About
LIX Colloquium 2015 conferences
As for GSI’13, the objective of this SEE Conference GSI’15, hosted by Ecole Polytechnique, is to bring together pure/applied mathematicians and engineers, with common interest for Geometric tools and their applications for Information analysis.
It emphasizes an active participation of young researchers to discuss emerging areas of collaborative research on “Information Geometry Manifolds and Their Advanced Applications”.
Current and ongoing uses of Information Geometry Manifolds in applied mathematics are the following: Advanced Signal/Image/Video Processing, Complex Data Modeling and Analysis, Information Ranking and Retrieval, Coding, Cognitive Systems, Optimal Control, Statistics on Manifolds, Machine Learning, Speech/sound recognition, natural language treatment, etc., which are also substantially relevant for industry.
The Conference will be therefore held in areas of priority/focused themes and topics of mutual interest with the aim to:

Provide an overview on the most recent stateoftheart

Exchange mathematical information/knowledge/expertise in the area

Identify research areas/applications for future collaboration

Identify academic & industry labs expertise for further collaboration
This conference will be an interdisciplinary event and will unify skills from Geometry, Probability and Information Theory. The conference proceedings are published in Springer's Lecture Note in Computer Science (LNCS) series.
Authors will be solicited to submit a paper in a special Issue "Differential Geometrical Theory of Statistics” in ENTROPY Journal, an international and interdisciplinary open access journal of entropy and information studies published monthly online by MDPI 
Provisional Topics of Special Sessions:

Manifold/Topology Learning

Riemannian Geometry in Manifold Learning

Optimal Transport theory and applications in Imagery/Statistics

Shape Space & Diffeomorphic mappings

Geometry of distributed optimization

Random Geometry/Homology

Hessian Information Geometry

Topology and Information

Information Geometry Optimization

Divergence Geometry

Optimization on Manifold

Lie Groups and Geometric Mechanics/Thermodynamics

Quantum Information Geometry

Infinite Dimensional Shape spaces

Geometry on Graphs

Bayesian and Information geometry for inverse problems

Geometry of Time Series and Linear Dynamical Systems

Geometric structure of Audio Processing

Lie groups in Structural Biology

Computational Information Geometry
Committees
Secrétaire
 Valérie Alidor  SEE, France https://www.see.asso.fr
Webmestre
 Jean Vieille  SyntropicFactory http://www.syntropicfactory.com
Program chairs
 Frédéric Barbaresco  Thales, France http://www.thalesgroup.com
 Frank Nielsen  Ecole Polytechnique, France http://www.lix.polytechnique.fr/~nielsen/
Scientific committee
 PierreAntoine Absil  University of Louvain, Belgium http://sites.uclouvain.be/absil/
 Bijan Afsari  Johns Hopkins University, USA http://www.cis.jhu.edu/~bijan/
 Stéphanie Allassonnière  Ecole Polytechnique, France https://sites.google.com/site/stephanieallassonniere/home
 Shunichi Amari  RIKEN, Japan http://www.brain.riken.jp/labs/mns/amari/homeE.html
 Jesus Angulo  Mines ParisTech, France http://cmm.ensmp.fr/~angulo/
 JeanPhilippe Anker  Université d'Orléans, France http://www.univorleans.fr/mapmo/membres/anker/
 Sylvain Arguillère  John Hopkins University, USA http://www.cis.jhu.edu/~arguille/
 Marc Arnaudon  Université de Bordeaux, France http://www.math.ubordeaux1.fr/~marnaudo/
 Dena Asta  Carnegie Mellon University, USA http://www.stat.cmu.edu/~dasta/
 Michael Aupetit  Qatar Computing Research Institute, Quatar http://michael.aupetit.free.fr/
 Roger Balian  Academy of Sciences, France https://en.wikipedia.org/wiki/Roger_Balian
 Trivellato Barbara  Politecnico di Torino, Italy http://calvino.polito.it/~trivellato/
 Frédéric Barbaresco  Thales, France http://www.thalesgroup.com
 Michèle Basseville  IRISA, France http://people.irisa.fr/Michele.Basseville/
 Pierre Baudot  Max Planck Institute for Mathematic in the Sciences http://www.mis.mpg.de/jjost/members/pierrebaudot.html
 Martin Bauer  University of Vienna, Austria http://mat.univie.ac.at/~bauerm/Home_Page_of_Martin_Bauer/Home.html
 Roman Belavkin  Middlesex University, UK http://www.eis.mdx.ac.uk/staffpages/rvb/
 Daniel Bennequin  ParisDiderot University http://webusers.imjprg.fr/~daniel.bennequin/
 Jérémy Bensadon  LRI, France https://www.lri.fr/~bensadon/
 JeanFrançois Bercher  ESIEE, France http://perso.esiee.fr/~bercherj/
 Yannick Berthoumieu  IMS Université de Bordeaux, France https://sites.google.com/site/berthoumieuims/
 Jérémie Bigot  Université de Bordeaux, France https://sites.google.com/site/webpagejbigot/
 Michael Blum  IMAG, France http://membrestimc.imag.fr/Michael.Blum/
 Lionel Bombrun  IMS, Université de Bordeaux, France https://www.imsbordeaux.fr/fr/annuaire/4158bombrunlionel
 Silvère Bonnabel  MinesParistech http://www.silverebonnabel.com/
 Ugo Boscain  Ecole polytechnique, France http://www.cmapx.polytechnique.fr/~boscain/
 Nicolas Boumal  Inria & ENS Paris, France http://perso.uclouvain.be/nicolas.boumal/
 Charles Bouveyron  University Paris Descartes, France http://w3.mi.parisdescartes.fr/~cbouveyr/
 Michel Boyom  Université de Montpellier, France http://www.i3m.univmontp2.fr/
 Michel Broniatowski  University of Pierre and Marie Curie, France http://www.lsta.upmc.fr/Broniatowski/
 Martins Bruveris  Brunel University London, UK http://www.brunel.ac.uk/~mastmmb/
 Olivier Cappé  Telecom Paris, France http://perso.telecomparistech.fr/~cappe/
 Charles Cavalcante  Federal University of Ceará, Brazil http://www.ppgeti.ufc.br/charles/
 Antonin Chambolle  Ecole Polytechnique, France http://www.cmap.polytechnique.fr/~antonin/
 Frédéric Chazal  INRIA, France http://geometrica.saclay.inria.fr/team/Fred.Chazal/
 Emmanuel Chevallier  Mines ParisTech, France http://cmm.ensmp.fr/~chevallier/
 Sylvain Chevallier  IUT de Vélizy, France https://sites.google.com/site/sylvchev/
 Arshia Cont  Ircam, France http://repmus.ircam.fr/arshiacont
 Benjamin Couéraud  LAREMA Université d'Angers, France
 Philippe Cuvillier  Ircam, France http://repmus.ircam.fr/cuvillier
 Laurent Decreusefond  Telecom ParisTech, France http://www.infres.enst.fr/~decreuse/
 Alexis Decurninge  Huawei Technologies, Paris, France http://www.huawei.com/en/
 Michel Deza  Ecole Normale Supérieure Paris, CNRS, France http://www.liga.ens.fr/~deza/
 Stanley Durrleman  INRIA, France https://who.rocq.inria.fr/Stanley.Durrleman/index.html
 Patrizio Frosini  Università di Bologna, Italy http://www.dm.unibo.it/~frosini/
 Alfred Galichon  New York University, USA http://alfredgalichon.com/
 JeanPaul Gauthier  University of Toulon, France http://www.lsis.org/gauthierjp/
 Alexis Glaunès  Mines ParisTech, France http://www.mi.parisdescartes.fr/~glaunes/
 PierreYves Gousenbourger  Ecole Polytechnique de Louvain, Belgium http://www.uclouvain.be/pierreyves.gousenbourger
 Piotr Graczyk  University of Angers, France math.univangers.fr
 Peter Grunwald  CWI, Amsterdam, The Netherlands http://homepages.cwi.nl/~pdg/
 Nikolaus Hansen  INRIA, France www.lri.fr
 K V Harsha  Indian Institute of Space Science & Technology, India http://www.iist.ac.in/departments/
 Susan Holmes  Stanford University, USA http://statweb.stanford.edu/~susan/
 Wen Huang  University of Louvain, Belgium
 Stephan Huckemann  Institut für Mathematische Stochastik, Göttingen, Germany http://www.stochastik.math.unigoettingen.de/index.php?id=huckemann
 Shiro Ikeda  ISM, Japan http://www.ism.ac.jp/~shiro/
 Alexander Ivanov  Lomonosov Moscow State University, Russia  Imperial College, UK http://www.imperial.ac.uk/people/a.ivanov
 Jérémie Jakubowicz  Institut Mines Telecom, France http://wwwpublic.itsudparis.eu/~jakubowi/
 Martin Kleinsteuber  Technische Universität München, Germany http://www.professoren.tum.de/en/kleinsteubermartin/
 Ryszard Kostecki  Perimeter Institute for Theoretical Physics, Canada http://www.fuw.edu.pl/~kostecki/
 Hong Van Le  Mathematical Institute of ASCR, Czech Republik http://users.math.cas.cz/~hvle/
 Nicolas Le Bihan  Université de Grenoble, CNRS, France  University of Melbourne, Australia http://www.gipsalab.grenobleinp.fr/~nicolas.lebihan/
 Christian Léonard  Ecole Polytechnique, France http://www.cmap.polytechnique.fr/~leonard/
 Hervé Lombaert  INRIA, France http://step.polymtl.ca/~rv101/
 Jeanmichel Loubes  Toulouse University, France http://perso.math.univtoulouse.fr/loubes/
 Luigi Malagò  Shinshu University, Japan http://malago.di.unimi.it/
 Jonathan Manton  The University of Melbourne http://people.eng.unimelb.edu.au/jmanton/
 Matilde Marcolli  Caltech, USA http://www.its.caltech.edu/~matilde/
 JeanFrançois Marcotorchino  Thales, France https://www.thalesgroup.com/
 CharlesMichel Marle  Université Pierre et Marie Curie, France http://charlesmichel.marle.pagespersoorange.fr/
 Juliette Mattioli  THALES, France https://www.thalesgroup.com/en
 Bertrand Maury  Université Paris Sud, France http://www.math.upsud.fr/~maury/
 Quentin Mérigot  Université ParisDauphine / CNRS, France http://quentin.mrgt.fr/
 Fernand Meyer  Mines ParisTech, France fernandmeyer
 Klas Modin  Chalmers University of Technology, Göteborg, Sweden https://klasmodin.wordpress.com/
 Ali MohammadDjafari  Supelec, CNRS, France http://djafari.free.fr/
 Guido Montufar  Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany http://personalhomepages.mis.mpg.de/montufar/
 Subrahamanian Moosath  Indian Institute of Space Science and Technology, India http://www.iist.ac.in
 Eric Moulines  Telecom ParisTech, France http://perso.telecomparistech.fr/~moulines/
 Jan Naudts  Universiteit Antwerpen, Belgium https://www.uantwerpen.be/en/staff/jannaudts/mywebsite/
 Frank Nielsen  Ecole Polytechnique, France http://www.lix.polytechnique.fr/~nielsen/
 Richard Nock  Université des Antilles et de la Guyane, France  NICTA, Australia http://www.univag.fr/rnock/index.html
 Yann Ollivier  Université Paris Sud, France http://www.yannollivier.org/
 JeanPhilippe Ovarlez  ONERA & SONDRA Lab, France http://www.jeanphilippeovarlez.com
 Bruno Pelletier  University of Rennes, France http://pelletierb.perso.math.cnrs.fr/
 Xavier Pennec  INRIA, France http://wwwsop.inria.fr/members/Xavier.Pennec/
 Michel Petitjean  Université Paris Diderot, CNRS, France http://petitjeanmichel.free.fr/itoweb.petitjean.html
 Gabriel Peyre  Université Paris Dauphine, CNRS, France http://gpeyre.github.io/
 Giovanni Pistone  Collegio Carlo Alberto, Moncalieri, Italie http://www.giannidiorestino.it/
 Julien Rabin  Université de Caen, France https://sites.google.com/site/rabinjulien/
 Tudor Ratiu  Ecole Polytechnique Federale de Lausanne, Swiss http://cag.epfl.ch/page39504en.html
 Johannes Rauh  Leibniz Universität hannover, Germany http://www2.iag.unihannover.de/~jrauh/index.php
 Olivier Rioul  Telecom ParisTech, France http://perso.telecomparistech.fr/~rioul/
 Said Salem  Université de Bordeaux, France https://www.imsbordeaux.fr/fr/annuaire/4069saidsalem
 Alessandro Sarti  Ecole des hautes études en sciences sociales, France http://cams.ehess.fr/document.php?id=1194
 Gery de Saxcé  Université des Sciences et des Technologies de Lille, France http://www.univlille1.fr/
 Olivier Schwander  Ecole Polytechnique, France http://www.lix.polytechnique.fr/~schwander/en/
 Rodolphe Sepulchre  Cambridge University, Department of Engineering, UK http://wwwcontrol.eng.cam.ac.uk/Main/RodolpheSepulchre
 Hichem Snoussi  Université de Technologie de Troyes, France http://h.snoussi.free.fr/
 Anuj Srivastava  Florida State University, USA http://stat.fsu.edu/~anuj/
 Udo von Toussaint  MaxPlanckInstitut fuer Plasmaphysik, Garching, Germany http://home.rzg.mpg.de/~udt/
 Emmanuel Trelat  UPMC, France https://www.ljll.math.upmc.fr/trelat/
 Alain Trouvé  ENS Cachan, France http://atrouve.perso.math.cnrs.fr/
 Corinne Vachier  Université Paris Est Créteil, France www.upec.fr
 Claude Vallée  Poitiers University, France http://www.univpoitiers.fr/
 Geert Verdoolaege  Ghent University, Belgium http://www.ugent.be/ea/appliedphysics/en/research/fusion/personal_pages.htm/verdoolaege.htm
 JeanPhilippe Vert  Mines ParisTech, France http://cbio.ensmp.fr/~jvert/
 FrançoisXavier Vialard  Ceremade, Paris, France https://www.ceremade.dauphine.fr/~vialard/
 Rui Vigelis  Universidade Federal do ceará, Brazil
 Stephan Weis  Unicamp, Brazil http://www.stephanweis.info
 Laurent Younes  John Hopkins University, USA www.cis.jhu.edu
 Jun Zhang  University of Michigan, Ann Arbor, USA http://www.lsa.umich.edu/psych/junz/
Links
Documents
Opening Session (chaired by Frédéric Barbaresco)
Keynote speach Matilde Marcolli (chaired by Daniel Bennequin)
Random Geometry/Homology (chaired by Laurent Decreusefond/Frédéric Chazal)
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Let m be a random tessellation in R d , d ≥ 1, observed in the window W p = ρ1/d[0, 1] d , ρ > 0, and let f be a geometrical characteristic. We investigate the asymptotic behaviour of the maximum of f(C) over all cells C ∈ m with nucleus W p as ρ goes to infinity.When the normalized maximum converges, we show that its asymptotic distribution depends on the socalled extremal index. Two examples of extremal indices are provided for PoissonVoronoi and PoissonDelaunay tessellations.

