• Geo-Sci-Info


    Michel Ledoux (Université de Toulouse, France) Mokshay Madiman (University of Delaware, USA)


    Mokshay Madiman (University of Delaware, USA)


    Thomas Courtade (University of California, Berkeley, USA) Nathael Gozlan (Université Paris-Est, France) Oliver Johnson (University of Bristol, UK) Jan Maas (IST, Austria) Karl-Theodor Sturm (Institut für Angewandte Mathematik, Germany)

    This session will explore the geometry of particular instances as well as general classes of metric measure spaces as captured using the notion of entropy. In the general context, the ground-breaking work of Lott-Villani-Sturm has provided a notion of Ricci curvature lower bounds that make sense for a large class of non-discrete metric measure spaces. Particular contexts such as discrete and Euclidean spaces benefit from separate development, either because the Lott-Villani-Sturm notion does not apply (discrete) or because much more can be said than in the general setting (Euclidean). The session will explore the swirl of ideas that have seen rapid development in these areas in recent years, including optimal transport, concentration of measure, and isoperimetry for general settings, and entropy power inequalities, Stein’s method, and connections/parallels with convex geometry in particular settings.

  • Pierre Baudot, Daniel Bennequin, Michel Boyom, Herbert Gangl, Matilde Marcolli, John Terrila


    Organisers: Pierre Baudot, Daniel Bennequin, Michel Boyom, Herbert Gangl, Matilde Marcolli, John Terilla

    List of speakers (TBA):

    Daniel Bennequin
    Université Paris Diderot-Paris 7, UFR de Mathématiques, Equipe Géométrie et Dynamique, Paris, France. José Ignacio Burgos Gil
    ICMAT (CSIC), Madrid, Spain Michel Boyom
    Université du Languedoc-Montpellier II, France. Philippe Elbaz-Vincent
    Institut Fourier, Grenoble, France. Tom Leinster
    School of Mathematics, University of Edinburgh, Edinburgh. Matilde Marcolli
    Mathematics Department, Caltech, Pasadena, USA. John Terilla, Queens College, USA.

    A classical result of Cencov [1] in information geometry established that the Fisher-Rao metric is the unique metric (up to a constant) on the space of finite probability densities that is invariant under the action of the diffeomorphism group. This result was extended to infinite sample spaces via n-tensorisation by Ay, Jost, Le and Schwachhofer [2]. Entropy first appeared in the computation of the degree one homology of the discrete group SL2 over C with coefficients in the adjoint action by choosing a pertinent definition of the derivative of the Bloch-Wigner dilogarithm [3]. It could be shown that the functional equation with 5-terms of the dilogarithm implies the functional equation of information with 4-terms. Then it was discovered that a finite truncated version of the logarithm appearing in cyclotomic studies also satisfied the functional equation of entropy, suggesting a higher degree generalisation of information, analog to polylogarithm [4]. This information generalisation was achieved by algebraic means that yet holds over finite fields [5], and further developed into framework where information functions appears as derivation [6]. a more geometric version in terms of algebraic cycles was found [7], introducing the notion of additive dilogarithm with respect to the notion of multiplicative structures [8]. On another side, after that entropy appeared in tropical and idempotent semi-ring analysis in the study of the extension of Witt semiring to the characteristic 1 limit [9], thermodynamic semiring and entropy operad could be constructed as deformation of the tropical semiring [10]. Introducing Rota-Baxter algebras, It allowed to derive a renormalisation procedure. Further completing the entropic landscape in number theory, entropy was encountered as height-degree function in the context of Arakelov geometry while following the program of relating the arithmetic geometry of toric varieties and convex analysis [11]. Defining the category of finite probability and using Fadeev axiomatization, it could be shown that the only family of function which has functorial property is Shannon information loss [12]. Introducing a deformation theoretic framework, and chain complex of random variables, a homotopy probability theory could be constructed for which the cumulants coincide with the morphisms of the homotopy algebras [13]. Using the geometrical and combinatorial structure of probability and random variable seen as partitions, the basement of the cohomology, topos and of a (quasi)-operad of information could be constructed in a probabilistic setting, allowing to retrieve entropy uniquely as the first cohomology group. Mutual-informations arise from the consideration of the trivial action, and appear as differential operator, while their "non-positive" extrema are giving rise to homotopy links [14]. In a series of papers and courses [15], M.Gromov reviews results on the subject and introduces to some geometrical formulation of information inequalities, as well as a complete program in current development rooted on Morse cohomology and homotopy.

