Keynote Speakers

Invited talks will be scheduled with 3 Keynote speakers that will be invited for plenary sessions:

  • Yann OLLIVIER (Paris-Sud Univ.): “Information-geometric optimization: The interest of information theory for discrete and continuous optimization”
  • Hirohiko SHIMA (Yamaguchi Univ.): “Geometry of Hessian Structures” dedicated to Prof. J.L. KOSZUL
  • Giovanni PISTONE (Collegio Carlo Alberto): “Nonparametric Information Geometry”

Yann Ollivier, Paris-Sud University, France

 Information-geometric optimization: The interest of information theory for discrete and continuous optimization

Abstract

Black box optimization is the problem of searching for the minimum of a function on a given space (discrete or continuous), without any prior knowledge about the function. Information geometry provides a systematic method, IGO (information-geometric optimization) to easily build optimization algorithms having nice properties; in particular it minimizes the influence of arbitrary choices such as how the space of solutions is represented. In some situations IGO recovers known and widely used algorithms, thus providing theoretical justification for them.

Specific properties of information geometry and the Kullback-Leibler divergence guarantee, at each step, minimal diversity loss in the exploration of possible solutions; this suggests IGO algorithms automatically tune the simultaneous exploration of different regions.

References
  • L. Arnold, A. Auger, N. Hansen, Y. Ollivier, Information-geometric optimization: A unifying picture via invariance principles, preprint, arXiv:1106.3708 Y. Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct.  Anal. 256 (2009), no. 3, 810-864.
  • A. Joulin, Y. Ollivier, Curvature, concentration, and error estimates for Markov chain Monte Carlo, Ann. Probab. 38 (2010), no. 6, 2418-2442.
  • Y. Ollivier, A January 2005 Invitation to Random Groups, Ensaios Matemáticos 10, Sociedade Brasileira de Matemática, Rio de Janeiro (2005).
  • Y. Ollivier, Sharp phase transition theorems for hyperbolicity of random groups, GAFA, Geom. Funct. Anal. 14 (2004), no. 3, 595-679.
  • C. Chevalier, F. Debbasch, Y. Ollivier, Multiscale cosmological dynamics, Physica A 388 (2009), 5029-5035.
  • Y. Ollivier, P. Senellart, Finding related pages using Green measures: An illustration with Wikipedia, Proc. of the Twenty-Second Conference on Artificial Intelligence (AAAI 2007), 1427-1433.
Biography

Yann's research generally focuses on the introduction of probabilistic models on structured objects, and more particularly addresses the interplay between probability and differential geometry.

He is currently Research scientist at the CNRS, currently in the Computer Science department at Paris-Sud Orsay University, previously in the Mathematics department at the École Normale Supérieure in Lyon (2004–2010).

He graduated to his PhD in Mathematics, under the supervision of M. Gromov and P. Pansu in 2003 and is accredited to supervise research since 2009

http://www.yann-ollivier.org/rech/index 


Hirohiko Shima, Yamaguchi University, Japan

Geometry of Hessian Structures

Abstract

A Riemannian metric g on a flat manifold M with flat connection D is called a Hessian metric if it is locally expressed by the Hessian of local functions φ with respect to the affine coordinate systems, that is, g = Ddφ. Such pair (D, g), g, and M are called a Hessian structure, a Hessian metric, and a Hessian manifold, respectively [S7]. Typical examples of these manifolds include homogeneous regular convex cones [V] (e.g. the space of all positive definite real symmetric matrices). J.L. Koszul studied a flat manifold endowed with a closed 1-form α  such that  is positive definite [K1][K2]. Then g = Dα is exactly a Hessian metric. Hence this is the ultimate origin of the notion of Hessian structures. On the other hand, a Riemannian metric on a complex manifold is said to be a Kählerian metric if it is locally expressed by the complex Hessian of functions with respect to holomorphic coordinate systems. For this reason S.Y. Cheng and S.T. Yau called Hessian metrics affine Kähler metrics [CY]. These two types of metrics are not only formally similar, but also intimately related. In fact, the tangent bundle of a Hessian manifold is a Kählerian manifold. Hessian geometry (the geometry of Hessian structures) is thus a very close relative of Kählerian geometry, and may be placed among, and finds connection with important pure mathematical fields such as affine differential geometry, homogeneous spaces, cohomology, non-associative algebras (e.g. left symmetric algebras, Jordan algebras) and others. Moreover, Hessian geometry, as well as being connected with these pure mathematical areas, also, perhaps surprisingly, finds deep connections with information geometry. The notion of flat dual connections, which plays an important role in information geometry, appears in precisely the same way for our Hessian structures [A][AN]. Thus Hessian geometry offers both an interesting and fruitful area of research.

A Hessian structure is characterized by the Codazzi equation; (DX g)(Y, Z) = (DY g)(X, Z). Using this equation the notion of Hessian structure is easily generalized as follows. A pair (D, g) of a torsion free connection D and a Riemannian metric g on M is called a Codazzi structure if it satisfies the Codazzi equation;(DX g)(Y, Z) = (DY g)(X, Z) [Del]. For a Codazzi structure (D, g) we can define a new torsion-free connection D' by Xg(Y, Z) = g(DXY, Z) + g(Y, D'XZ). Then we have D' = 2 -  D where   is the Levi-Civita connection of g. The pair (D', g) is also a Codazzi structure. The connection D' and the pair (D', g) are called the dual connection of D and the dual Codazzi structure of (D, g), respectively.