A twocolor interacting random balls model for colocalization analysis of proteins Frederic Lavancier, Charles Kervrann
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A model of twotype (or twocolor) interacting random balls is introduced. Each colored random set is a union of random balls and the interaction relies on the volume of the intersection between the two random sets. This model is motivated by the detection and quantification of colocalization between two proteins. Simulation and inference are discussed. Since all individual balls cannot been identified, e.g. a ball may contain another one, standard methods of inference as likelihood or pseudolikelihood are not available and we apply the TakacsFiksel method with a specific choice of test functions.

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Ginibre point process, Poisson point process, Stein’s method, Stochastic geometry, βGinibre point process
The characteristic independence property of Poisson point processes gives an intuitive way to explain why a sequence of point processes becoming less and less repulsive can converge to a Poisson point process. The aim of this paper is to show this convergence for sequences built by superposing, thinning or rescaling determinantal processes. We use Papangelou intensities and Stein’s method to prove this result with a topology based on total variation distance.

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Random polytopes have constituted some of the central objects of stochastic geometry for more than 150 years. They are in general generated as convex hulls of a random set of points in the Euclidean space. The study of such models requires the use of ingredients coming from both convex geometry and probability theory. In the last decades, the study has been focused on their asymptotic properties and in particular expectation and variance estimates. In several joint works with Tomasz Schreiber and J. E. Yukich, we have investigated the scaling limit of several models (uniform model in the unitball, uniform model in a smooth convex body, Gaussian model) and have deduced from it limiting variances for several geometric characteristics including the number of kdimensional faces and the volume. In this paper, we survey the most recent advances on these questions and we emphasize the particular cases of random polytopes in the unitball and Gaussian polytopes.

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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 KullbackLeibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.

Computational Information Geometry (chaired by Frank Nielsen, Paul Marriott)
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We introduce a new approach to goodnessoffit testing in the high dimensional, sparse extended multinomial context. The paper takes a computational information geometric approach, extending classical higher order asymptotic theory. We show why the Wald – equivalently, the Pearson X2 and score statistics – are unworkable in this context, but that the deviance has a simple, accurate and tractable sampling distribution even for moderate sample sizes. Issues of uniformity of asymptotic approximations across model space are discussed. A variety of important applications and extensions are noted.

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Computational information geometry, Computing boundaries, Embedded manifolds, Local mixture models, Polytopes, Ruled and developable surfaces
Local mixture models give an inferentially tractable but still flexible alternative to general mixture models. Their parameter space naturally includes boundaries; near these the behaviour of the likelihood is not standard. This paper shows how convex and differential geometries help in characterising these boundaries. In particular the geometry of polytopes, ruled and developable surfaces is exploited to develop efficient inferential algorithms.

Approximating Covering and Minimum Enclosing Balls in Hyperbolic Geometry Frank Nielsen, Gaëtan Hadjeres
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We generalize the O(dnϵ2)time (1 + ε)approximation algorithm for the smallest enclosing Euclidean ball [2,10] to point sets in hyperbolic geometry of arbitrary dimension. We guarantee a O(1/ϵ2) convergence time by using a closedform formula to compute the geodesic αmidpoint between any two points. Those results allow us to apply the hyperbolic kcenter clustering for statistical locationscale families or for multivariate spherical normal distributions by using their Fisher information matrix as the underlying Riemannian hyperbolic metric.

From Euclidean to Riemannian Means Information Geometry for SSVEP Classification Emmanuel Kalunga, Sylvain Chevallier, Quentin Barthélemy, Karim Djouani, Yskandar Hamam, Eric Monacelli
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Brain Computer, Information geometry, Interfaces, Riemannian means, Steady State, Visually Evoked Potentials
Brain Computer Interfaces (BCI) based on electroencephalography (EEG) rely on multichannel brain signal processing. Most of the stateoftheart approaches deal with covariance matrices, and indeed Riemannian geometry has provided a substantial framework for developing new algorithms. Most notably, a straightforward algorithm such as Minimum Distance to Mean yields competitive results when applied with a Riemannian distance. This applicative contribution aims at assessing the impact of several distances on real EEG dataset, as the invariances embedded in those distances have an influence on the classification accuracy. Euclidean and Riemannian distances and means are compared both in term of quality of results and of computational load.

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We introduce a class of paths or oneparameter models connecting arbitrary two probability density functions (pdf’s). The class is derived by employing the KolmogorovNagumo average between the two pdf’s. There is a variety of such path connectedness on the space of pdf’s since the KolmogorovNagumo average is applicable for any convex and strictly increasing function. The information geometric insight is provided for understanding probabilistic properties for statistical methods associated with the path connectedness. The oneparameter model is extended to a multidimensional model, on which the statistical inference is characterized by sufficient statistics.

Bayesian and Information Geometry for Inverse Problems (chaired by Ali MohammadDjafari, Olivier Swander)
Stochastic PDE projection on manifolds AssumedDensity and Galerkin Filters Damiano Brigo, John Armstrong
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We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L2 direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finitedimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (KushnerStratonovich) or a specific unnormalized (Zakai) density of the optimal filter.

Variational Bayesian Approximation method for Classification and Clustering with a mixture of Studen Ali MohammadDjafari
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Clustering, classification and Pattern Recognition in a set of data are between the most important tasks in statistical researches and in many applications. In this paper, we propose to use a mixture of Studentt distribution model for the data via a hierarchical graphical model and the Bayesian framework to do these tasks. The main advantages of this model is that the model accounts for the uncertainties of variances and covariances and we can use the Variational Bayesian Approximation (VBA) methods to obtain fast algorithms to be able to handle large data sets.

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The textile plot proposed by Kumasaka and Shibata (2008) is a method for data visualization. The method transforms a data matrix in order to draw a parallel coordinate plot. In this paper, we investigate a set of matrices induced by the textile plot, which we call the textile set, from a geometrical viewpoint. It is shown that the textile set is written as the union of two differentiable manifolds if data matrices are restricted to be fullrank.

A generalization of independence and multivariate Student's tdistributions Monta Sakamoto, Hiroshi Matsuzoe
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In anomalous statistical physics, deformed algebraic structures are important objects. Heavily tailed probability distributions, such as Student’s tdistributions, 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 tdistribution is studied in this paper. Even if two random variables which follow to univariate Student’s tdistributions are independent, the joint probability distribution of these two distributions is not a bivariate Student’s tdistribution. It is shown that a bivariate Student’s tdistribution is obtained from two univariate Student’s tdistributions under qdeformed independence.