    It is appealing and a very good sign that an important part of those developments has been achieved in parallel and independently with very different approaches and methods; the session will be one of the first occasion to gather the actors of the domain, and hence to promote the convergence and discuss open problems. Moreover, following these important extensions of the information and probability framework, It appears natural to ask now for a proper logical foundation of information theory, extending the Boolean world of information digit. Given those developments, motivic integration, topos and homotopy type, that are inherently geometrical logics, provide some candidate for such a program that we propose to discuss.

    [1] Cencov, N.N. Statistical Decision Rules and Optimal Inference. Translations of Mathematical Monographs. 1982.
    [2] Ay, N. and Jost, J. and Lê, H.V. and Schwachhöfer, L. Information geometry and sufficient statistics. Probability Theory and Related Fields 2015 PDF
    [3] Cathelineau, J. Sur l’homologie de sl2 a coefficients dans l’action adjointe, Math. Scand., 63, 51-86, 1988. PDF
    [4] Kontsevitch, M. The 1+1/2 logarithm. Unpublished note, Reproduced in Elbaz-Vincent & Gangl, 2002, 1995 PDF
    [5] Elbaz-Vincent, P., Gangl, H. On poly(ana)logs I., Compositio Mathematica, 130(2), 161-214. 2002. PDF
    [6] Elbaz-Vincent, P., Gangl, H., Finite polylogarithms, their multiple analogues and the Shannon entropy. Vol. 9389 Lecture Notes in Computer Science. 277-285, Archiv.
    [7] Bloch S.; Esnault, H. An additive version of higher Chow groups, Annales Scientifiques de l’École Normale Supérieure. Volume 36, Issue 3, May–June 2003, Pages 463–477. PDF
    [8] Bloch S.; Esnault, H. The Additive Dilogarithm, Documenta Mathematica Extra Volume : in Kazuya Kato’s Fiftieth Birthday., 131-155. 2003. PDF
    [9] Connes, A., Consani, C., Characteristic 1, entropy and the absolute point. preprint arXiv:0911.3537v1. 2009.
    [10] Marcolli, M. & Thorngren, R. Thermodynamic Semirings, arXiv 10.4171 / JNCG/159, Vol. abs/1108.2874, 2011.
    Marcolli, M. & Tedeschi, R. Entropy algebras and Birkhoff factorization, arXiv, Vol. abs/1108.2874, 2014.
    [11] Burgos Gil J.I., Philippon P., Sombra M., Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque 360. 2014 . PDF
    [12] Baez, J.; Fritz, T. & Leinster, T. A Characterization of Entropy in Terms of Information Loss Entropy, 13, 1945-1957, 2011. PDF
    Baez J. C.; Fritz, T. A Bayesian characterization of relative entropy. Theory and Applications of Categories, Vol. 29, No. 16, p. 422-456. 2014. PDF
    [13] Drummond-Cole, G.-C., Park., J.-S., Terrila, J., Homotopy probability theory I. J. Homotopy Relat. Struct. November 2013. PDF
    Drummond-Cole, G.-C., Park., J.-S., Terrila, J., Homotopy probability theory II. J. Homotopy Relat. Struct. April 2014. PDF
    Park., J.-S., Homotopy theory of probability spaces I: classical independence and homotopy Lie algebras. Archiv . 2015
    Tomasic, I., Independence, measure and pseudofinite fields. Selecta Mathematica, 12 271-306. Archiv. 2006.
    [14] Baudot P., Bennequin D. The homological nature of entropy. Entropy, 17, 1-66; 2015. PDF
    [15] Gromov, M. In a Search for a Structure, Part 1: On Entropy, unpublished manuscript, 2013. PDF
    Gromov, M. Symmetry, probability, entropy. Entropy 2015. PDF
    Gromov, M. Morse Spectra, Homology Measures, Spaces of Cycles and Parametric Packing Problems, april 2015. PDF
    [16]McMullen, C.T., Entropy and the clique polynomial, 2013. PDF

  • Geo-Sci-Info

    Organisers: Frédéric Barbaresco, Joël Bensoam

    List of speakers (TBA):

    Frédéric Barbaresco
    Thales Air Systems. Limours, France. Joël Bensoam
    Ircam, centre G. Pompidou, CNRS UMR 9912, Paris, France. François Gay-Balmaz
    LMD - Ecole Normale Supérieure de Paris, Paris, France. Frédéric Hélein
    Institut de Mathématiques de Jussieu, Paris, France. Bernhard Maschke
    Claude Bernard University Lyon 1, LAGEP, Villeurbanne, France.

