.fr/node/14342
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Abstract:
Riemannian symmetric spaces play an important role in many areas that are interrelated to information geometry. For instance, in image processing one of the most elementary tasks is image interpolation. Since a set of images may be represented by a point in the Graßmann manifold, image interpolation can be formulated as an interpolation problem on that symmetric space. It turns out that rolling motions, subject to nonholonomic constraints of no-slip and no-twist, provide efficient algorithms to generate interpolating curves on certain Riemannian manifolds, in particular on symmetric spaces. The main goal of this paper is to study rolling motions on symmetric spaces. It is shown that the natural decomposition of the Lie algebra associated to a symmetric space provides the structure of the kinematic equations that describe the rolling motion of that space upon its affine tangent space at a point. This generalizes what can be observed in all the particular cases that are known to the authors. Some of these cases illustrate the general results.

Author: Danielle Fortune, Francois Dubois, Juan Antonio Rojas Quintero
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_58
Video: http://www.youtube.com/watch?v=uiOKLhoh3IA
Slides: Dubois Pontryagin calculus Riemannian.pdf
Presentation: https://www.see.asso.fr/node/14340
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Abstract:
In this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin’s framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor.

Author: Frédéric Barbaresco
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_57
Video: http://www.youtube.com/watch?v=wTxIdmALzjo
Slides: Barbaresco_symplectic structure.pdf
Presentation: https://www.see.asso.fr/node/14339
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Abstract:
We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat, and momentum as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. Finally, we conclude on Balian Gauge theory of Thermodynamics compatible with Souriau’s Model.

Author: Géry de Saxcé
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_56
Video: http://www.youtube.com/watch?v=7czAJrfFfgo
Slides: DeSaxce_Entropy structures thermodynamical.pdf
Presentation: https://www.see.asso.fr/node/14338
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Abstract:
With respect to the concept of affine tensor, we analyse in this work the underlying geometric structure of the theories of Lie group statistical mechanics and relativistic thermodynamics of continua, formulated by Souriau independently one of each other. We reveal the link between these ones in the classical Galilean context. These geometric structures of the thermodynamics are rich and we think they might be source of inspiration for the geometric theory of information based on the concept of entropy.

Author: Federico Renda, Frederic Boyer
DIO URL: http://dx.doi.org/10.1007/978-3-319-25040-3_55
Video: http://www.youtube.com/watch?v=RXF12CcGOwU
Slides: Boyer_ Poincaré equations Cosserat shells.pdf
Presentation: https://www.see.asso.fr/node/14356
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Abstract:
In 1901 Henri Poincaré proposed a new set of equations for mechanics. These equations are a generalization of Lagrange equations to a system whose configuration space is a Lie group, which is not necessarily commutative. Since then, this result has been extensively refined by the Lagrangian reduction theory. In this article, we show the relations between these equations and continuous Cosserat media, i.e. media for which the conventional model of point particle is replaced by a rigid body of small volume named microstructure. In particular, we will see that the usual shell balance equations of nonlinear structural dynamics can be easily derived from the Poincaré’s result. This framework is illustrated through the simulation of a simplified model of cephalopod swimming.

Author: Silvere Bonnabel, Axel Barrau
DIO URL: http://dx.doi.org/10.1007/978-3-319-25040-3_71
Video: http://www.youtube.com/watch?v=w7nIx16lIYA
Slides: Bonnabel_Intrinsic Kramer-Rao bound on Lie group.pdf
Presentation: https://www.see.asso.fr/en/node/14594
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Abstract:
In his 2005 paper, S.T. Smith proposed an intrinsic Cramér- Rao bound on the variance of estimators of a parameter defined on a Riemannian manifold. In the present technical note, we consider the special case where the parameter lives in a Lie group. In this case, by choosing, e.g., the right invariant metric, parallel transport becomes very simple, which allows a more straightforward and natural derivation of the bound in terms of Lie bracket, albeit for a slightly different definition of the estimation error. For bi-invariant metrics, the Lie group exponential map we use to define the estimation error, and the Riemannian exponential map used by S.T. Smith coincide, and we prove in this case that both results are identical indeed.

Author: Jesús Angulo, Santiago Velasco-Forero
DIO URL: http://dx.doi.org/10.1007/978-3-319-25040-3_70
Video: http://www.youtube.com/watch?v=NtWV_wO_R5M
Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiVUFyQjZMQnVZaUU
Presentation: https://www.see.asso.fr/node/14336
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Abstract:
We consider a framework for nonlinear operators on functions evaluated on graphs via stacks of level sets. We investigate a family of transformations on functions evaluated on graph which includes adaptive flat and non-flat erosions and dilations in the sense of mathematical morphology. Additionally, the connection to mean motion curvature on graphs is noted. Proposed operators are illustrated in the cases of functions on graphs, textured meshes and graphs of images.