Hessian Information Geometry (chaired by ShunIchi Amari, Michel Boyom)
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We define a metric and a family of αconnections in statistical manifolds, based on ϕdivergence, which emerges in the framework of ϕfamilies of probability distributions. This metric and αconnections generalize the Fisher information metric and Amari’s αconnections. We also investigate the parallel transport associated with the αconnection for α = 1.

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Affine connection, Curvature tensor, Laplacian Bochner’s technique, Ricci tensor, Sectional curvature
Curvature properties for statistical structures are studied. The study deals with the curvature tensor of statistical connections and their duals as well as the Ricci tensor of the connections, Laplacians and the curvature operator. Two concepts of sectional curvature are introduced. The meaning of the notions is illustrated by presenting few exemplary theorems.

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We show that Hessian manifolds of dimensions 4 and above must have vanishing Pontryagin forms. This gives a topological obstruction to the existence of Hessian metrics. We find an additional explicit curvature identity for Hessian 4manifolds. By contrast, we show that all analytic Riemannian 2manifolds are Hessian.

Topological forms and Information (chaired by Daniel Bennequin, Pierre Baudot)
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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.

Finite polylogarithms, their multiple analogues and the Shannon entropy Philippe ElbazVincent, Herbert Gangl
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We show that the entropy function–and hence the finite 1logarithm–behaves a lot like certain derivations. We recall its cohomological interpretation as a 2cocycle and also deduce 2ncocycles for any n. Finally, we give some identities for finite multiple polylogarithms together with number theoretic applications.

Heights of toric varieties, integration over polytopes and entropy José Ignacio Burgos Gil, Patrice Philippon, Martin Sombra
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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.

Characterization and Estimation of the Variations of a Random Convex Set by it's Mean $n$Variogram : Application to the Boolean Model Saïd Rahmani, JeanCharles Pinoli, Johan Debayle
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Boolean model, Geometric covariogram, Mixed volumes, n points set probability, Particle size distribution, Random set Shape variations
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 nvariogram γ(n)Ξ0:(u1⋯un)↦E[νd(Ξ0∩(Ξ0−u1)⋯∩(Ξ0−un))] of a random convex set Ξ0 on ℝ d reveals information on the n th order structure of Ξ0. Especially we will show that considering the mean nvariograms 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 germgrain model is stable by convex dilatations, furthermore the mean nvariogram of the primary grain is estimable in several type of stationary germgrain models by the so called npoints 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)).

Short course (chaired by Roger Balian)
Keynote speach Marc Arnaudon (chaired by Frank Nielsen)
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We will prove a EulerPoincaré 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 NavierStokes equation on a bounded domain and the CamassaHolm equation.

Information Geometry Optimization (chaired by Giovanni Pistone, Yann Ollivier)
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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 logloss scoring rule, this results in infinite loss the first time “heads” appears.)

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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 nonflat 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.

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The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsionfree 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 nondegenerate twoform 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 Ktranslations, various transformations that generalize traditional projective and dualprojective transformations, and study their commutativity with Lperturbation and hconjugation 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 conformalprojective transformation.

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This paper address the problem of online learning finite statistical mixtures of exponential families. A short review of the ExpectationMaximization (EM) algorithm and its online extensions is done. From these extensions and the description of the kMaximum Likelihood Estimator (kMLE), 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.

Secondorder Optimization over the Multivariate Gaussian Distribution Luigi Malagò, Giovanni Pistone
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We discuss the optimization of the stochastic relaxation of a realvalued 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 secondorder geometry of the exponential family, with applications to the multivariate Gaussian distributions, to generalize secondorder 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 secondorder methods, such as the Newton method.

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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 nonEuclidean 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 nonEuclidean 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 loglikelihood loss applied to parameter estimation in exponential families asymptotically achieves the classical CramérRao lower bound.

Geometry of Time Series and Linear Dynamical systems (chaired by Bijan Afsari, Arshia Cont)
TSGNPR Clustering Random Walk Time Series Gautier Marti, Frank Nielsen, Philippe Very, Philippe Donnat
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We present in this paper a novel nonparametric approach useful for clustering independent identically distributed stochastic processes. We introduce a preprocessing step consisting in mapping multivariate independent and identically distributed samples from random variables to a generic nonparametric 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 .

New model search for nonlinear recursive models, regressions and autoregressions Wolfgang Stummer, AnnaLena Kißlinger
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3D score surface, AR SARIMA NARX, Autorecursions, Model selection, Nonlinear regression, Scaled Bregman distances
Scaled Bregman distances SBD have turned out to be useful tools for simultaneous estimation and goodnessoffittesting 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 informationgeometric 3D computergraphical selection procedure. Some samplesize asymptotics is given as well.

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In the context of sensor networks, gossip algorithms are a popular, well established technique, for achieving consensus when sensor data are encoded in linear spaces. Gossip algorithms also have several extensions to non linear data spaces. Most of these extensions deal with Riemannian manifolds and use Riemannian gradient descent. This paper, instead, studies gossip in a broader CAT(k) metric setting, encompassing, but not restricted to, several interesting cases of Riemannian manifolds. As it turns out, convergence can be guaranteed as soon as the data lie in a small enough ball of a mere CAT(k) metric space. We also study convergence speed in this setting and establish linear rates of convergence.

Optimal Transport (chaired by JeanFrançois Marcotorchino, Alfred Galichon)
The nonlinear BernsteinSchrodinger equation in Economics Alfred Galichon, Scott Kominers, Simon Weber
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In this paper we relate the Equilibrium Assignment Problem (EAP), which is underlying in several economics models, to a system of nonlinear equations that we call the “nonlinear BernsteinSchrödinger system”, which is wellknown in the linear case, but whose nonlinear extension does not seem to have been studied. We apply this connection to derive an existence result for the EAP, and an efficient computational method.

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Displacement interpolations, Entropic interpolations, LottSturmVillani theory, Lower bounded curvature of metric spaces, Optimal transport, Schrödinger problem
This note presents a short review of the Schrödinger problem and of the first steps that might lead to interesting consequences in terms of geometry. We stress the analogies between this entropy minimization problem and the renowned optimal transport problem, in search for a theory of lower bounded curvature for metric spaces, including discrete graphs.

Optimal Transport, Independance versus Indetermination duality, impact on a new Copula Design Benoit Huyot, Yves Mabiala, JeanFrançois Marcotorchino
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Condorcet and relational analysis, Copula theory, Indetermination and independance structures, MKP problem, Optimal transport
This article leans on some previous results already presented in [10], based on the Fréchet’s works,Wilson’s entropy and Minimal Trade models in connectionwith theMKPtransportation problem (MKP, stands for MongeKantorovich Problem). Using the duality between “independance” and “indetermination” structures, shown in this former paper, we are in a position to derive a novel approach to design a copula, suitable and efficient for anomaly detection in IT systems analysis.

Information Geometry in Image Analysis (chaired by Yannick Berthoumieu, Geert Verdoolaege)
Texture classification using Rao's distance on the space of covariance matrices Salem Said, Lionel Bombrun, Yannick Berthoumieu
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EM algorithm, Information geometry, Mixture estimation, Riemannian centre of mass, Texture classification
The current paper introduces new prior distributions on the zeromean multivariate Gaussian model, with the aim of applying them to the classification of covariance matrices populations. These new prior distributions are entirely based on the Riemannian geometry of the multivariate Gaussian model. More precisely, the proposed Riemannian Gaussian distribution has two parameters, the centre of mass ˉY and the dispersion parameter σ. Its density with respect to Riemannian volume is proportional to exp(−d2(Y;ˉY)), where d2(Y;ˉY) is the square of Rao’s Riemannian distance. We derive its maximum likelihood estimators and propose an experiment on the VisTex database for the classification of texture images.

Color Texture Discrimination using the Principal Geodesic Distance on a Multivariate Generalized Gau Geert Verdoolaege, Aqsa Shabbir
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We present a new texture discrimination method for textured color images in the wavelet domain. In each wavelet subband, the correlation between the color bands is modeled by a multivariate generalized Gaussian distribution with fixed shape parameter (Gaussian, Laplacian). On the corresponding Riemannian manifold, the shape of texture clusters is characterized by means of principal geodesic analysis, specifically by the principal geodesic along which the cluster exhibits its largest variance. Then, the similarity of a texture to a class is defined in terms of the Rao geodesic distance on the manifold from the texture’s distribution to its projection on the principal geodesic of that class. This similarity measure is used in a classification scheme, referred to as principal geodesic classification (PGC). It is shown to perform significantly better than several other classifiers.

Bagofcomponents an online algorithm for batch learning of mixture models Olivier Schwander, Frank Nielsen
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Practical estimation of mixture models may be problematic when a large number of observations are involved: for such cases, online versions of ExpectationMaximization may be preferred, avoiding the need to store all the observations before running the algorithms. We introduce a new online method wellsuited when both the number of observations is large and lots of mixture models need to be learned from different sets of points. Inspired by dictionary methods, our algorithm begins with a training step which is used to build a dictionary of components. The next step, which can be done online, amounts to populating the weights of the components given each arriving observation. The usage of the dictionary of components shows all its interest when lots of mixtures need to be learned using the same dictionary in order to maximize the return on investment of the training step. We evaluate the proposed method on an artificial dataset built from random Gaussian mixture models.

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Stochastic watershed is an image segmentation technique based on mathematical morphology which produces a probability density function of image contours. Estimated probabilities depend mainly on local distances between pixels. This paper introduces a variant of stochastic watershed where the probabilities of contours are computed from a gaussian model of image regions. In this framework, the basic ingredient is the distance between pairs of regions, hence a distance between normal distributions. Hence several alternatives of statistical distances for normal distributions are compared, namely Bhattacharyya distance, Hellinger metric distance and Wasserstein metric distance.

Quantization of hyperspectral image manifold using probabilistic distances Gianni Franchi, Jesús Angulo
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Hyperspectral images, Information geometry, Mathematical morphology, Probabilistic distances, Quantization
A technique of spatialspectral 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 αconnected 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.

Optimal Transport and applications in Imagery/Statistics (chaired by Bertrand Maury, Jérémie Bigot)
Nonconvex relaxation of optimal transport for color transfer between images Julien Rabin, Nicolas Papadakis
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Optimal transport (OT) is a major statistical tool to measure similarity between features or to match and average features. However, OT requires some relaxation and regularization to be robust to outliers. With relaxed methods, as one feature can be matched to several ones, important interpolations between different features arise. This is not an issue for comparison purposes, but it involves strong and unwanted smoothing for transfer applications. We thus introduce a new regularized method based on a nonconvex formulation that minimizes transport dispersion by enforcing the onetoone matching of features. The interest of the approach is demonstrated for color transfer purposes.

