Subcategories


  • Marc Arnaudon - Matilde Marcolli - Tudor Ratiu

    cschreiner

    Author(s): Marc Arnaudon
    Institution: Institut de Mathématiques de Bordeaux (IMB), CNRS : UMR 5251, Université de Bordeaux, France
    Website: http://www.math.u-bordeaux1.fr/~marnaudo/
    Video: http://www.youtube.com/watch?v=1mKs_akkEuw
    Slides: Arnaudon_Stochastic EulerPoincare reduction.pdf
    Presentation: https://www.see.asso.fr/node/13650
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We will prove a Euler-Poincaré reduction theorem for stochastic processes taking values in a Lie group, which is a generalization of the Lagrangian version of reduction and its associated variational principles. We will also show examples of its application to the rigid body and to the group of diffeomorphisms, which includes the Navier-Stokes equation on a bounded domain and the Camassa-Holm equation.

    References:

    M. Arnaudon, A.B. Cruzeiro and X. Chen, "Stochastic Euler-Poincaré Reduction", Journal of Mathematical Physics, to appear V. I. Arnold and B. Khesin, "Topological methods in hydrodynamics", Applied Math. Series 125, Springer (1998). J. M. Bismut, "Mécanique aléatoire", Lecture Notes in Mathematics, 866, Springer (1981). D.G. Ebin and J.E. Marsden, "Groups of diffeomorphisms and the motion of an incompressible fluid", Ann of Math. 92 (1970), 102--163. J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry: a basic exposition of classical mechanical systems", Springer, Texts in Applied Math. (2003).

    Bio:
    Marc Arnaudon was born in France in 1965. He graduated from Ecole Normale Supérieure de Paris, France, in 1991. He received the PhD degree in mathematics and the Habilitation à diriger des Recherches degree from Strasbourg University, France, in January 1994 and January 1998 respectively. After postdoctoral research and teaching at Strasbourg, he began in September 1999 a full professor position in the Department of Mathematics at Poitiers University, France, where he was the head of the Probability Research Group. In January 2013 he left Poitiers and joined the Department of Mathematics of Bordeaux University, France, where he is a full professor in mathematics.
    Prof. Arnaudon is an expert in stochastic differential geometry and stochastic calculus in manifolds, he has published over 50 articles in mathematical and physical journals.

    Arnaudon.jpg


  • Laurent Decreusefond, Frédéric Chazal

    pcardosi

    Author : Roman Belavkin
    DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_23
    Video : http://www.youtube.com/watch?v=1vURZX7plVk
    Slides: Belavkin_Asymetric topologies on statistical manifolds.pdf
    Presentation : https://www.see.asso.fr/node/14261
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.


  • Paul Marriott, Frank Nielsen

    pcardosi

    Authors : Frank Critchley, Germain Van Bever, Paul Marriott, Radka Sabolova
    Video : http://www.youtube.com/watch?v=lpzCTiW1QHI
    Sides: https://drive.google.com/open?id=0B0QKxsVtOaTiMjVPTWowM2J5U2M
    Presentation : https://www.see.asso.fr/node/14268
    Creative Commons Attribution-ShareAlike 4.0 International


  • Ali Mohammad-Djafari, Olivier Swander

    pcardosi

    Authors : Hiroshi Matsuzoe, Monta Sakamoto
    DOI URL : http://dx.doi.org/10.1007/978-3-319-25040-3_79
    Video : http://www.youtube.com/watch?v=hGc1z8EYR24
    Slides: Matsuzoe-generalization independence Student.pdf
    Presentation : https://www.see.asso.fr/node/14272
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    In anomalous statistical physics, deformed algebraic structures are important objects. Heavily tailed probability distributions, such as Student’s t-distributions, are characterized by deformed algebras. In addition, deformed algebras cause deformations of expectations and independences of random variables. Hence, a generalization of independence for multivariate Student’s t-distribution is studied in this paper. Even if two random variables which follow to univariate Student’s t-distributions are independent, the joint probability distribution of these two distributions is not a bivariate Student’s t-distribution. It is shown that a bivariate Student’s t-distribution is obtained from two univariate Student’s t-distributions under q-deformed independence.


  • Shun-Ichi Amari, Michel Boyom

    cschreiner

    Author(s): Jamali Mohammed, Michel Boyom, Shahid Hasan
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_29
    Video: http://www.youtube.com/watch?v=9qTs052AfX8
    Slides: Hasan_Multiply CR-warped.pdf
    Presentation: https://www.see.asso.fr/node/14282
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    In this article, we derive an inequality satisfied by the squared norm of the imbedding curvature tensor of Multiply CR-warped product statistical submanifolds N of holomorphic statistical space forms M. Furthermore, we prove that under certain geometric conditions, N and M become Einstein.