    This session will explore a unifying framework for Classical/Stochastic Geometric Mechanics and Statistical Physics, based on Symplectic and Multi-Symplectic Geometry and Lie Group Theories. Jean- Marie Souriau has introduced a “Lie Group Thermodynamics” in Statistical Physics and “Continuous Medium Thermodynamics” that could beneficiate of most recent works from Geometric Mechanics and its most recent stochastic developments. In particular, the conservation laws (or more generally the circulation invariants: Noether’s currents) that derive from the underlying symmetries of Lie groups allow reducing the Partial Differential Equations (PDEs) problems. Other approaches will be considered as Euler-Poincaré reduction theorem for stochastic processes taking values in a Lie group, which is a generalization of the Lagrangian version of reduction and its associated variational principles. The objective is to embrace common principles for all the changes of state of the body , both the change of place and the change of physical qualities extended to the case of incomplete knowledge and random uncertainties.

  • Geo-Sci-Info

    Organiser: Dominique Spehner
    List of Speakers :

    Madalin Guta, University of Nottingham, UK. Dominique Spehner, Institut Fourier, Saint Martin d'Hères, France. Karol Zyczkowski, Jagiellonian University, Cracow, Poland.

    One of the central questions in quantum information theory is to characterize quantum correlations in composite quantum systems. The latter play the role of resources in quantum information processing tasks and are believed to be at the origin of the higher efficiencies with respect to classical information processors.
    A promising approach to the characterization and quantification of these correlations has emerged in recent years. It is based on Riemannian geometries on the convex cone of quantum states. In particular, the amount of quantum correlations in a given state has been quantified by the distance between this state and a distinguished subset of states, formed e.g. by all disentangled states or by all classical states. The properties and examples of Riemannian metrics such that good measures of entanglement and quantum discord are obtained, have been studied. The explicit determination of these geometric measures in concrete physical systems has attracted a lot of attention in the physics community. The aim of the conference is to try to gather together the various attempts proposed so far within a single general geometric framework, and to discuss its link with the more traditional entropic approach.

  • Geo-Sci-Info

    Organisers: Carlo Rovelli

    Speakers :

    Livine Etera, Laboratoire de Physique, ENS Lyon, Lyon, France. Antonino Marcianò, Fudan University, China. Carlo Rovelli, Centre de Physique Theorique de Luminy, Marseille, France.

    The objective of the session is to make the point on the way geometry and quantum correlations are related in quantum gravity. The focus would be on the recent developments in the possibility of describing quantum states of the geometry with long distance correlations.

  • Stéphanie Allasonnière, Xavier Pennec



    Stéphanie Allasonnière (School of medicine, Paris Descartes university, France) Xavier Pennec (INRIA sophia, France)


    Xavier Pennec (INRIA sophia, France) Alain Trouvé (ENS Cachan, France)


    Marc Arnaudon (IMB, France) Aasa Feragen (DIKU, Denmark) Stanley Durrleman (ARAMIS lab, France) Ian Dryden (University of Nottingham, UK) Alice Le Brigant (IMB, Université de Bordeaux, France)

    This session presents recent progresses in geometric statistics. In many applications domains such as computational anatomy, computer vision, structural biology, computational phylogenetics, etc one models data as elements of a manifold which is quotiented by a proper and isometric Lie group action (a shape space). For instance, Kendall shape spaces encode the invariants of the configuration of a fixed number of points under the action of Euclidean or similarity transformations. One can also construct (infinite dimensional) spaces of curves or surfaces by quotienting out the parametrization. Shape spaces are stratified spaces whose manifold part is Riemannian. Looking for summary, explanatory or discriminative statistics on such measurements is first complicated by the presence of the curvature. The theory of statistics on Riemannian manifolds now begins to be powerful enough to provide simple notions (mean values, moments) and some (limited) central limit theorems, although the role of the curvature is not fully elucidated. However, subspace estimation methods going beyond the mean value have only recently emerged. A second complication arise in quotient spaces when data are scattered over several regular strata or close to singular points of the stratification. Other differential geometric structures such as affine connection spaces or fiber spaces raise new questions for the generalization of statistical notions and estimation methods.

