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
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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

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
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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.

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
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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.

Author(s): Alain Sarlette
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_73
Video: http://www.youtube.com/watch?v=tbUBxuawaMg
Slides: Sarlette_operational viewpoint.pdf
Presentation: https://www.see.asso.fr/node/14295
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Abstract:
This paper highlights some more examples of maps that follow a recently introduced “symmetrization” structure behind the average consensus algorithm. We review among others some generalized consensus settings and coordinate descent optimization.

Author(s): Frank Nielsen, Gautier Marti, Philippe Donnat, Philippe Very
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_72
Video: http://www.youtube.com/watch?v=wgb6olnmO50
Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiYUpPWkc2WkdRaDA
Presentation: https://www.see.asso.fr/node/14294
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Abstract:
We present in this paper a novel non-parametric approach useful for clustering independent identically distributed stochastic processes. We introduce a pre-processing step consisting in mapping multivariate independent and identically distributed samples from random variables to a generic non-parametric representation which factorizes dependency and marginal distribution apart without losing any information. An associated metric is defined where the balance between random variables dependency and distribution information is controlled by a single parameter. This mixing parameter can be learned or played with by a practitioner, such use is illustrated on the case of clustering financial time series. Experiments, implementation and results obtained on public financial time series are online on a web portal http://www.datagrapple.com .

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
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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.

Author(s): Giovanni Pistone, Luigi Malagò
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_38
Video: http://www.youtube.com/watch?v=L_pYet3UoPs
Slides: Pistone_second-order optimization multivariate Gaussian.pdf
Presentation: https://www.see.asso.fr/node/14292
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Abstract:
We discuss the optimization of the stochastic relaxation of a real-valued function, i.e., we introduce a new search space given by a statistical model and we optimize the expected value of the original function with respect to a distribution in the model. From the point of view of Information Geometry, statistical models are Riemannian manifolds of distributions endowed with the Fisher information metric, thus the stochastic relaxation can be seen as a continuous optimization problem defined over a differentiable manifold. In this paper we explore the second-order geometry of the exponential family, with applications to the multivariate Gaussian distributions, to generalize second-order optimization methods. Besides the Riemannian Hessian, we introduce the exponential and the mixture Hessians, which come from the dually flat structure of an exponential family. This allows us to obtain different Taylor formulæ according to the choice of the Hessian and of the geodesic used, and thus different approaches to the design of second-order methods, such as the Newton method.

Author(s): Christophe Saint-Jean, Frank Nielsen
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_37
Video: http://www.youtube.com/watch?v=W4H5ccCNqck
Slides: Saint-Jean_Online kMLE.pdf
Presentation: https://www.see.asso.fr/node/14291
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Abstract:
This paper address the problem of online learning finite statistical mixtures of exponential families. A short review of the Expectation-Maximization (EM) algorithm and its online extensions is done. From these extensions and the description of the k-Maximum Likelihood Estimator (k-MLE), three online extensions are proposed for this latter. To illustrate them, we consider the case of mixtures of Wishart distributions by giving details and providing some experiments.

Author(s): James Tao, Jun Zhang
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_36
Video: http://www.youtube.com/watch?v=ebmReSVXZ1E
Slides: Tao_Transformation coupling relations.pdf
Presentation: https://www.see.asso.fr/node/14290
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Abstract:
The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇ g is totally symmetric, forming, by definition, a “Codazzi pair” { ∇ , g}. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.

Author(s): Nihat Ay, Shun-Ichi Amari
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_35
Video: http://www.youtube.com/watch?v=Pu8M9Fu7_fM
Slides: Amari_standard divergence.pdf
Presentation: https://www.see.asso.fr/node/14289
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Abstract:
A divergence function defines a Riemannian metric G and dually coupled affine connections (∇, ∇  ∗ ) with respect to it in a manifold M. When M is dually flat, a canonical divergence is known, which is uniquely determined from {G, ∇, ∇  ∗ }. We search for a standard divergence for a general non-flat M. It is introduced by the magnitude of the inverse exponential map, where α = -(1/3) connection plays a fundamental role. The standard divergence is different from the canonical divergence.

Author(s): Yann Ollivier
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_34
Video: http://www.youtube.com/watch?v=hBzvSaA9yRU
Slides: Ollivier_Laplace rule succession.pdf
Presentation: https://www.see.asso.fr/node/14288
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Abstract:
When observing data x1, . . . , x t modelled by a probabilistic distribution pθ(x), the maximum likelihood (ML) estimator θML = arg max θ Σti=1 ln pθ(x i ) cannot, in general, safely be used to predict xt + 1. For instance, for a Bernoulli process, if only “tails” have been observed so far, the probability of “heads” is estimated to 0. (Thus for the standard log-loss scoring rule, this results in infinite loss the first time “heads” appears.)

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
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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)).

Author(s): José Ignacio Burgos Gil, Martin Sombra, Patrice Philippon
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_32
Video: http://www.youtube.com/watch?v=7FfYJ94VhAk
Slides: Philippon_Heights of toric.pdf
Presentation: https://www.see.asso.fr/node/14286
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Abstract:
We present a dictionary between arithmetic geometry of toric varieties and convex analysis. This correspondence allows for effective computations of arithmetic invariants of these varieties. In particular, combined with a closed formula for the integration of a class of functions over polytopes, it gives a number of new values for the height (arithmetic analog of the degree) of toric varieties, with respect to interesting metrics arising from polytopes. In some cases these heights are interpreted as the average entropy of a family of random processes.

Author(s): Herbert Gangl, Philippe Elbaz-Vincent
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_31
Video: http://www.youtube.com/watch?v=Pcu3oKN0Xnk
Slides: Elbaz-Vincent_finite polylogarithm.pdf
Presentation: https://www.see.asso.fr/node/14285
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Abstract:
We show that the entropy function–and hence the finite 1-logarithm–behaves a lot like certain derivations. We recall its cohomological interpretation as a 2-cocycle and also deduce 2n-cocycles for any n. Finally, we give some identities for finite multiple polylogarithms together with number theoretic applications.

Author(s): Matilde Marcolli
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_30
Video: http://www.youtube.com/watch?v=iUPjhUcHvxM
Slides: Marcolli_information algebra.pdf
Presentation: https://www.see.asso.fr/node/14284
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Abstract:
In this lecture we will present joint work with Ryan Thorngren on thermodynamic semirings and entropy operads, with Nicolas Tedeschi on Birkhoff factorization in thermodynamic semirings, ongoing work with Marcus Bintz on tropicalization of Feynman graph hypersurfaces and Potts model hypersurfaces, and their thermodynamic deformations, and ongoing work by the author on applications of thermodynamic semirings to models of morphology and syntax in Computational Linguistics.

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
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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.

Author(s): Hideyuki Ishi
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_28
Video: http://www.youtube.com/watch?v=sdB0NTRSuUE
Slides: Ishi_Matrix realisation homogenous cone.pdf
Presentation: https://www.see.asso.fr/node/14281

Abstract:
Based on the theory of compact normal left-symmetric algebra (clan), we realize every homogeneous cone as a set of positive definite real symmetric matrices, where homogeneous Hessian metrics as well as a transitive group action on the cone are described efficiently.