Generalized Pareto Distributions, Image Statistics and Autofocusing in Automated Microscopy Reiner Lenz
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We introduce the generalized Pareto distributions as a statistical model to describe thresholded edgemagnitude image filter results. Compared to the more commonWeibull or generalized extreme value distributions these distributions have at least two important advantages, the usage of the high threshold value assures that only the most important edge points enter the statistical analysis and the estimation is computationally more efficient since a much smaller number of data points have to be processed. The generalized Pareto distributions with a common threshold zero form a twodimensional Riemann manifold with the metric given by the Fisher information matrix. We compute the Fisher matrix for shape parameters greater than 0.5 and show that the determinant of its inverse is a product of a polynomial in the shape parameter and the squared scale parameter. We apply this result by using the determinant as a sharpness function in an autofocus algorithm. We test the method on a large database of microscopy images with given ground truth focus results. We found that for a vast majority of the focus sequences the results are in the correct focal range. Cases where the algorithm fails are specimen with too few objects and sequences where contributions from different layers result in a multimodal sharpness curve. Using the geometry of the manifold of generalized Pareto distributions more efficient autofocus algorithms can be constructed but these optimizations are not included here.

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We study barycenters in the Wasserstein space Pp(E) of a locally compact geodesic space (E, d). In this framework, we define the barycenter of a measure ℙ on Pp(E) as its Fréchet mean. The paper establishes its existence and states consistency with respect to ℙ. We thus extends previous results on ℝ d , with conditions on ℙ or on the sequence converging to ℙ for consistency.

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Univariate Lmoments are expressed as projections of the quantile function onto an orthogonal basis of univariate polynomials. We present multivariate versions of Lmoments 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 Lmoments. The properties of estimated Lmoments are illustrated for heavytailed distributions.

Probability Density Estimation (chaired by Jesús Angulo, S. Said)
Probability density estimation on the hyperbolic space applied to radar processing Emmanuel Chevallier, Jesús Angulo, Frédéric Barbaresco
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The two main techniques of probability density estimation on symmetric spaces are reviewed in the hyperbolic case. For computational reasons we chose to focus on the kernel density estimation and we provide the expression of Pelletier estimator on hyperbolic space. The method is applied to density estimation of reflection coefficients derived from radar observations.

Histograms of images valued in the manifold of colours endowed with perceptual metrics Emmanuel Chevallier, Jesús Angulo, Ivar Farup
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We address here the problem of perceptual colour histograms. The Riemannian structure of perceptual distances is measured through standards sets of ellipses, such as Macadam ellipses. We propose an approach based on local Euclidean approximations that enables to take into account the Riemannian structure of perceptual distances, without introducing computational complexity during the construction of the histogram.

Entropy minimizing curves with application to automated flight path design Stephane Puechmorel, Florence Nicol
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Air traffic management (ATM) aims at providing companies with a safe and ideally optimal aircraft trajectory planning. Air traffic controllers act on flight paths in such a way that no pair of aircraft come closer than the regulatory separation norm. With the increase of traffic, it is expected that the system will reach its limits in a near future: a paradigm change in ATM is planned with the introduction of trajectory based operations. This paper investigate a mean of producing realistic air routes from the output of an automated trajectory design tool. For that purpose, an entropy associated with a system of curves is defined and a mean of iteratively minimizing it is presented. The network produced is suitable for use in a semiautomated ATM system with human in the loop.

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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 Hölderclass assumptions on the densities. A main tool used in proving the convergence rate is the HelgasonFourier 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 2dimensional hyperboloid.

Keynote speach Tudor Ratiu (chaired by Xavier Pennec)
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The goal of these lectures is to show the influence of symmetry in various aspects of theoretical mechanics. Canonical actions of Lie groups on Poisson manifolds often give rise to conservation laws, encoded in modern language by the concept of momentum maps. Reduction methods lead to a deeper understanding of the dynamics of mechanical systems. Basic results in singular Hamiltonian reduction will be presented. The Lagrangian version of reduction and its associated variational principles will also be discussed. The understanding of symmetric bifurcation phenomena in for Hamiltonian systems are based on these reduction techniques. Time permitting, discrete versions of these geometric methods will also be discussed in the context of examples from elasticity.

Dimension reduction on Riemannian manifolds (chaired by Xavier Pennec, Alain Trouvé)
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This paper presents derivations of evolution equations for the family of paths that in the Diffusion PCA framework are used for approximating data likelihood. The paths that are formally interpreted as most probable paths generalize geodesics in extremizing an energy functional on the space of differentiable curves on a manifold with connection. We discuss how the paths arise as projections of geodesics for a (non bracketgenerating) subRiemannian metric on the frame bundle. Evolution equations in coordinates for both metric and cometric formulations of the subRiemannian geometry are derived. We furthermore show how rankdeficient metrics can be mixed with an underlying Riemannian metric, and we use the construction to show how the evolution equations can be implemented on finite dimensional LDDMM landmark manifolds.

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This paper addresses the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. Current methods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) minimize the distance to a “Geodesic subspace”. This allows to build sequences of nested subspaces which are consistent with a forward component analysis approach. However, these methods cannot be adapted to a backward analysis and they are not symmetric in the parametrization of the subspaces. We propose in this paper a new and more general type of family of subspaces in manifolds: barycentric subspaces are implicitly defined as the locus of points which are weighted means of k + 1 reference points. Depending on the generalization of the mean that we use, we obtain the Fréchet/Karcher barycentric subspaces (FBS/KBS) or the affine span (with exponential barycenter). This definition restores the full symmetry between all parameters of the subspaces, contrarily to the geodesic subspaces which intrinsically privilege one point. We show that this definition defines locally a submanifold of dimension k and that it generalizes in some sense geodesic subspaces. Like PGA, barycentric subspaces allow the construction of a forward nested sequence of subspaces which contains the Fréchet mean. However, the definition also allows the construction of backward nested sequence which may not contain the mean. As this definition relies on points and do not explicitly refer to tangent vectors, it can be extended to non Riemannian geodesic spaces. For instance, principal subspaces may naturally span over several strata in stratified spaces, which is not the case with more classical generalizations of PCA.

Dimension Reduction on Polyspheres with Application to Skeletal Representations Benjamin Eltzner, Sungkyu Jung, Stephan Huckemann
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We present a novel method that adaptively deforms a polysphere (a product of spheres) into a single high dimensional sphere which then allows for principal nested spheres (PNS) analysis. Applying our method to skeletal representations of simulated bodies as well as of data from real human hippocampi yields promising results in view of dimension reduction. Specifically in comparison to composite PNS (CPNS), our method of principal nested deformed spheres (PNDS) captures essential modes of variation by lower dimensional representations.

Affineinvariant Riemannian Distance Between Infinitedimensional Covariance Operators Minh Ha Quang
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This paper studies the affineinvariant Riemannian distance on the RiemannHilbert manifold of positive definite operators on a separable Hilbert space. This is the generalization of the Riemannian manifold of symmetric, positive definite matrices to the infinitedimensional setting. In particular, in the case of covariance operators in a Reproducing Kernel Hilbert Space (RKHS), we provide a closed form solution, expressed via the corresponding Gram matrices.

A subRiemannian modular approach for diffeomorphic deformations Barbara Gris, Stanley Durrleman, Alain Trouvé
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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 subRiemannian 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.

Optimization on Manifold (chaired by PierreAntoine Absil, Rodolphe Sepulchre)
Riemannian trust regions with finitedifference Hessian approximations are globally convergent Nicolas Boumal
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The Riemannian trustregion algorithm (RTR) is designed to optimize differentiable cost functions on Riemannian manifolds. It proceeds by iteratively optimizing local models of the cost function. When these models are exact up to second order, RTR boasts a quadratic convergence rate to critical points. In practice, building such models requires computing the Riemannian Hessian, which may be challenging. A simple idea to alleviate this difficulty is to approximate the Hessian using finite differences of the gradient. Unfortunately, this is a nonlinear approximation, which breaks the known convergence results for RTR. We propose RTRFD: a modification of RTR which retains global convergence when the Hessian is approximated using finite differences. Importantly, RTRFD reduces gracefully to RTR if a linear approximation is used. This algorithm is available in the Manopt toolbox.

BlockJacobi methods with Newtonsteps and nonunitary joint matrix diagonalization Martin Kleinsteuber, Hao Shen
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In this work, we consider blockJacobi methods with Newton steps in each subspace search and prove their local quadratic convergence to a local minimum with nondegenerate Hessian under some orthogonality assumptions on the search directions. Moreover, such a method is exemplified for nonunitary joint matrix diagonalization, where we present a blockJacobitype method on the oblique manifold with guaranteed local quadratic convergence.

Weakly Correlated Sparse Components with Nearly Orthonormal Loadings Matthieu Genicot, Wen Huang, Nickolay Trendafilov
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Dantzig selector, LASSO, Optimization on matrix manifolds, Orthonormal and oblique component loadings matrices
There is already a great number of highly efficient methods producing components with sparse loadings which significantly facilitates the interpretation of principal component analysis (PCA). However, they produce either only orthonormal loadings, or only uncorrelated components, or, most frequently, neither of them. To overcome this weakness, we introduce a new approach to define sparse PCA similar to the Dantzig selector idea already employed for regression problems. In contrast to the existing methods, the new approach makes it possible to achieve simultaneously nearly uncorrelated sparse components with nearly orthonormal loadings. The performance of the new method is illustrated on real data sets. It is demonstrated that the new method outperforms one of the most popular available methods for sparse PCA in terms of preservation of principal components properties.

Fitting Smooth Paths on Riemannian Manifolds  Endometrial Surface Antoine Arnould, PierreYves Gousenbourger, Chafik Samir, PierreAntoine Absil, Michel Canis
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Bézier functions, Endometrial surface reconstruction, MRIbased navigation, Optimization on manifolds, Path fitting on Riemannian manifolds
We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using C1 piecewiseBézier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group SO(3) and the special Euclidean group SE(3) for preoperative MRIbased navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.

Shape Space & Diffeomorphic mappings (chaired by Stanley Durrleman, Stéphanie Allassonnière)
Spherical parameterization for genus zero surfaces using LaplaceBeltrami eigenfunctions Julien Lefèvre, Guillaume Auzias
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In this work, we propose a fast and simple approach to obtain a spherical parameterization of a certain class of closed surfaces without holes. Our approach relies on empirical findings that can be mathematically investigated, to a certain extent, by using LaplaceBeltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first nontrivial eigenfunctions of the LaplaceBeltrami Operator. Our approach requires a topological condition on those eigenfunctions, whose nodal domains must be 2. We show the efficiency of the approach through numerical experiments performed on cortical surface meshes.

