(2015) GSI2015

Finite polylogarithms, their multiple analogues and the Shannon entropy Herbert Gangl, Philippe Elbaz-Vincent GSI2015
Détails de l'article
We show that the entropy function–and hence the finite 1-logarithm–behaves a lot like certain derivations. We recall its cohomological interpretation as a 2-cocycle and also deduce 2n-cocycles for any n. Finally, we give some identities for finite multiple polylogarithms together with number theoretic applications.
Finite polylogarithms, their multiple analogues and the Shannon entropy
Some geometric consequences of the Schrödinger problem Christian Leonard GSI2015
Détails de l'article
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.
Some geometric consequences of the Schrödinger problem
Barycentric Subspaces and Affine Spans in Manifolds Xavier Pennec GSI2015
Détails de l'article
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.
Barycentric Subspaces and Affine Spans in Manifolds
Universal, non-asymptotic confidence sets for circular means Florian Kelma, Johannes Wieditz, Thomas Hotz GSI2015
Détails de l'article
Based on Hoeffding’s mass concentration inequalities, nonasymptotic confidence sets for circular means are constructed which are universal in the sense that they require no distributional assumptions. These are then compared with asymptotic confidence sets in simulations and for a real data set.
Universal, non-asymptotic confidence sets for circular means
Standard Divergence in Manifold of Dual Affine Connections Nihat Ay, Shun-Ichi Amari GSI2015
Détails de l'article
A divergence function defines a Riemannian metric G and dually coupled affine connections (∇, ∇  ∗ ) with respect to it in a manifold M. When M is dually flat, a canonical divergence is known, which is uniquely determined from {G, ∇, ∇  ∗ }. We search for a standard divergence for a general non-flat M. It is introduced by the magnitude of the inverse exponential map, where α = -(1/3) connection plays a fundamental role. The standard divergence is different from the canonical divergence.
Standard Divergence in Manifold of Dual Affine Connections
Color Texture Discrimination using the Principal Geodesic Distance on a Multivariate Generalized Gau Aqsa Shabbir, Geert Verdoolaege GSI2015
Détails de l'article
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.
Color Texture Discrimination using the Principal Geodesic Distance on a Multivariate Generalized Gau
Block-Jacobi methods with Newton-steps and non-unitary joint matrix diagonalization Hao Shen, Martin Kleinsteuber GSI2015
Détails de l'article
In this work, we consider block-Jacobi methods with Newton steps in each subspace search and prove their local quadratic convergence to a local minimum with non-degenerate Hessian under some orthogonality assumptions on the search directions. Moreover, such a method is exemplified for non-unitary joint matrix diagonalization, where we present a block-Jacobi-type method on the oblique manifold with guaranteed local quadratic convergence.
Block-Jacobi methods with Newton-steps and non-unitary joint matrix diagonalization
Entropy and structure of the thermodynamical systems Géry de Saxcé GSI2015
Détails de l'article
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.
Entropy and structure of the thermodynamical systems
Generalized Pareto Distributions, Image Statistics and Autofocusing in Automated Microscopy Reiner Lenz GSI2015
Détails de l'article
We introduce the generalized Pareto distributions as a statistical model to describe thresholded edge-magnitude 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 two-dimensional 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 multi-modal 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.
Generalized Pareto Distributions, Image Statistics and Autofocusing in Automated Microscopy
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
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
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
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
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
Barycenter in Wasserstein space existence and consistency Jean-Michel Loubes, Thibaut Le Gouic GSI2015
Détails de l'article
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.
Barycenter in Wasserstein space existence and consistency
Uniqueness of the Fisher-Rao Metric on the Space of Smooth Densities Martin Bauer, Martins Bruveris, Peter Michor 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.
Uniqueness of the Fisher-Rao Metric on the Space of Smooth Densities
New model search for nonlinear recursive models, regressions and autoregressions Anna-Lena Kißlinger, Wolfgang Stummer GSI2015
Détails de l'article
Scaled Bregman distances SBD have turned out to be useful tools for simultaneous estimation and goodness-of-fit-testing in parametric models of random data (streams, clouds). We show how SBD can additionally be used for model preselection (structure detection), i.e. for finding appropriate candidates of model (sub)classes in order to support a desired decision under uncertainty. For this, we exemplarily concentrate on the context of nonlinear recursive models with additional exogenous inputs; as special cases we include nonlinear regressions, linear autoregressive models (e.g. AR, ARIMA, SARIMA time series), and nonlinear autoregressive models with exogenous inputs (NARX). In particular, we outline a corresponding information-geometric 3D computer-graphical selection procedure. Some sample-size asymptotics is given as well.