  • Pierre Baudot, Daniel Bennequin

    cschreiner

    Author(s): Jean-Charles Pinoli, Johan Debayle, Saïd Rahmani
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_33
    Video: http://www.youtube.com/watch?v=2ebliJYcWEA
    Slides: https://drive.google.com/open?id=0B0QKxsVtOaTicEphTng2VE9wclU
    Presentation: https://www.see.asso.fr/node/14287
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    In this paper we propose a method to characterize and estimate the variations of
    a random convex set Ξ0 in terms of shape, size and direction. The mean n-variog
    ram γ(n)Ξ0:(u1⋯un)↦E[νd(Ξ0∩(Ξ0−u1)⋯∩(Ξ0−un))] of a random convex set Ξ0 on ℝ d r
    eveals information on the n th order structure of Ξ0. Especially we will show that considering the mean n-variograms of the dilated random sets Ξ0 ⊕ rK by an homothetic convex family rKr > 0, it’s possible to estimate some characteristic of the n th order structure of Ξ0. If we make a judicious choice of K, it provides relevant measures of Ξ0. Fortunately the germ-grain model is stable by convex dilatations, furthermore the mean n-variogram of the primary grain is estimable in several type of stationary germ-grain models by the so called n-points probability function. Here we will only focus on the Boolean model, in the planar case we will show how to estimate the n th order structure of the random vector composed by the mixed volumes t (A(Ξ0),W(Ξ0,K)) of the primary grain, and we will describe a procedure to do it from a realization of the Boolean model in a bounded window. We will prove that this knowledge for all convex body K is sufficient to fully characterize the so called difference body of the grain Ξ0⊕˘Ξ0. we will be discussing the choice of the element K, by choosing a ball, the mixed volumes coincide with the Minkowski’s functional of Ξ0 therefore we obtain the moments of the random vector composed of the area and perimeter t (A(Ξ0),U(Ξ)). By choosing a segment oriented by θ we obtain estimates for the moments of the random vector composed by the area and the Ferret’s diameter in the direction θ, t((A(Ξ0),HΞ0(θ)). Finally, we will evaluate the performance of the method on a Boolean model with rectangular grain for the estimation of the second order moments of the random vectors t (A(Ξ0),U(Ξ0)) and t((A(Ξ0),HΞ0(θ)).


  • Yann Ollivier, Giovanni Pistone

    cschreiner

    Author(s): Garvesh Raskutti, Sayan Mukherjee
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_39
    Video: http://www.youtube.com/watch?v=PKujdGuu5Bc
    Slides: Monod_Information geomerty mirror descent.pdf
    Presentation: https://www.see.asso.fr/node/14293
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We prove the equivalence of two online learning algorithms, mirror descent and natural gradient descent. Both mirror descent and natural gradient descent are generalizations of online gradient descent when the parameter of interest lies on a non-Euclidean manifold. Natural gradient descent selects the steepest descent direction along a Riemannian manifold by multiplying the standard gradient by the inverse of the metric tensor. Mirror descent induces non-Euclidean structure by solving iterative optimization problems using different proximity functions. In this paper, we prove that mirror descent induced by a Bregman divergence proximity functions is equivalent to the natural gradient descent algorithm on the Riemannian manifold in the dual coordinate system.We use techniques from convex analysis and connections between Riemannian manifolds, Bregman divergences and convexity to prove this result. This equivalence between natural gradient descent and mirror descent, implies that (1) mirror descent is the steepest descent direction along the Riemannian manifold corresponding to the choice of Bregman divergence and (2) mirror descent with log-likelihood loss applied to parameter estimation in exponential families asymptotically achieves the classical Cramér-Rao lower bound.

  • cschreiner

    Author(s): Anna-Lena Kißlinger, Wolfgang Stummer
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_74
    Video: http://www.youtube.com/watch?v=lmmIXF0ziCk
    Slides: https://drive.google.com/open?id=0B0QKxsVtOaTib0V6RTNodGlKbUk
    Presentation: https://www.see.asso.fr/node/14296
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    Scaled Bregman distances SBD have turned out to be useful tools for simultaneous estimation and goodness-of-fit-testing in parametric models of random data (streams, clouds). We show how SBD can additionally be used for model preselection (structure detection), i.e. for finding appropriate candidates of model (sub)classes in order to support a desired decision under uncertainty. For this, we exemplarily concentrate on the context of nonlinear recursive models with additional exogenous inputs; as special cases we include nonlinear regressions, linear autoregressive models (e.g. AR, ARIMA, SARIMA time series), and nonlinear autoregressive models with exogenous inputs (NARX). In particular, we outline a corresponding information-geometric 3D computer-graphical selection procedure. Some sample-size asymptotics is given as well.