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Usual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc? How can we describe the interaction between statistics and geometry? The structure of Quotient space in particular is widely used to model data, for example every time one deals with shape data. These can be shapes of constellations in Astronomy, shapes of human organs in Computational Anatomy, shapes of skulls in Palaeontology, etc. Given this broad field of applications, statistics on shapes and more generally on observations belonging to quotient spaces have been studied since the 1980’s. However, most theories model the variability in the shapes but do not take into account the noise on the observations themselves. In this paper, we show that statistics on quotient spaces are biased and even inconsistent when one takes into account the noise. In particular, some algorithms of template estimation in Computational Anatomy are biased and inconsistent. Our development thus gives a first theoretical geometric explanation of an experimentally observed phenomenon. A biased estimator is not necessarily a problem. In statistics, it is a general rule of thumb that a bias can be neglected for example when it represents less than 0.25 of the variance of the estimator. We can also think about neglecting the bias when it is low compared to the signal we estimate. In view of the applications, we thus characterize geometrically the situations when the bias can be neglected with respect to the situations when it must be corrected.

Uniqueness of the FisherRao Metric on the Space of Smooth Densities Martin Bauer, Martins Bruveris, Peter Michor
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We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L2 direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finitedimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (KushnerStratonovich) or a specific unnormalized (Zakai) density of the optimal filter.

Reparameterization invariant metric on the space of curves Alice Le Brigant, Marc Arnaudon, Frédéric Barbaresco
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This paper focuses on the study of open curves in a manifold M, and its aim is to define a reparameterization invariant distance on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions =Imm([0,1],M) by pullback of a metric on the tangent bundle T derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on T induces a firstorder Sobolev metric on with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the image curves by the SRVF parallel transported to a same vector space, with an added curvature term. This provides a generalized theoretical SRV framework for curves lying in a general manifold M.

Invariant geometric structures on statistical models Lorenz Schwachhöfer, Nihat Ay, Jürgen Jost, Hong Van Le
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We review the notion of parametrized measure models and tensor fields on them, which encompasses all statistical models considered by Chentsov [6], Amari [3] and PistoneSempi [10]. We give a complete description of ntensor fields that are invariant under sufficient statistics. In the cases n = 2 and n = 3, the only such tensors are the Fisher metric and the AmariChentsov tensor. While this has been shown by Chentsov [7] and Campbell [5] in the case of finite measure spaces, our approach allows to generalize these results to the cases of infinite sample spaces and arbitrary n. Furthermore, we give a generalisation of the monotonicity theorem and discuss its consequences.

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This paper aims to define a unified setting for shape registration and LDDMM methods for shape analysis. This setting turns out to be subRiemannian, 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.

Lie Groups: Novel Statistical and Computational Frontiers (chaired by Jérémie Jakubowicz, Bijan Afsari)
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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 biinvariant 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.

Image processing in the semidiscrete group of rototranslations Dario Prandi, Ugo Boscain, JeanPaul Gauthier
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It is wellknown, since [12], that cells in the primary visual cortex V1 do much more than merely signaling position in the visual field: most cortical cells signal the local orientation of a contrast edge or bar . they are tuned to a particular local orientation. This orientation tuning has been given a mathematical interpretation in a subRiemannian model by Petitot, Citti, and Sarti [6,14]. According to this model, the primary visual cortex V1 lifts greyscale images, given as functions f : ℝ2 → [0, 1], to functions Lf defined on the projectivized tangent bundle of the plane PTℝ2 = ℝ2 ×ℙ1. Recently, in [1], the authors presented a promising semidiscrete variant of this model where the Euclidean group of rototranslations SE(2), which is the double covering of PTℝ2, is replaced by SE(2,N), the group of translations and discrete rotations. In particular, in [15], an implementation of this model allowed for stateoftheart image inpaintings. In this work, we review the inpainting results and introduce an application of the semidiscrete model to image recognition. We remark that both these applications deeply exploit the Moore structure of SE(2,N) that guarantees that its unitary representations behaves similarly to those of a compact group. This allows for nice properties of the Fourier transform on SE(2,N) exploiting which one obtains numerical advantages.

Universal, nonasymptotic confidence sets for circular means Thomas Hotz, Florian Kelma, Johannes Wieditz
Deblurring and Recovering Conformational States in 3D Single Particle Electron Bijan Afsari, Gregory S. Chirikjian
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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 multiunit 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 spatialmotional convolution.We discuss the illconditioned 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 multiunit 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.

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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 nonflat 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.

Lie Groups and Geometric Mechanics/Thermodynamics (chaired by Frédéric Barbaresco, Géry de Saxcé)
Poincaré's equations for Cosserat shells  application to locomotion of cephalopods Frederic Boyer, Federico Renda
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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.

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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.

Symplectic Structure of Information Geometry: Fisher Metric and EulerPoincaré Equation of Souriau Lie Group Thermodynamics Frédéric Barbaresco
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CartanPoincaré Integral, EulerPoincaré Equation, Fisher Metric, Gauge Theory, Geometric Mechanics, Gibbs Equilibrium, Information geometry, Invariant Lie Group, Maximum Entropy, Momentum Map, Symplectic Geometry, Thermodynamics
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 coadjoint 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 EulerPoincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant PoincaréCartanSouriau integral. Finally, we conclude on Balian Gauge theory of Thermodynamics compatible with Souriau’s Model.

Pontryagin calculus in Riemannian geometry Francois Dubois, Danielle Fortune, Juan Antonio Rojas Quintero
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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.

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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 noslip and notwist, 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.

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Collective behavior, Liepoisson reduction, Nonholonomic integrator, Subriemannian geodesics, Subriemannian geometry
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 driftfree particle dynamics, the coupling terms in the cost functionals lead to a novel class of problems in subriemannian geometry of product Lie groups.

Divergence Geometry (chaired by Michel Broniatowski, Imre Csiszár)
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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 twocomponent gaussian mixture. Simulations on Gaussian andWeibull mixtures are performed to compare the results with the MLE.

Extension of information geometry to nonstatistical systems some examples Jan Naudts, Ben Anthonis, Michel Broniatowski
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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.

An Information Geometry Problem in Mathematical Finance Imre Csiszár, Thomas Breuer, Michel Broniatowski
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Almost worst case densities, Bregman distance, Convex integral functional, fdivergence Idivergence, Payoff function, Pythagorean identity, Risk measure, Worst case density
Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family Γ of plausible risk factor distributions IP. We consider Γ determined by a bound on a convex integral functional of the density of IP, thus Γ may be an Idivergence (relative entropy) ball or some other fdivergence 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∈Γ), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When Γ is an fdivergence 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 fdivergence balls.

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We introduce what we will call multivariate divergences between K, K ≥ 1, signed finite measures (Q1, . . . , Q K ) and a given reference probability measure P on a σfield (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 multigroup classification.

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We derive independence tests by means of dependence measures thresholding in a semiparametric context. Precisely, estimates of mutual information associated to ϕdivergences are derived through the dual representations of ϕdivergences. 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.

Invited speaker CharlesMichel Marle (chaired by Frédéric Barbaresco)
Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems CharlesMichel Marle
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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.

Authors
Frédéric Barbaresco
Symplectic Structure of Information Geometry: Fisher Metric and EulerPoincaré Equation of Souriau Lie Group Thermodynamics Frédéric Barbaresco
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CartanPoincaré Integral, EulerPoincaré Equation, Fisher Metric, Gauge Theory, Geometric Mechanics, Gibbs Equilibrium, Information geometry, Invariant Lie Group, Maximum Entropy, Momentum Map, Symplectic Geometry, Thermodynamics
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 coadjoint 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 EulerPoincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant PoincaréCartanSouriau integral. Finally, we conclude on Balian Gauge theory of Thermodynamics compatible with Souriau’s Model.

Frederic Lavancier, Charles Kervrann
A twocolor interacting random balls model for colocalization analysis of proteins Frederic Lavancier, Charles Kervrann
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A model of twotype (or twocolor) interacting random balls is introduced. Each colored random set is a union of random balls and the interaction relies on the volume of the intersection between the two random sets. This model is motivated by the detection and quantification of colocalization between two proteins. Simulation and inference are discussed. Since all individual balls cannot been identified, e.g. a ball may contain another one, standard methods of inference as likelihood or pseudolikelihood are not available and we apply the TakacsFiksel method with a specific choice of test functions.

Roman Belavkin
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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 KullbackLeibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.

Pierre Calka
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Random polytopes have constituted some of the central objects of stochastic geometry for more than 150 years. They are in general generated as convex hulls of a random set of points in the Euclidean space. The study of such models requires the use of ingredients coming from both convex geometry and probability theory. In the last decades, the study has been focused on their asymptotic properties and in particular expectation and variance estimates. In several joint works with Tomasz Schreiber and J. E. Yukich, we have investigated the scaling limit of several models (uniform model in the unitball, uniform model in a smooth convex body, Gaussian model) and have deduced from it limiting variances for several geometric characteristics including the number of kdimensional faces and the volume. In this paper, we survey the most recent advances on these questions and we emphasize the particular cases of random polytopes in the unitball and Gaussian polytopes.

Laurent Decreusefond, Aurélien Vasseur
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Ginibre point process, Poisson point process, Stein’s method, Stochastic geometry, βGinibre point process
The characteristic independence property of Poisson point processes gives an intuitive way to explain why a sequence of point processes becoming less and less repulsive can converge to a Poisson point process. The aim of this paper is to show this convergence for sequences built by superposing, thinning or rescaling determinantal processes. We use Papangelou intensities and Stein’s method to prove this result with a topology based on total variation distance.

Nicolas Chenavier
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Let m be a random tessellation in R d , d ≥ 1, observed in the window W p = ρ1/d[0, 1] d , ρ > 0, and let f be a geometrical characteristic. We investigate the asymptotic behaviour of the maximum of f(C) over all cells C ∈ m with nucleus W p as ρ goes to infinity.When the normalized maximum converges, we show that its asymptotic distribution depends on the socalled extremal index. Two examples of extremal indices are provided for PoissonVoronoi and PoissonDelaunay tessellations.

Frank Nielsen, Gaëtan Hadjeres
Approximating Covering and Minimum Enclosing Balls in Hyperbolic Geometry Frank Nielsen, Gaëtan Hadjeres
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We generalize the O(dnϵ2)time (1 + ε)approximation algorithm for the smallest enclosing Euclidean ball [2,10] to point sets in hyperbolic geometry of arbitrary dimension. We guarantee a O(1/ϵ2) convergence time by using a closedform formula to compute the geodesic αmidpoint between any two points. Those results allow us to apply the hyperbolic kcenter clustering for statistical locationscale families or for multivariate spherical normal distributions by using their Fisher information matrix as the underlying Riemannian hyperbolic metric.

Vahed Maroufy, Paul Marriott
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Computational information geometry, Computing boundaries, Embedded manifolds, Local mixture models, Polytopes, Ruled and developable surfaces
Local mixture models give an inferentially tractable but still flexible alternative to general mixture models. Their parameter space naturally includes boundaries; near these the behaviour of the likelihood is not standard. This paper shows how convex and differential geometries help in characterising these boundaries. In particular the geometry of polytopes, ruled and developable surfaces is exploited to develop efficient inferential algorithms.