  • Geo-Sci-Info

    Organisers: Nihat Ay, František Matúš

    Speakers :

    Nihat Ay
    Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. Tobias Fritz
    Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada. Luigi Malago
    Romanian Institute of Science and Technology - RIST, Romania. František Matúš
    Institute of Information Theory and Automation, Academy of Sciences,Prague, Czech Republic. Guido Montúfar
    Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. Johannes Rauh
    Leibniz Universität Hannover, Institut für Algebraische Geometrie, Hannover, Germany. Milan Studený
    Institute of Information Theory and Automation, Academy of Sciences, Prague, Czech Republic.

    Abstract: This session will review the role in network analysis within the fields of artificial intelligence and machine learning, of geometric objects defined in terms of information equalities as well as information inequalities.

  • Geo-Sci-Info

    The poster sessions will be held every afternoon after the sessions and after the diners in the Lounge room, such that discussions are encouraged:

    Goffredo Chirco (Albert Einstein Institute Potsdam, Germany) We study the extension of Souriau's formalism of Lie group thermodynamics to the case of reparametrization invariant systems, in the perspective of a general covariant reformulation of statistical mechanics compatible with the conceptual scheme of General Relativity. We explore the extension of such a formalism in the quantum regime, in the context of the group field theory approach to quantum gravity.

    Maël Dugast (Univ Lyon, UJM-Saint-Etienne, INSA Lyon), Guillaume Bouleux and Eric Marcon. The geometry of periodically correlated stochastic processes induced by the dilation. We investigate hidden structure of non-stationary signals, and describes them in a geometrical frame- work. Using the specific structure of positive-definite matrices, that arise as correlation matrices, we arrive to some dilation type results. The dilation theory states that the coefficients of a positive-definite matrice can be expressed in term of rotation matrices. These dilations matrices are composed by partial correlation coefficients, and not by the correlations coefficients. Therefore, we can map the set of correlation matrices to the Lie group which consist of rotation matrices, namely the special orthogonal or unitary group. In this settings, a stochastic process can be represented in SO(n), and methods such as classification can be reformulated in SO(n). The main strength in obtaining dilation matrices resides in its stationarity independence assumption. When dealing with stationary processes, the dilation result is the so-called Naimark dilation and a single dilation matrix is obtained. When the process is non-stationary, the dilation leads to the Kolmogorov decomposition, and a set of dilation matrices is provided. An interesting case is when the process is periodically correlated. In this case, we can show that only a finite number of dilation matrices is required to represent the whole process. Features of the underlying process can then be explored through these dilation matrices. Typically, in non-stationary operations, the dilation matrices are time ordered and we can interpolate a trajectory between them, using splines. For periodically correlated processes, this trajectory is a closed loop in the Lie group SO(n), and can be characterized by its length, and other geometrical features, like the torsion of the curve. The idea is to show that a higher level of variability in the signal is associated to a more erratic curve in the associated Lie group.

