(2015) GSI2015

Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunctions Guillaume Auzias, Julien Lefèvre GSI2015
Détails de l'article
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 Laplace-Beltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first non-trivial eigenfunctions of the Laplace-Beltrami 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.
Spherical parameterization for genus zero surfaces using Laplace-Beltrami eigenfunctions
PDE Constrained Shape Optimization as Optimization on Shape Manifolds Kathrin Welker, Martin Siebenborn, Volker Schulz GSI2015
Détails de l'article
The novel Riemannian view on shape optimization introduced in [14] is extended to a Lagrange–Newton as well as a quasi–Newton approach for PDE constrained shape optimization problems.
PDE Constrained Shape Optimization as Optimization on Shape Manifolds
Fitting Smooth Paths on Riemannian Manifolds - Endometrial Surface Antoine Arnould, Chafik Samir, Michel Canis, Pierre-Antoine Absil, Pierre-Yves Gousenbourger GSI2015
Détails de l'article
We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using C1 piecewise-Bézier 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 MRI-based 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.
Fitting Smooth Paths on Riemannian Manifolds - Endometrial Surface
A two-color interacting random balls model for co-localization analysis of proteins Charles Kervrann, Frederic Lavancier GSI2015
Détails de l'article
A model of two-type (or two-color) 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 co-localization 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 Takacs-Fiksel method with a specific choice of test functions.
A two-color interacting random balls model for co-localization analysis of proteins
The extremal index for a random tessellation Nicolas Chenavier GSI2015
Détails de l'article
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 so-called extremal index. Two examples of extremal indices are provided for Poisson-Voronoi and Poisson-Delaunay tessellations.
The extremal index for a random tessellation
From Geometry and Physics to Computational Linguistics Matilde Marcolli GSI2015
Détails de l'article
I will show how techniques from geometry (algebraic geometry and topology) and physics (statistical physics) can be applied to Linguistics, in order to provide a computational approach to questions of syntactic 
From Geometry and Physics to Computational Linguistics
Multiply CR-Warped Product Statistical Submanifolds of a Holomorphic Stastistical Space Form Hasan Shahid, Jamali Mohammed, Michel Nguiffo Boyom GSI2015
Détails de l'article
In this article, we derive an inequality satisfied by the squared norm of the imbedding curvature tensor of Multiply CR-warped product statistical submanifolds N of holomorphic statistical space forms M. Furthermore, we prove that under certain geometric conditions, N and M become Einstein.
Multiply CR-Warped Product Statistical Submanifolds of a Holomorphic Stastistical Space Form
Matrix realization of a homogeneous cone Hideyuki Ishi GSI2015
Détails de l'article
Based on the theory of compact normal left-symmetric algebra (clan), we realize every homogeneous cone as a set of positive definite real symmetric matrices, where homogeneous Hessian metrics as well as a transitive group action on the cone are described efficiently.
Matrix realization of a homogeneous cone
The Pontryagin Forms of Hessian Manifolds John Armstrong, Shun-Ichi Amari GSI2015
Détails de l'article
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 4-manifolds. By contrast, we show that all analytic Riemannian 2-manifolds are Hessian.
The Pontryagin Forms of Hessian Manifolds
Curvatures of Statistical Structures Barbara Opozda GSI2015
Détails de l'article
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.
Curvatures of Statistical Structures
New metric and connections in statistical manifolds Charles Casimiro Cavalcante, David de Souza, Rui F. Vigelis GSI2015
Détails de l'article
We define a metric and a family of α-connections in statistical manifolds, based on ϕ-divergence, which emerges in the framework of ϕ-families of probability distributions. This metric and α-connections generalize the Fisher information metric and Amari’s α-connections. We also investigate the parallel transport associated with the α-connection for α = 1.
New metric and connections in statistical manifolds
Geometry on the set of quantum states and quantum correlations Dominique Spehner GSI2015
Symetry methods in geometrics mechanics Tudor Ratiu GSI2015
Détails de l'article
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. 