Emmanuel Kalunga, Sylvain Chevallier, Quentin Barthélemy, Karim Djouani, Yskandar Hamam, Eric Monacelli
From Euclidean to Riemannian Means Information Geometry for SSVEP Classification Emmanuel Kalunga, Sylvain Chevallier, Quentin Barthélemy, Karim Djouani, Yskandar Hamam, Eric Monacelli
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Brain Computer, Information geometry, Interfaces, Riemannian means, Steady State, Visually Evoked Potentials
Brain Computer Interfaces (BCI) based on electroencephalography (EEG) rely on multichannel brain signal processing. Most of the stateoftheart approaches deal with covariance matrices, and indeed Riemannian geometry has provided a substantial framework for developing new algorithms. Most notably, a straightforward algorithm such as Minimum Distance to Mean yields competitive results when applied with a Riemannian distance. This applicative contribution aims at assessing the impact of several distances on real EEG dataset, as the invariances embedded in those distances have an influence on the classification accuracy. Euclidean and Riemannian distances and means are compared both in term of quality of results and of computational load.

Shinto Eguchi, Osamu Komori
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We introduce a class of paths or oneparameter models connecting arbitrary two probability density functions (pdf’s). The class is derived by employing the KolmogorovNagumo average between the two pdf’s. There is a variety of such path connectedness on the space of pdf’s since the KolmogorovNagumo average is applicable for any convex and strictly increasing function. The information geometric insight is provided for understanding probabilistic properties for statistical methods associated with the path connectedness. The oneparameter model is extended to a multidimensional model, on which the statistical inference is characterized by sufficient statistics.

Monta Sakamoto, Hiroshi Matsuzoe
A generalization of independence and multivariate Student's tdistributions Monta Sakamoto, Hiroshi Matsuzoe
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In anomalous statistical physics, deformed algebraic structures are important objects. Heavily tailed probability distributions, such as Student’s tdistributions, 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 tdistribution is studied in this paper. Even if two random variables which follow to univariate Student’s tdistributions are independent, the joint probability distribution of these two distributions is not a bivariate Student’s tdistribution. It is shown that a bivariate Student’s tdistribution is obtained from two univariate Student’s tdistributions under qdeformed independence.

Tomonari Sei, Ushio Tanaka
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The textile plot proposed by Kumasaka and Shibata (2008) is a method for data visualization. The method transforms a data matrix in order to draw a parallel coordinate plot. In this paper, we investigate a set of matrices induced by the textile plot, which we call the textile set, from a geometrical viewpoint. It is shown that the textile set is written as the union of two differentiable manifolds if data matrices are restricted to be fullrank.

Damiano Brigo, John Armstrong
Stochastic PDE projection on manifolds AssumedDensity and Galerkin Filters Damiano Brigo, John Armstrong
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We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L2 direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finitedimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (KushnerStratonovich) or a specific unnormalized (Zakai) density of the optimal filter.

Ali MohammadDjafari
Variational Bayesian Approximation method for Classification and Clustering with a mixture of Studen Ali MohammadDjafari
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Clustering, classification and Pattern Recognition in a set of data are between the most important tasks in statistical researches and in many applications. In this paper, we propose to use a mixture of Studentt distribution model for the data via a hierarchical graphical model and the Bayesian framework to do these tasks. The main advantages of this model is that the model accounts for the uncertainties of variances and covariances and we can use the Variational Bayesian Approximation (VBA) methods to obtain fast algorithms to be able to handle large data sets.

Barbara Opozda
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Affine connection, Curvature tensor, Laplacian Bochner’s technique, Ricci tensor, Sectional curvature
Curvature properties for statistical structures are studied. The study deals with the curvature tensor of statistical connections and their duals as well as the Ricci tensor of the connections, Laplacians and the curvature operator. Two concepts of sectional curvature are introduced. The meaning of the notions is illustrated by presenting few exemplary theorems.

Michel Boyom, Jamali Mohammed, Shahid Hasan
Rui Vigelis, David de Souza, Charles Cavalcante
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We define a metric and a family of αconnections in statistical manifolds, based on ϕdivergence, which emerges in the framework of ϕfamilies of probability distributions. This metric and αconnections generalize the Fisher information metric and Amari’s αconnections. We also investigate the parallel transport associated with the αconnection for α = 1.

John Armstrong, ShunIchi Amari
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We show that Hessian manifolds of dimensions 4 and above must have vanishing Pontryagin forms. This gives a topological obstruction to the existence of Hessian metrics. We find an additional explicit curvature identity for Hessian 4manifolds. By contrast, we show that all analytic Riemannian 2manifolds are Hessian.

Saïd Rahmani, JeanCharles Pinoli, Johan Debayle
Characterization and Estimation of the Variations of a Random Convex Set by it's Mean $n$Variogram : Application to the Boolean Model Saïd Rahmani, JeanCharles Pinoli, Johan Debayle
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Boolean model, Geometric covariogram, Mixed volumes, n points set probability, Particle size distribution, Random set Shape variations
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 nvariogram γ(n)Ξ0:(u1⋯un)↦E[νd(Ξ0∩(Ξ0−u1)⋯∩(Ξ0−un))] of a random convex set Ξ0 on ℝ d reveals information on the n th order structure of Ξ0. Especially we will show that considering the mean nvariograms 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 germgrain model is stable by convex dilatations, furthermore the mean nvariogram of the primary grain is estimable in several type of stationary germgrain models by the so called npoints 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)).

Philippe ElbazVincent, Herbert Gangl
Finite polylogarithms, their multiple analogues and the Shannon entropy Philippe ElbazVincent, Herbert Gangl
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We show that the entropy function–and hence the finite 1logarithm–behaves a lot like certain derivations. We recall its cohomological interpretation as a 2cocycle and also deduce 2ncocycles for any n. Finally, we give some identities for finite multiple polylogarithms together with number theoretic applications.

José Ignacio Burgos Gil, Patrice Philippon, Martin Sombra
Heights of toric varieties, integration over polytopes and entropy José Ignacio Burgos Gil, Patrice Philippon, Martin Sombra
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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.

Matilde Marcolli
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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.

Yann Ollivier
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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 logloss scoring rule, this results in infinite loss the first time “heads” appears.)

Christophe SaintJean, Frank Nielsen
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This paper address the problem of online learning finite statistical mixtures of exponential families. A short review of the ExpectationMaximization (EM) algorithm and its online extensions is done. From these extensions and the description of the kMaximum Likelihood Estimator (kMLE), 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.

Luigi Malagò, Giovanni Pistone
Secondorder Optimization over the Multivariate Gaussian Distribution Luigi Malagò, Giovanni Pistone
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We discuss the optimization of the stochastic relaxation of a realvalued 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 secondorder geometry of the exponential family, with applications to the multivariate Gaussian distributions, to generalize secondorder 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 secondorder methods, such as the Newton method.

ShunIchi Amari, Nihat Ay
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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 nonflat 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.

Garvesh Raskutti, Sayan Mukherjee
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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 nonEuclidean 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 nonEuclidean 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 loglikelihood loss applied to parameter estimation in exponential families asymptotically achieves the classical CramérRao lower bound.

James Tao, Jun Zhang
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The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsionfree 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 nondegenerate twoform 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 Ktranslations, various transformations that generalize traditional projective and dualprojective transformations, and study their commutativity with Lperturbation and hconjugation 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 conformalprojective transformation.

Wolfgang Stummer, AnnaLena Kißlinger
New model search for nonlinear recursive models, regressions and autoregressions Wolfgang Stummer, AnnaLena Kißlinger
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3D score surface, AR SARIMA NARX, Autorecursions, Model selection, Nonlinear regression, Scaled Bregman distances
Scaled Bregman distances SBD have turned out to be useful tools for simultaneous estimation and goodnessoffittesting 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 informationgeometric 3D computergraphical selection procedure. Some samplesize asymptotics is given as well.

Anass Bellachehab, Jérémie Jakubowicz
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In the context of sensor networks, gossip algorithms are a popular, well established technique, for achieving consensus when sensor data are encoded in linear spaces. Gossip algorithms also have several extensions to non linear data spaces. Most of these extensions deal with Riemannian manifolds and use Riemannian gradient descent. This paper, instead, studies gossip in a broader CAT(k) metric setting, encompassing, but not restricted to, several interesting cases of Riemannian manifolds. As it turns out, convergence can be guaranteed as soon as the data lie in a small enough ball of a mere CAT(k) metric space. We also study convergence speed in this setting and establish linear rates of convergence.

Gautier Marti, Frank Nielsen, Philippe Very, Philippe Donnat
TSGNPR Clustering Random Walk Time Series Gautier Marti, Frank Nielsen, Philippe Very, Philippe Donnat
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We present in this paper a novel nonparametric approach useful for clustering independent identically distributed stochastic processes. We introduce a preprocessing step consisting in mapping multivariate independent and identically distributed samples from random variables to a generic nonparametric 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 .

Yonxin Chen, Tryphon Georgiou, Michele Pavon
Benoit Huyot, Yves Mabiala, JeanFrançois Marcotorchino
Optimal Transport, Independance versus Indetermination duality, impact on a new Copula Design Benoit Huyot, Yves Mabiala, JeanFrançois Marcotorchino
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Condorcet and relational analysis, Copula theory, Indetermination and independance structures, MKP problem, Optimal transport
This article leans on some previous results already presented in [10], based on the Fréchet’s works,Wilson’s entropy and Minimal Trade models in connectionwith theMKPtransportation problem (MKP, stands for MongeKantorovich Problem). Using the duality between “independance” and “indetermination” structures, shown in this former paper, we are in a position to derive a novel approach to design a copula, suitable and efficient for anomaly detection in IT systems analysis.

Christian Leonard
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Displacement interpolations, Entropic interpolations, LottSturmVillani theory, Lower bounded curvature of metric spaces, Optimal transport, Schrödinger problem
This note presents a short review of the Schrödinger problem and of the first steps that might lead to interesting consequences in terms of geometry. We stress the analogies between this entropy minimization problem and the renowned optimal transport problem, in search for a theory of lower bounded curvature for metric spaces, including discrete graphs.

Alfred Galichon, Scott Kominers, Simon Weber
The nonlinear BernsteinSchrodinger equation in Economics Alfred Galichon, Scott Kominers, Simon Weber
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In this paper we relate the Equilibrium Assignment Problem (EAP), which is underlying in several economics models, to a system of nonlinear equations that we call the “nonlinear BernsteinSchrödinger system”, which is wellknown in the linear case, but whose nonlinear extension does not seem to have been studied. We apply this connection to derive an existence result for the EAP, and an efficient computational method.