    Simon Fong (School of Computer Science, The University of Birmingham, UK) Joshua Knowles, and Peter Tiňo. Induced Dualistic Structure and Probability Densities on Riemannian Manifolds. Stochastic optimization algorithms over Euclidean spaces has been studied in the framework of Information Geometry [1, 4]. In this work we aim to establish a foundation for computable stochastic optimization algorithms on Riemannian manifolds M by analysing induced pobability densities over M . In particular we focus on the relationship between the manifold M of the search space and the manifold of probability densities, by constructing a locally dually flat family of probability distributions on Riemannian manifolds. This relates the notions of volume elements on manifolds, dually flat Hessian Riemannian statistical manifolds [7], and probability distributions on a measurable space.
    We extend the construction of Pennec [5] and Said et al. [6] to density bundle over M and to dually flat families of probability densities. We begin with dually flat family of probability densities supported on the closed ball of injectivity radius in tangent spaces of M . Amari [2] showed that such a family of probability densities supported on a subset of a topological vector space is a Riemannian manifold equipped with Fisher-Rao metric. A family of probability densities over M can thus be defined locally via the pullback line bundle induced by the inverse of Riemannian exponential map, which then allows us to construct a globally supported family of probability densities over M as mixture densities using partition of unity. The pulled-back family of densities locally satisfies Amari’s classical model of statistical manifolds [2], while it also satisfies globally the conditions of k- integrable parametrized measure models by Ay et al. [3] when M has finite Riemannian volume. Finally we also discuss the pullback dualistic structure between Hessian Riemannian manifolds. This allows us to analyse and compute induced probability densities on Riemannian manifolds explicitly, and study convergence results in stochastic optimization algorithms via the induced dualistic structure.
    The contents of this abstract have been included in an extended paper submitted to
    GSI2017 (currently under review): Induced Dualistic Structure and Locally Dually Flat Densities on Riemannian Manifolds.
    [1] Akimoto, Y., Nagata, Y., Ono, I., Kobayashi, S.: Theoretical foundation for CMA-ES from information geometry perspective. Algorithmica 64(4), 698–716 (2012)
    [2] Amari, S., Nagaoka, H.: Methods of Information Geometry, Translations of Mathematical monographs, vol. 191. Oxford University Press (2000)
    [3] Ay, N., Jost, J., Vân Lê, H., Schwachhöfer, L.: Information geometry and sufficient statistics. Probability Theory and Related Fields 162(1–2), 327– 364 (2015)
    [4] Malagò, L., Matteucci, M., Pistone, G.: Towards the geometry of estimation of distribution algorithms based on the exponential family. In: Proceedings of the 11th Workshop Proceedings on Foundations of Genetic Algorithms. pp. 230–242. ACM (2011)
    [5] Pennec, X.: Probabilities and statistics on Riemannian manifolds: A geometric approach. Ph.D. thesis, INRIA (2004)
    [6] Said, S., Bombrun, L., Berthoumieu, Y., Manton, J.H.: Riemannian Gaussian distributions on the space of symmetric positive definite matrices. IEEE Transactions on Information Theory 63(4), 2153–2170 (2017)
    [7] Shima, H., Yagi, K.: Geometry of Hessian manifolds. Differential Geometry and its Applications 7(3), 277–290 (1997)

    Antonia M Frassino (Frankfurt Institute for Advanced Studies, Germany) and Dimitri Marinelli (Romanian Institute of Science and Technology): Deep learning and the universal quantum simulator, as meant by Feynmann, are machines able to build representations and can be used as efficient generative models. The representational power of the deep neural networks has been experimentally verified while its theoretical reasons behind their power are still unclear. The generative power of the universal quantum simulators has been proved rigorously, while their actual construction is a topic of active research in Physics. Quantum states and the relative entropy (KL-divergence) allow unveiling the geometrical structure of the two computational models and how information shapes their representation space. Here we introduce the two systems, addressing some problems and the open questions. In particular, we focus on an example presented in the literature that shows the representational power of neural network for quantum systems. The purpose of the analysis is to study how information geometry models their links with Nature.

    Edgar Guzman-Gonzalez (Institute of Nuclear Sciences, UNAM, Mexico) Given a spin j, we define the spin shape space as the quotient of the space of the spin space under the action of the rotation group SO(3). This permits us to decompose the spin space as a fiber bundle, where the base is shape space an the fiber is generically isomorphic to SO(3). Using the Fubini-Study metric, we can define a connection and a metric in the fibers and in shape space in a natural way. Here we will present some applications of this decomposition to quantum mechanics, particularly, to the Berry's curvature, along with some mathematical results of the induced geometry in spin shape space.

    Hideyuki Ishi (Nagoya University, Japan) We present a new class of convex cones consisting of real symmetric matrices. The class contains both affine homogeneous cones and cones corresponding to decomposable graphs. The affine homogeneous cone is a natural generalization of a symmetric cone. Harmonic analysis and differential geometry on the cone has been developed over the homogeneous cones, and the results can be applied to information geometry, convex programming, and statistics. On the other hand, the convex cones of positive definite matrices with prescribed zeros are intensively studied in the so-called graphical model theory, where the location of zeros ared assigned by a simple graph. If the graph is decomposable (chordal), analysis over the corresponding cone becomes quite feasible. In particular, the cone admits some analytic formulas which are quite similar to the ones for homogeneous cones. In our previous works, a new method for the study of homogeneous cone is established, where the cone is realized as the set of positive definite symmetric matrices with specific block decomposition. Extending this method, we now succeed to find a new class of convex cones admitting matrix realizations, and as is stated at the beginning, the class includes the cones corresponding to decomposable graphs. Then we can discuss the formulas mentioned above in a unified way. In particular, we have explicit formulas for Siegel-type integrals as well as the Koszul-Vinberg characteristic function of the cone in this class. Moreover, we introduce a family of Hessian metrics over the cone with explicit Legendre transforms. These results will be applied to statistics and information geometry.