  • Alfred Galichon, Jean-François Marcotorchino

    Pawamoy

    Author: Michele Pavon, Tryphon Georgiou, Yonxin Chen
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_9
    Video: http://www.youtube.com/watch?v=gxFFKApXJZU
    Slides: Pavon_optimal mass transport.pdf
    Presentation: https://www.see.asso.fr/node/14301
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We present an overview of our recent work on implementable solutions to the Schrödinger bridge problem and their potential application to optimal transport and various generalizations.


  • Yannick Berthoumieu, Geert Verdoolaege

    Pawamoy

    Author: Gianni Franchi, Jesús Angulo
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_44
    Video: http://www.youtube.com/watch?v=Xwav5x4AWEY
    Slides: Franchi_Quantization hyperspectral image.pdf
    Presentation: https://www.see.asso.fr/node/14306
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    A technique of spatial-spectral quantization of hyperspectral images is introduced. Thus a quantized hyperspectral image is just summarized by K spectra which represent the spatial and spectral structures of the image. The proposed technique is based on α-connected components on a region adjacency graph. The main ingredient is a dissimilarity metric. In order to choose the metric that best fit the hyperspectral data manifold, a comparison of different probabilistic dissimilarity measures is achieved.

  • Pawamoy

    Author: Alexis Decurninge
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_13
    Video: http://www.youtube.com/watch?v=lF0PbFujrGs
    Slides: Decurninge_MultivariateLmoments.pdf
    Presentation: https://www.see.asso.fr/node/14310
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    Univariate L-moments are expressed as projections of the quantile function onto an orthogonal basis of univariate polynomials. We present multivariate versions of L-moments expressed as collections of orthogonal projections of a multivariate quantile function on a basis of multivariate polynomials. We propose to consider quantile functions defined as transports from the uniform distribution on [0; 1] d onto the distribution of interest and present some properties of the subsequent L-moments. The properties of estimated L-moments are illustrated for heavy-tailed distributions.


  • Jesús Angulo, S. Said

    Pawamoy

    Author: Dena Asta
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_83
    Video: Not available
    Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiX0lnY3YzZDV1Y0E
    Presentation: https://www.see.asso.fr/node/14314
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We introduce a novel kernel density estimator for a large class of symmetric spaces and prove a minimax rate of convergence as fast as the minimax rate on Euclidean space. We prove a minimax rate of convergence proven without any compactness assumptions on the space or Hölder-class assumptions on the densities. A main tool used in proving the convergence rate is the Helgason-Fourier transform, a generalization of the Fourier transform for semisimple Lie groups modulo maximal compact subgroups. This paper obtains a simplified formula in the special case when the symmetric space is the 2-dimensional hyperboloid.

  • pparrend

    Author(s): Alain Trouvé, Barbara Gris, Stanley Durrleman
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_5
    Video: http://www.youtube.com/watch?v=-8ZAdmUEyLY
    Slides: Gris_subriemannian modular approach.pdf
    Presentation: https://www.see.asso.fr/node/14320
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    We develop a generic framework to build large deformations from a combination of base modules. These modules constitute a dynamical dictionary to describe transformations. The method, built on a coherent sub-Riemannian framework, defines a metric on modular deformations and characterises optimal deformations as geodesics for this metric. We will present a generic way to build local affine transformations as deformation modules, and display examples.


  • Pierre-Antoine Absil, Rodolphe Sepulchre

    pparrend

    Author(s): Kathrin Welker, Martin Siebenborn, Volker Schulz
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_54
    Video: http://www.youtube.com/watch?v=720pF5M093U
    Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiVzdHanNsOEcxOXM
    Presentation: https://www.see.asso.fr/node/14326
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    The novel Riemannian view on shape optimization introduced in [14] is extended to a Lagrange–Newton as well as a quasi–Newton approach for PDE constrained shape optimization problems.


  • Stéphanie Allassonnière, Stanley Durrleman

    pparrend

    Author(s): Sylvain Arguillere
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_18
    Video: http://www.youtube.com/watch?v=ZJmjDlsRqDs
    Slides: Arguillere_General setting shape deformation.pdf
    Presentation: https://www.see.asso.fr/node/14332
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    This paper aims to define a unified setting for shape registration and LDDMM methods for shape analysis. This setting turns out to be sub-Riemannian, and not Riemannian. An abstract definition of a space of shapes in ℝ d is given, and the geodesic flow associated to any reproducing kernel Hilbert space of sufficiently regular vector fields is showed to exist for all time.