Olivier Schwander, Frank Nielsen
Bagofcomponents an online algorithm for batch learning of mixture models Olivier Schwander, Frank Nielsen
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Practical estimation of mixture models may be problematic when a large number of observations are involved: for such cases, online versions of ExpectationMaximization may be preferred, avoiding the need to store all the observations before running the algorithms. We introduce a new online method wellsuited when both the number of observations is large and lots of mixture models need to be learned from different sets of points. Inspired by dictionary methods, our algorithm begins with a training step which is used to build a dictionary of components. The next step, which can be done online, amounts to populating the weights of the components given each arriving observation. The usage of the dictionary of components shows all its interest when lots of mixtures need to be learned using the same dictionary in order to maximize the return on investment of the training step. We evaluate the proposed method on an artificial dataset built from random Gaussian mixture models.

Geert Verdoolaege, Aqsa Shabbir
Color Texture Discrimination using the Principal Geodesic Distance on a Multivariate Generalized Gau Geert Verdoolaege, Aqsa Shabbir
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We present a new texture discrimination method for textured color images in the wavelet domain. In each wavelet subband, the correlation between the color bands is modeled by a multivariate generalized Gaussian distribution with fixed shape parameter (Gaussian, Laplacian). On the corresponding Riemannian manifold, the shape of texture clusters is characterized by means of principal geodesic analysis, specifically by the principal geodesic along which the cluster exhibits its largest variance. Then, the similarity of a texture to a class is defined in terms of the Rao geodesic distance on the manifold from the texture’s distribution to its projection on the principal geodesic of that class. This similarity measure is used in a classification scheme, referred to as principal geodesic classification (PGC). It is shown to perform significantly better than several other classifiers.

Gianni Franchi, Jesús Angulo
Quantization of hyperspectral image manifold using probabilistic distances Gianni Franchi, Jesús Angulo
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Hyperspectral images, Information geometry, Mathematical morphology, Probabilistic distances, Quantization
A technique of spatialspectral 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 αconnected 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.

Jesús Angulo
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Stochastic watershed is an image segmentation technique based on mathematical morphology which produces a probability density function of image contours. Estimated probabilities depend mainly on local distances between pixels. This paper introduces a variant of stochastic watershed where the probabilities of contours are computed from a gaussian model of image regions. In this framework, the basic ingredient is the distance between pairs of regions, hence a distance between normal distributions. Hence several alternatives of statistical distances for normal distributions are compared, namely Bhattacharyya distance, Hellinger metric distance and Wasserstein metric distance.

Salem Said, Lionel Bombrun, Yannick Berthoumieu
Texture classification using Rao's distance on the space of covariance matrices Salem Said, Lionel Bombrun, Yannick Berthoumieu
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EM algorithm, Information geometry, Mixture estimation, Riemannian centre of mass, Texture classification
The current paper introduces new prior distributions on the zeromean multivariate Gaussian model, with the aim of applying them to the classification of covariance matrices populations. These new prior distributions are entirely based on the Riemannian geometry of the multivariate Gaussian model. More precisely, the proposed Riemannian Gaussian distribution has two parameters, the centre of mass ˉY and the dispersion parameter σ. Its density with respect to Riemannian volume is proportional to exp(−d2(Y;ˉY)), where d2(Y;ˉY) is the square of Rao’s Riemannian distance. We derive its maximum likelihood estimators and propose an experiment on the VisTex database for the classification of texture images.

Thibaut Le Gouic, JeanMichel Loubes
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We study barycenters in the Wasserstein space Pp(E) of a locally compact geodesic space (E, d). In this framework, we define the barycenter of a measure ℙ on Pp(E) as its Fréchet mean. The paper establishes its existence and states consistency with respect to ℙ. We thus extends previous results on ℝ d , with conditions on ℙ or on the sequence converging to ℙ for consistency.

Reiner Lenz
Generalized Pareto Distributions, Image Statistics and Autofocusing in Automated Microscopy Reiner Lenz
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We introduce the generalized Pareto distributions as a statistical model to describe thresholded edgemagnitude image filter results. Compared to the more commonWeibull or generalized extreme value distributions these distributions have at least two important advantages, the usage of the high threshold value assures that only the most important edge points enter the statistical analysis and the estimation is computationally more efficient since a much smaller number of data points have to be processed. The generalized Pareto distributions with a common threshold zero form a twodimensional Riemann manifold with the metric given by the Fisher information matrix. We compute the Fisher matrix for shape parameters greater than 0.5 and show that the determinant of its inverse is a product of a polynomial in the shape parameter and the squared scale parameter. We apply this result by using the determinant as a sharpness function in an autofocus algorithm. We test the method on a large database of microscopy images with given ground truth focus results. We found that for a vast majority of the focus sequences the results are in the correct focal range. Cases where the algorithm fails are specimen with too few objects and sequences where contributions from different layers result in a multimodal sharpness curve. Using the geometry of the manifold of generalized Pareto distributions more efficient autofocus algorithms can be constructed but these optimizations are not included here.

Alexis Decurninge
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Univariate Lmoments are expressed as projections of the quantile function onto an orthogonal basis of univariate polynomials. We present multivariate versions of Lmoments 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 Lmoments. The properties of estimated Lmoments are illustrated for heavytailed distributions.

Julien Rabin, Nicolas Papadakis
Nonconvex relaxation of optimal transport for color transfer between images Julien Rabin, Nicolas Papadakis
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Optimal transport (OT) is a major statistical tool to measure similarity between features or to match and average features. However, OT requires some relaxation and regularization to be robust to outliers. With relaxed methods, as one feature can be matched to several ones, important interpolations between different features arise. This is not an issue for comparison purposes, but it involves strong and unwanted smoothing for transfer applications. We thus introduce a new regularized method based on a nonconvex formulation that minimizes transport dispersion by enforcing the onetoone matching of features. The interest of the approach is demonstrated for color transfer purposes.

Stephane Puechmorel, Florence Nicol
Entropy minimizing curves with application to automated flight path design Stephane Puechmorel, Florence Nicol
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Air traffic management (ATM) aims at providing companies with a safe and ideally optimal aircraft trajectory planning. Air traffic controllers act on flight paths in such a way that no pair of aircraft come closer than the regulatory separation norm. With the increase of traffic, it is expected that the system will reach its limits in a near future: a paradigm change in ATM is planned with the introduction of trajectory based operations. This paper investigate a mean of producing realistic air routes from the output of an automated trajectory design tool. For that purpose, an entropy associated with a system of curves is defined and a mean of iteratively minimizing it is presented. The network produced is suitable for use in a semiautomated ATM system with human in the loop.

Emmanuel Chevallier, Jesús Angulo, Ivar Farup
Histograms of images valued in the manifold of colours endowed with perceptual metrics Emmanuel Chevallier, Jesús Angulo, Ivar Farup
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We address here the problem of perceptual colour histograms. The Riemannian structure of perceptual distances is measured through standards sets of ellipses, such as Macadam ellipses. We propose an approach based on local Euclidean approximations that enables to take into account the Riemannian structure of perceptual distances, without introducing computational complexity during the construction of the histogram.

Dena Asta
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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 Hölderclass assumptions on the densities. A main tool used in proving the convergence rate is the HelgasonFourier 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 2dimensional hyperboloid.

Emmanuel Chevallier, Jesús Angulo, Frédéric Barbaresco
Probability density estimation on the hyperbolic space applied to radar processing Emmanuel Chevallier, Jesús Angulo, Frédéric Barbaresco
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The two main techniques of probability density estimation on symmetric spaces are reviewed in the hyperbolic case. For computational reasons we chose to focus on the kernel density estimation and we provide the expression of Pelletier estimator on hyperbolic space. The method is applied to density estimation of reflection coefficients derived from radar observations.

Barbara Gris, Stanley Durrleman, Alain Trouvé
A subRiemannian modular approach for diffeomorphic deformations Barbara Gris, Stanley Durrleman, Alain Trouvé
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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 subRiemannian 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.

Minh Ha Quang
Affineinvariant Riemannian Distance Between Infinitedimensional Covariance Operators Minh Ha Quang
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This paper studies the affineinvariant Riemannian distance on the RiemannHilbert manifold of positive definite operators on a separable Hilbert space. This is the generalization of the Riemannian manifold of symmetric, positive definite matrices to the infinitedimensional setting. In particular, in the case of covariance operators in a Reproducing Kernel Hilbert Space (RKHS), we provide a closed form solution, expressed via the corresponding Gram matrices.

Xavier Pennec
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This paper addresses the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. Current methods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) minimize the distance to a “Geodesic subspace”. This allows to build sequences of nested subspaces which are consistent with a forward component analysis approach. However, these methods cannot be adapted to a backward analysis and they are not symmetric in the parametrization of the subspaces. We propose in this paper a new and more general type of family of subspaces in manifolds: barycentric subspaces are implicitly defined as the locus of points which are weighted means of k + 1 reference points. Depending on the generalization of the mean that we use, we obtain the Fréchet/Karcher barycentric subspaces (FBS/KBS) or the affine span (with exponential barycenter). This definition restores the full symmetry between all parameters of the subspaces, contrarily to the geodesic subspaces which intrinsically privilege one point. We show that this definition defines locally a submanifold of dimension k and that it generalizes in some sense geodesic subspaces. Like PGA, barycentric subspaces allow the construction of a forward nested sequence of subspaces which contains the Fréchet mean. However, the definition also allows the construction of backward nested sequence which may not contain the mean. As this definition relies on points and do not explicitly refer to tangent vectors, it can be extended to non Riemannian geodesic spaces. For instance, principal subspaces may naturally span over several strata in stratified spaces, which is not the case with more classical generalizations of PCA.

Benjamin Eltzner, Sungkyu Jung, Stephan Huckemann
Dimension Reduction on Polyspheres with Application to Skeletal Representations Benjamin Eltzner, Sungkyu Jung, Stephan Huckemann
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We present a novel method that adaptively deforms a polysphere (a product of spheres) into a single high dimensional sphere which then allows for principal nested spheres (PNS) analysis. Applying our method to skeletal representations of simulated bodies as well as of data from real human hippocampi yields promising results in view of dimension reduction. Specifically in comparison to composite PNS (CPNS), our method of principal nested deformed spheres (PNDS) captures essential modes of variation by lower dimensional representations.

Stefan Sommer
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This paper presents derivations of evolution equations for the family of paths that in the Diffusion PCA framework are used for approximating data likelihood. The paths that are formally interpreted as most probable paths generalize geodesics in extremizing an energy functional on the space of differentiable curves on a manifold with connection. We discuss how the paths arise as projections of geodesics for a (non bracketgenerating) subRiemannian metric on the frame bundle. Evolution equations in coordinates for both metric and cometric formulations of the subRiemannian geometry are derived. We furthermore show how rankdeficient metrics can be mixed with an underlying Riemannian metric, and we use the construction to show how the evolution equations can be implemented on finite dimensional LDDMM landmark manifolds.