New model search for nonlinear recursive models, regressions and autoregressions
An Information Geometry Problem in Mathematical Finance Imre Csiszár, Michel Broniatowski, Thomas Breuer GSI2015
Détails de l'article
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 I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution IPo. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing EIP(X) subject to IP∈Γ), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When Γ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.
An Information Geometry Problem in Mathematical Finance
Entropy minimizing curves with application to automated flight path design Florence Nicol, Stephane Puechmorel GSI2015
Détails de l'article
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 semi-automated ATM system with human in the loop.
Entropy minimizing curves with application to automated flight path design
Heights of toric varieties, integration over polytopes and entropy José Ignacio Burgos Gil, Martin Sombra, Patrice Philippon GSI2015
Détails de l'article
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.
Heights of toric varieties, integration over polytopes and entropy
Optimal Transport, Independance versus Indetermination duality, impact on a new Copula Design Benoit Huyot, Jean-François Marcotorchino, Yves Mabiala GSI2015
Détails de l'article
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 Monge-Kantorovich 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.
Optimal Transport, Independance versus Indetermination duality, impact on a new Copula Design
Dimension Reduction on Polyspheres with Application to Skeletal Representations Benjamin Eltzner, Stephan Huckemann, Sungkyu Jung GSI2015
Détails de l'article
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.
Dimension Reduction on Polyspheres with Application to Skeletal Representations
Deblurring and Recovering Conformational States in 3D Single Particle Electron Bijan Afsari, Gregory S. Chirikjian GSI2015
Détails de l'article
In this paper we study two forms of blurring effects that may appear in the reconstruction of 3D Electron Microscopy (EM), specifically in single particle reconstruction from random orientations of large multi-unit biomolecular complexes. We model the blurring effects as being due to independent contributions from: (1) variations in the conformation of the biomolecular complex; and (2) errors accumulated in the reconstruction process. Under the assumption that these effects can be separated and treated independently, we show that the overall blurring effect can be expressed as a special form of a convolution operation of the 3D density with a kernel defined on SE(3), the Lie group of rigid body motions in 3D. We call this form of convolution mixed spatial-motional convolution.We discuss the ill-conditioned nature of the deconvolution needed to deblur the reconstructed 3D density in terms of parameters associated with the unknown probability in SE(3). We provide an algorithm for recovering the conformational information of large multi-unit biomolecular complexes (essentially deblurring) under certain biologically plausible prior structural knowledge about the subunits of the complex in the case the blurring kernel has a special form.
Deblurring and Recovering Conformational States in 3D Single Particle Electron
Transformations and Coupling Relations for Affine Connections James Tao, Jun Zhang GSI2015
Détails de l'article
The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on the TM, such that ∇ g is totally symmetric, forming, by definition, a “Codazzi pair” { ∇ , g}. In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on TM), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.
Transformations and Coupling Relations for Affine Connections
Bag-of-components an online algorithm for batch learning of mixture models Frank Nielsen, Olivier Schwander GSI2015
Détails de l'article
Practical estimation of mixture models may be problematic when a large number of observations are involved: for such cases, online versions of Expectation-Maximization may be preferred, avoiding the need to store all the observations before running the algorithms. We introduce a new online method well-suited 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.
Bag-of-components an online algorithm for batch learning of mixture models
Weakly Correlated Sparse Components with Nearly Orthonormal Loadings Matthieu Genicot, Nickolay Trendafilov, Wen Huang GSI2015
Détails de l'article
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.
Weakly Correlated Sparse Components with Nearly Orthonormal Loadings
Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics Frédéric Barbaresco GSI2015
Détails de l'article
We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat, and momentum as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. Finally, we conclude on Balian Gauge theory of Thermodynamics compatible with Souriau’s Model.
Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics
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
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
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