    Marco Laudato (University of Naples Federico II) Quantum and Tomographic Metrics from Relative Entropies. Starting form the relative Tsallis q-entropy as a potential function for the metric, we derive a one parameter family of quantum metrics for N-level systems and analyze in detail the cases N = 2,3. By using a two-parameters generalization of the Tsallis relative entropy, we derive a general expression for quantum metrics depending on the parameters of the entropy. Then we construct explicitly the Fisher-Rao tomographic metric for qubit and qutrit states in different reference frames on the Hilbert space of quantum states. We thus address the problem of reconstructing quantum metrics from tomographic ones, in relation to the uniqueness properties of the latter. Finally, we show that there exists a bijective relation between the choice of the tomographic scheme and the particular operator monotone function identifying a unique quantum metric tensor.

    Leo Liberti (CNRS LIX, Ecole Polytechnique, France) Random projections in optimization. The well-known Johnson-Lindenstrauss lemma (JLL) shows that Euclidean distances can be preserved with low distortion via random projections. We show that such projections also preserve many other quantities, such as angles, separating hyperplanes, and distances to cones and convex sets. This makes it possible to approximately solve very large linear programs with a much smaller number of constraints (or variables). Some of our work in progress aims at solving harder problems, such as trust region subproblems and integer linear programs. Ultimately, we hope to show that the practical application of JLL-type random projections will exceed the realm of algorithms which are purely based on Euclidean distances, such as k-means or k-nearest neighbors.

    Anton Mallasto (University of Copenhagen, Danemark) : We introduce a novel framework for statistical analysis of populations of non-degenerate Gaussian processes (GPs), which are natural representations of uncertain curves. This allows inherent variation or uncertainty in function-valued data to be properly incorporated in the population analysis. Using the 2-Wasserstein metric we geometrize the space of GPs with L^2 mean and covariance functions over compact index spaces. We present results on existence and uniqueness of the barycenter of a population of GPs, as well as convergence of the metric and the barycenter of their finite-dimensional counterparts. This justifies practical computations. Finally, we demonstrate our framework through experimental validation on GP datasets representing brain connectivity and climate development.

    Fabio Maria Mele (Naples University Federico II, Italy) Fisher Metric, Geometric Entanglement and Spin Networks. Motivated by the idea that, in the background-independent framework of a quantum theory of gravity, entanglement is expected to play a key role in the reconstruction of spacetime geometry, we introduce the geometric formulation of Quantum Mechanics in the quantum gravity context, and we use it to give a tensorial characterization of entanglement on spin network states of quantum geometry. Starting from the simplest case of a single-link graph (Wilson line), we define a dictionary to construct a Riemannian metric tensor and a symplectic structure on the space of spin network states. The manifold of (pure) quantum states is then stratified in terms of orbits of equally entangled states and the block-coefficient matrices of the corresponding pulled-back tensors fully encode the information about separability and entanglement. In particular, the off-diagonal blocks define an entanglement monotone interpreted as a distance with respect to the separable state. In the maximally entangled gauge-invariant case, the entanglement monotone is proportional to a power of the area of the surface dual to the link thus supporting a connection between entanglement and the (simplicial) geometric properties of spin network states. We finally extend then such analysis to the study of non-local correlations between two non-adjacent regions of a generic spin network. Our analysis shows that the same spin network graph can be understood as an information graph whose connectivity encodes, both at the local and non-local level, the quantum correlations among its parts. This gives a further connection between entanglement and geometry.

    Hiroshi Matsuzoe (Nagoya Institute of Technology, Japan) : A survey on infinite dimensional affine differential geometry and information geometry. The main object of affine differential geometry is to study hypersurfaces or immersions that are affinely congruent in an affine space. It is known that dual affine connections and statistical manifold structures naturally arise in this framework. For example, a statistical manifold structure of an exponential family is realized by an affine hypersurface immersion, and the Kullback-Leibler divergence coincides with the affine support function. The Legendre transformation can be discussed by the codimension two affine immersion of a special kind. Therefore, the geometry of dually flat spaces can be generalized by affine differential geometry. In this presentation, we would like to give a short survey about the relations between the infinite dimensional framework of affine differential geometry and information geometry. In particular, we would like to discuss the infinite dimensional affine differential geometry of the maximal exponential families and the alpha-families.