Martin Kleinsteuber, Hao Shen
BlockJacobi methods with Newtonsteps and nonunitary joint matrix diagonalization Martin Kleinsteuber, Hao Shen
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In this work, we consider blockJacobi methods with Newton steps in each subspace search and prove their local quadratic convergence to a local minimum with nondegenerate Hessian under some orthogonality assumptions on the search directions. Moreover, such a method is exemplified for nonunitary joint matrix diagonalization, where we present a blockJacobitype method on the oblique manifold with guaranteed local quadratic convergence.

Antoine Arnould, PierreYves Gousenbourger, Chafik Samir, PierreAntoine Absil, Michel Canis
Fitting Smooth Paths on Riemannian Manifolds  Endometrial Surface Antoine Arnould, PierreYves Gousenbourger, Chafik Samir, PierreAntoine Absil, Michel Canis
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Bézier functions, Endometrial surface reconstruction, MRIbased navigation, Optimization on manifolds, Path fitting on Riemannian manifolds
We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using C1 piecewiseBézier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group SO(3) and the special Euclidean group SE(3) for preoperative MRIbased navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.

Kathrin Welker, Volker Schulz, Martin Siebenborn
Nicolas Boumal
Riemannian trust regions with finitedifference Hessian approximations are globally convergent Nicolas Boumal
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The Riemannian trustregion algorithm (RTR) is designed to optimize differentiable cost functions on Riemannian manifolds. It proceeds by iteratively optimizing local models of the cost function. When these models are exact up to second order, RTR boasts a quadratic convergence rate to critical points. In practice, building such models requires computing the Riemannian Hessian, which may be challenging. A simple idea to alleviate this difficulty is to approximate the Hessian using finite differences of the gradient. Unfortunately, this is a nonlinear approximation, which breaks the known convergence results for RTR. We propose RTRFD: a modification of RTR which retains global convergence when the Hessian is approximated using finite differences. Importantly, RTRFD reduces gracefully to RTR if a linear approximation is used. This algorithm is available in the Manopt toolbox.

Matthieu Genicot, Wen Huang, Nickolay Trendafilov
Weakly Correlated Sparse Components with Nearly Orthonormal Loadings Matthieu Genicot, Wen Huang, Nickolay Trendafilov
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Dantzig selector, LASSO, Optimization on matrix manifolds, Orthonormal and oblique component loadings matrices
There is already a great number of highly efficient methods producing components with sparse loadings which significantly facilitates the interpretation of principal component analysis (PCA). However, they produce either only orthonormal loadings, or only uncorrelated components, or, most frequently, neither of them. To overcome this weakness, we introduce a new approach to define sparse PCA similar to the Dantzig selector idea already employed for regression problems. In contrast to the existing methods, the new approach makes it possible to achieve simultaneously nearly uncorrelated sparse components with nearly orthonormal loadings. The performance of the new method is illustrated on real data sets. It is demonstrated that the new method outperforms one of the most popular available methods for sparse PCA in terms of preservation of principal components properties.

Xavier Pennec, Nina Miolane
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Usual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc? How can we describe the interaction between statistics and geometry? The structure of Quotient space in particular is widely used to model data, for example every time one deals with shape data. These can be shapes of constellations in Astronomy, shapes of human organs in Computational Anatomy, shapes of skulls in Palaeontology, etc. Given this broad field of applications, statistics on shapes and more generally on observations belonging to quotient spaces have been studied since the 1980’s. However, most theories model the variability in the shapes but do not take into account the noise on the observations themselves. In this paper, we show that statistics on quotient spaces are biased and even inconsistent when one takes into account the noise. In particular, some algorithms of template estimation in Computational Anatomy are biased and inconsistent. Our development thus gives a first theoretical geometric explanation of an experimentally observed phenomenon. A biased estimator is not necessarily a problem. In statistics, it is a general rule of thumb that a bias can be neglected for example when it represents less than 0.25 of the variance of the estimator. We can also think about neglecting the bias when it is low compared to the signal we estimate. In view of the applications, we thus characterize geometrically the situations when the bias can be neglected with respect to the situations when it must be corrected.

Lorenz Schwachhöfer, Nihat Ay, Jürgen Jost, Hong Van Le
Invariant geometric structures on statistical models Lorenz Schwachhöfer, Nihat Ay, Jürgen Jost, Hong Van Le
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We review the notion of parametrized measure models and tensor fields on them, which encompasses all statistical models considered by Chentsov [6], Amari [3] and PistoneSempi [10]. We give a complete description of ntensor fields that are invariant under sufficient statistics. In the cases n = 2 and n = 3, the only such tensors are the Fisher metric and the AmariChentsov tensor. While this has been shown by Chentsov [7] and Campbell [5] in the case of finite measure spaces, our approach allows to generalize these results to the cases of infinite sample spaces and arbitrary n. Furthermore, we give a generalisation of the monotonicity theorem and discuss its consequences.

Alice Le Brigant, Marc Arnaudon, Frédéric Barbaresco
Reparameterization invariant metric on the space of curves Alice Le Brigant, Marc Arnaudon, Frédéric Barbaresco
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This paper focuses on the study of open curves in a manifold M, and its aim is to define a reparameterization invariant distance on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions =Imm([0,1],M) by pullback of a metric on the tangent bundle T derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on T induces a firstorder Sobolev metric on with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the image curves by the SRVF parallel transported to a same vector space, with an added curvature term. This provides a generalized theoretical SRV framework for curves lying in a general manifold M.

Julien Lefèvre, Guillaume Auzias
Spherical parameterization for genus zero surfaces using LaplaceBeltrami eigenfunctions Julien Lefèvre, Guillaume Auzias
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In this work, we propose a fast and simple approach to obtain a spherical parameterization of a certain class of closed surfaces without holes. Our approach relies on empirical findings that can be mathematically investigated, to a certain extent, by using LaplaceBeltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first nontrivial eigenfunctions of the LaplaceBeltrami Operator. Our approach requires a topological condition on those eigenfunctions, whose nodal domains must be 2. We show the efficiency of the approach through numerical experiments performed on cortical surface meshes.

Sylvain Arguillere
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This paper aims to define a unified setting for shape registration and LDDMM methods for shape analysis. This setting turns out to be subRiemannian, 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.

Martin Bauer, Martins Bruveris, Peter Michor
Uniqueness of the FisherRao Metric on the Space of Smooth Densities Martin Bauer, Martins Bruveris, Peter Michor
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We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L2 direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finitedimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (KushnerStratonovich) or a specific unnormalized (Zakai) density of the optimal filter.

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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 biinvariant 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.

Bijan Afsari, Gregory S. Chirikjian
Deblurring and Recovering Conformational States in 3D Single Particle Electron Bijan Afsari, Gregory S. Chirikjian
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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 multiunit 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 spatialmotional convolution.We discuss the illconditioned 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 multiunit 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.

Dario Prandi, Ugo Boscain, JeanPaul Gauthier
Image processing in the semidiscrete group of rototranslations Dario Prandi, Ugo Boscain, JeanPaul Gauthier
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It is wellknown, since [12], that cells in the primary visual cortex V1 do much more than merely signaling position in the visual field: most cortical cells signal the local orientation of a contrast edge or bar . they are tuned to a particular local orientation. This orientation tuning has been given a mathematical interpretation in a subRiemannian model by Petitot, Citti, and Sarti [6,14]. According to this model, the primary visual cortex V1 lifts greyscale images, given as functions f : ℝ2 → [0, 1], to functions Lf defined on the projectivized tangent bundle of the plane PTℝ2 = ℝ2 ×ℙ1. Recently, in [1], the authors presented a promising semidiscrete variant of this model where the Euclidean group of rototranslations SE(2), which is the double covering of PTℝ2, is replaced by SE(2,N), the group of translations and discrete rotations. In particular, in [15], an implementation of this model allowed for stateoftheart image inpaintings. In this work, we review the inpainting results and introduce an application of the semidiscrete model to image recognition. We remark that both these applications deeply exploit the Moore structure of SE(2,N) that guarantees that its unitary representations behaves similarly to those of a compact group. This allows for nice properties of the Fourier transform on SE(2,N) exploiting which one obtains numerical advantages.

Santiago VelascoForero, Jesús Angulo
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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 nonflat 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.

Thomas Hotz, Florian Kelma, Johannes Wieditz
Eric Justh, P. S. Krishnaprasad
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Collective behavior, Liepoisson reduction, Nonholonomic integrator, Subriemannian geodesics, Subriemannian geometry
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 driftfree particle dynamics, the coupling terms in the cost functionals lead to a novel class of problems in subriemannian geometry of product Lie groups.

Géry de Saxcé
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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.

Frederic Boyer, Federico Renda
Poincaré's equations for Cosserat shells  application to locomotion of cephalopods Frederic Boyer, Federico Renda
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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.

Francois Dubois, Danielle Fortune, Juan Antonio Rojas Quintero
Pontryagin calculus in Riemannian geometry Francois Dubois, Danielle Fortune, Juan Antonio Rojas Quintero
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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.

Krzysztof Krakowski, Luís Machado, Fátima Leite
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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 noslip and notwist, 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.

Imre Csiszár, Thomas Breuer, Michel Broniatowski
An Information Geometry Problem in Mathematical Finance Imre Csiszár, Thomas Breuer, Michel Broniatowski
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Almost worst case densities, Bregman distance, Convex integral functional, fdivergence Idivergence, Payoff function, Pythagorean identity, Risk measure, Worst case density
Familiar approaches to risk and preferences involve minimizing the expectation EIP(X) of a payoff function X over a family Γ of plausible risk factor distributions IP. We consider Γ determined by a bound on a convex integral functional of the density of IP, thus Γ may be an Idivergence (relative entropy) ball or some other fdivergence 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∈Γ), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When Γ is an fdivergence 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 fdivergence balls.

Jan Naudts, Ben Anthonis, Michel Broniatowski
Extension of information geometry to nonstatistical systems some examples Jan Naudts, Ben Anthonis, Michel Broniatowski
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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.

Michel Broniatowski, Diaa Al Mohamad
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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 twocomponent gaussian mixture. Simulations on Gaussian andWeibull mixtures are performed to compare the results with the MLE.

Philippe Regnault, Amor Keziou
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We derive independence tests by means of dependence measures thresholding in a semiparametric context. Precisely, estimates of mutual information associated to ϕdivergences are derived through the dual representations of ϕdivergences. 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.

Amor Keziou
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We introduce what we will call multivariate divergences between K, K ≥ 1, signed finite measures (Q1, . . . , Q K ) and a given reference probability measure P on a σfield (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 multigroup classification.

CharlesMichel Marle
Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems CharlesMichel Marle
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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.

Marc Arnaudon
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We will prove a EulerPoincaré 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 NavierStokes equation on a bounded domain and the CamassaHolm equation.