Symetry methods in geometrics mechanics
A generalization of independence and multivariate Student's t-distributions Hiroshi Matsuzoe, Monta Sakamoto GSI2015
Détails de l'article
In anomalous statistical physics, deformed algebraic structures are important objects. Heavily tailed probability distributions, such as Student’s t-distributions, are characterized by deformed algebras. In addition, deformed algebras cause deformations of expectations and independences of random variables. Hence, a generalization of independence for multivariate Student’s t-distribution is studied in this paper. Even if two random variables which follow to univariate Student’s t-distributions are independent, the joint probability distribution of these two distributions is not a bivariate Student’s t-distribution. It is shown that a bivariate Student’s t-distribution is obtained from two univariate Student’s t-distributions under q-deformed independence.
A generalization of independence and multivariate Student's t-distributions
Differential geometric properties of textile plot Tomonari Sei, Ushio Tanaka GSI2015
Détails de l'article
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 full-rank.
Differential geometric properties of textile plot
Variational Bayesian Approximation method for Classification and Clustering with a mixture of Studen Ali Mohammad-Djafari GSI2015
Détails de l'article
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 Student-t 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.
Variational Bayesian Approximation method for Classification and Clustering with a mixture of Studen
Stochastic PDE projection on manifolds Assumed-Density and Galerkin Filters Damiano Brigo, John Armstrong GSI2015
Détails de l'article
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 finite-dimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (Kushner-Stratonovich) or a specific unnormalized (Zakai) density of the optimal filter.
Stochastic PDE projection on manifolds Assumed-Density and Galerkin Filters
Computational Information Geometry: mixture modelling Frank Critchley, Germain Van Bever, Paul Marriott, Radka Sabolova GSI2015
Path connectedness on a space of probability density functions Osamu Komori, Shinto Eguchi GSI2015
Détails de l'article
We introduce a class of paths or one-parameter models connecting arbitrary two probability density functions (pdf’s). The class is derived by employing the Kolmogorov-Nagumo average between the two pdf’s. There is a variety of such path connectedness on the space of pdf’s since the Kolmogorov-Nagumo 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 one-parameter model is extended to a multidimensional model, on which the statistical inference is characterized by sufficient statistics.
Path connectedness on a space of probability density functions
Group Theoretical Study on Geodesics for the Elliptical Models Hiroto Inoue GSI2015
Détails de l'article
We consider the geodesic equation on the elliptical model, which is a generalization of the normal model. More precisely, we characterize this manifold from the group theoretical view point and formulate Eriksen’s procedure to obtain geodesics on normal model and give an alternative proof for it.
Group Theoretical Study on Geodesics for the Elliptical Models
From Euclidean to Riemannian Means Information Geometry for SSVEP Classification Emmanuel Kalunga, Eric Monacelli, Karim Djouani, Quentin Barthélemy, Sylvain Chevallier, Yskandar Hamam GSI2015
Détails de l'article
Brain Computer Interfaces (BCI) based on electroencephalography (EEG) rely on multichannel brain signal processing. Most of the state-of-the-art 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.
From Euclidean to Riemannian Means Information Geometry for SSVEP Classification
Approximating Covering and Minimum Enclosing Balls in Hyperbolic Geometry Frank Nielsen, Gaëtan Hadjeres GSI2015
Détails de l'article
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 closed-form formula to compute the geodesic α-midpoint between any two points. Those results allow us to apply the hyperbolic k-center clustering for statistical location-scale families or for multivariate spherical normal distributions by using their Fisher information matrix as the underlying Riemannian hyperbolic metric.
Approximating Covering and Minimum Enclosing Balls in Hyperbolic Geometry
Computing Boundaries in Local Mixture Models Paul Marriott, Vahed Maroufy GSI2015
Détails de l'article
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.
Computing Boundaries in Local Mixture Models
Geometry of Goodness-of-Fit Testing in High Dimensional Low Sample Size Modelling Frank Critchley, Germain Van Bever, Paul Marriott, Radka Sabolova GSI2015
Détails de l'article
We introduce a new approach to goodness-of-fit 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.
Geometry of Goodness-of-Fit Testing in High Dimensional Low Sample Size Modelling
Asymmetric Topologies on Statistical Manifolds Roman Belavkin GSI2015
Détails de l'article
Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.
Asymmetric Topologies on Statistical Manifolds
Asymptotics of superposition of point processes Aurélien Vasseur, Laurent Decreusefond GSI2015
Détails de l'article
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.
Asymptotics of superposition of point processes