    Stéphane Puechmorel (Ecole nationale de l'aviation civile, France) Natural metrics on the tangent bundle to a statistical manifold. The Fischer information metric provides statistical manifolds with a riemanian structure and has been intensively studied in information geometry. The tangent bundle TM of a statistical manifold M is itself a manifold, and one may ask about metrics that can be defined on it. Generally speaking, a connection D on TM can be used to split the tangent space to TM into an horizontal HTM and a vertical VTM distribution, with TTM=HTM+VTM. Any vector field X on TM admits an horizontal (resp. vertical) lift Xh (resp. Xv) on HTM (resp. VTM). Assuming D to be the levi-civita connection with respect to the metric g on M, natural metrics G on TM are those satisfying G(Xh,Yh) = g(X,Y), G(Xh,Yv) = 0 for abritratry vector fields X,Y on M. Special cases are the Sasaki metric, for which G(Xv,Yv) = g(X,Y) and the Cheeger-Gromoll metric. In the context of statistical manifolds, the Levi-Civita connection is the 0-connection in the so-called family of alpha-connections of Amari. The first part of this work is devoted to the study of its associated Sasaki and Cheeger-Gromoll metrics, and their relations with derivatives of the score functions. In a second part, the question of general a-connections is discussed. In particular, e and m connections, which are non-metric but of major importance in information geometry, cannot be directly associated with a natural metric on TM due to the requirement G(Xh,Yv) = 0 that makes sense only with the Levi-Civita connection (orthogonality relation). Adapted metrics on TM will thus be considered, which fit within the frame of alpha-connections. Finally, a statistical interpretation will be given for the objects introduced.

    Ha Quang Minh (Istituto Italiano di Tecnologia, Italy) Infinite-dimensional Log-determinant Divergences between positive definite trace class operators. We present a novel parametrized family of Alpha Log-Determinant (Log-Det) divergences between positive definite unitized trace class operators on an infinite-dimensional Hilbert space. This is a generalization of the Alpha Log-Det diver-
    gences between symmetric, positive definite (SPD) matrices [1] to the infinite-dimensional setting. The generalization is carried out via the introduction of the extended Fredholm determinant for unitized trace class operators and the accompanying generalization of the log-concavity for determinants of SPD matrices. For the Log-Det divergences between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), we obtain closed form formulas via the corresponding Gram matrices. By employing the Log-Det divergences between positive definite unitized trace class operators, we then generalize the Bhattacharyya and Hellinger distances and the Kullback-Leibler and Rényi divergences between multivariate normal distributions to Gaussian measures on an infinite-dimensional Hilbert space. The current infinite-dimensional Log-Det formulation is appearing in [3]. It has recently been generalized to [4], which is the infinite-dimensional generalization of the Alpha-Beta Log-Det divergences between SPD matrices [2].
    [1] Z. Chebbi and M. Moakher. Means of Hermitian positive-definite matrices based on the Log-Determinant α-divergence function. Linear Algebra and its Applications, 436(7):1872–1889, 2012.
    [2] A. Cichocki, S. Cruces, and S. Amari. Log-Determinant divergences revisited: Alpha-Beta and
    Gamma Log-Det divergences. Entropy, 17(5):2988–3034, 2015.
    [3] H.Q. Minh. Infinite-dimensional Log-Determinant divergences between positive definite trace class operators. Linear Algebra and Its Applications, In Press, 2016.
    [4] H.Q. Minh. Infinite-dimensional Log-Determinant divergences II: Alpha-Beta divergences. arXiv preprint arXiv:1610.08087v2, 2016. Eduardo Serrano-Ensástiga (Institute for Nuclear Sciences, UNAM, Mexico) Study of the FS metric with the Majorana's stellar representation. The action of the group SO(3) in the Hilbert space of spin j H allows a principal fiber bundle structure, and split the tangent space in each point in two vectorial subspaces, the vectical and horizontal space, where the horizontal vectors are defined as the ortogonal vectors of the vertical ones with the FS metric. By the other hand, the Majorana’s stellar representation gives us a geometric view of H and its tangent bundle TH in terms of moving points on the sphere. In this representation, the vertical vectors have a clear interpretation, but the horizontal vectors don't. We expose some properties of the FS metric, the characteristics of the vertical vectors and a study of the horizontality condition in terms of the stellar representation.


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