  • pparrend

    Author(s): Bijan Afsari, Gregory S. Chirikjian
    DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_69
    Video: http://www.youtube.com/watch?v=a1WCMsM9HqE
    Slides: Afsari_Debluring and recovering conformational.pdf
    Presentation: https://www.see.asso.fr/node/14335
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    In this paper we study two forms of blurring effects that may appear in the reconstruction of 3D Electron Microscopy (EM), specifically in single particle reconstruction from random orientations of large multi-unit biomolecular complexes. We model the blurring effects as being due to independent contributions from: (1) variations in the conformation of the biomolecular complex; and (2) errors accumulated in the reconstruction process. Under the assumption that these effects can be separated and treated independently, we show that the overall blurring effect can be expressed as a special form of a convolution operation of the 3D density with a kernel defined on SE(3), the Lie group of rigid body motions in 3D. We call this form of convolution mixed spatial-motional convolution.We discuss the ill-conditioned nature of the deconvolution needed to deblur the reconstructed 3D density in terms of parameters associated with the unknown probability in SE(3). We provide an algorithm for recovering the conformational information of large multi-unit biomolecular complexes (essentially deblurring) under certain biologically plausible prior structural knowledge about the subunits of the complex in the case the blurring kernel has a special form.


  • Frédéric Barbaresco, Géry de Saxcé

    abotsi

    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
    Creative Commons Attribution-ShareAlike 4.0 International

    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.


  • Michel Broniatowski, Imre Csiszár

    abotsi

    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
    Creative Commons Attribution-ShareAlike 4.0 International

    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.

  • cschreiner

    Author: Dominique Spehner
    Institution: Université Joseph Fourier, Grenoble, Institut Fourier, France.
    Website: https://www-fourier.ujf-grenoble.fr/~spehner/
    Video: http://www.youtube.com/watch?v=5Nj5afyivI8
    Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiR3lWZ1dUSVlhakE
    Presentation: https://www.see.asso.fr/node/14277
    Creative Commons Attribution-ShareAlike 4.0 International

    Abstract:
    I will show that the set of states of a quantum system with a finite-dimensional Hilbert space can be equipped with various Riemannian distances having nice properties from a quantum information viewpoint, namely they are contractive under all physically allowed operations on the system. The corresponding metrics are quantum analogs of the Fisher metric and have been classified by D. Petz. Two distances are particularly relevant physically: the Bogoliubov-Kubo-Mori distance studied by R. Balian, Y. Alhassid and H. Reinhardt, and the Bures distance studied by A. Uhlmann and by S.L. Braunstein and C.M. Caves. The latter gives the quantum Fisher information playing an important role in quantum metrology. A way to measure the amount of quantum correlations (entanglement or quantum discord) in bipartite systems (that is, systems composed of two parties) with the help of these distances will be also discussed.

    References:

    D. Petz, Monotone Metrics on Matrix Spaces, Lin. Alg. and its Appl. 244, 81-96 (1996) R. Balian, Y. Alhassid, and H. Reinhardt, Dissipation in many-body systems: a geometric approach based on information theory, Phys. Rep. 131, 1 (1986) R. Balian, The entropy-based quantum metric, Entropy 2014 16(7), 3878-3888 (2014) A. Uhlmann, The ``transition probability'' in the state space of a *-algebra, Rep. Math. Phys. 9, 273-279 (1976) S.L. Braunstein and C.M. Caves, Statistical Distance and the Geometry of Quantum States, Phys. Rev. Lett. 72, 3439-3443 (1994) D. Spehner, Quantum correlations and Distinguishability of quantum states, J. Math. Phys. 55 (2014), 075211

    Bio:

    Diplôme d'Études Approfondies (DEA) in Theoretical Physics at the École Normale Supérieure de Lyon, 1994 Civil Service (Service National de la Coopération), Technion Institute of Technology, Haifa, Israel, 1995-1996 PhD in Theoretical Physics, Université Paul Sabatier, Toulouse, France, 1996-2000. Postdoctoral fellow, Pontificia Universidad Católica, Santiago, Chile, 2000-2001 Research Associate, University of Duisburg-Essen, Germany, 2001-2005 Maître de Conférences, Université Joseph Fourier, Grenoble, France, 2005-present Habilitation à diriger des Recherches (HDR), Université Grenoble Alpes, 2015 Member of the Institut Fourier (since 2005) and the Laboratoire de Physique et Modélisation des Milieux Condensés (since 2013) of the university Grenoble Alpes, France

    Spehner.jpg


  • Charles-Michel Marle

    abotsi

    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
    Creative Commons Attribution-ShareAlike 4.0 International

    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.

    Marle.jpg



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