Historically, the notion of dual connections was obtained by quite distinct approaches. In affine differential geometry the notion of dual connections was naturally obtained by considering a pair of a non-degenerate affine hypersurface immersion and its conormal immersion [NS]. In contrast, S. Amari and H. Nagaoka found that smooth families of probability distributions admit dual connections as their natural geometric structures. Information geometry aims to study information theory from the viewpoint of the dual connections [A][AN].

References
  • [A] Amari, S. (1985). Differential-geometrical methods in statistics, Springer Lecture Notes in Statistics.
  • [AN] Amari, S. and Nagaoka, H (2000). Methods of information geometry, Translation of Mathematical Mono-graphs, AMS, Oxford, Univ. Press.
  • [CY] Cheng, S. Y and Yau, S. T. (1982). The real Monge-Ampère equation and affine flat structures, Proc. the 1980 Beijing symposium of differential geometry and differential equations, Science Press, Beijing, hina, Gordon and Breach, Science Publishers, Inc., New York, pp. 339-370.
  • [Del] Delanoë, P. (1989). Remarques sur les variétés localement hessiennes, Osaka J. Math., pp. 65-69.
  • [K1] Koszul, J. L. (1961). Domaines bornés homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89, pp. 515-533.
  • [K2] Koszul, J. L. (1965). Variétés localement plates et convexité, Osaka J. Maht. 2, pp. 285-290.
  • [NS] Nomizu, K. and Sasaki, T. (1994). Affine Differential Geometry, Cambridge Univ. Press.
  • [S1] Shima, H. (1977). Symmetric spaces with invariant locally Hessian structures, J. Math. Soc. Japan,, pp. 581-589.
  • [S2] Shima, H. (1980). Homogeneous Hessian manifolds, Ann. Inst. Fourier, Grenoble, pp. 91-128.
  • [S3] Shima, H. (1986). Vanishing theorems for compact Hessian manifolds, Ann. Inst. Fourier, Grenoble, pp.183-205.
  • [S4] Shima, H. (1995). Harmonicity of gradient mappings of level surfaces in a real affine space, Geometriae Dedicata, pp. 177-184.
  • [S5] Shima, H. (1995). Hessian manifolds of constant Hessian sectional curvature, J. Math. Soc. Japan, pp. 735-753.
  • [S6] Shima, H. (1999). Homogeneous spaces with invariant projectively flat affine connections, Trans. Amer. Math. Soc., pp. 4713-4726.
  • [S7] Shima, H. (2007). The Geometry of Hessian Structures, World Scientific.
  • [V] Vinberg, E. B. (1963). The Theory of convex homogeneous cones, Trans. Moscow Math. Soc., pp. 340-403.
Biography

Emeritus Professor of Yamaguchi University
Degree of PhD (Osaka University 1970)
Osaka University (1966-1970)
Yamaguchi University (from 1970 to the present)

http://www.worldscientific.com/worldscibooks/10.1142/6241


Giovanni Pistone, Collegio Carlo Alberto, Italy

Nonparametric Information Geometry

Abstract

The differential-geometric structure of the set of positive densities on a given measure space has raised the interest of many mathematicians after the discovery by CR Rao of the geometric meaning of the Fisher information. Most of the research is focused on parametric statistical models. In series of papers [1-5] a particular version of the nonparametric case has been discussed. This minimalistic structure is modeled according the theory of exponential families: given a reference density other densities are represented by the centered log likelihood which is an element of an Orlicz space. This mappings give a system of charts of a Banach manifold. It has been observed that, while the construction is natural, the practical applicability is limited by the technical difficulty to deal with such a class of Banach spaces. It has been suggested recently [7-8] to  replace the exponential function with other functions with similar behavior but polynomial growth at infinity in order to obtain more tractable Banach spaces, e.g. Hilbert spaces. The aim of the talk is to present a state of the art of this issue and to discuss its connection with other approaches [9-10].

References
  • [1] G.Pistone and C. Sempi, An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one, The Annals of Statistics (1995) 1543-1561.
  • [2] P. Gibilisco, G. Pistone, Connections on non-parametric statistical manifolds by Orlicz space geometry, Infinite Dimensional Analysis, Quantum Probability and Related Topics (1998) 325-345.
  • [3] G. Pistone and M.-P. Rogantin, The exponential statistical manifold: mean parameters, orthogonality and space transformations, Bernoulli (1999) 721-760.
  • [4] A. Cena and G. Pistone, Exponential statistical manifold, Annals of the Institute of Statistical Mathematics (2007) 27-57.
  • [5] M. R. Grasselli, Dual connections in nonparametric classical information geometry, Ann Inst Stat Math (2010) 873-896.
  • [6] G. Pistone, K-exponential models from the geometrical viewpoint, Eur. Phys. J. B  (2009) 29-37.
  • [7] R.F. Vigelis and C.C. Cavalcante, On Phi-Families of Probability Distributions, Journal of Theoretical Probability (2011) in press.
  • [8] N.N. Newton, An infinite-dimensional statistical manifold modelled on Hilbert space, Journal of Functional Analysis (2012) in press.
  • [9] P. Gibilisco, E. Riccomagno, M.-P. Rogantin, H. Wynn, Algebraic and Geometric Methods in Statistics (2010) Cambridge University Press.
  • [10] N. Ay, J. Jost, Hông Vân Lê, and L. Schwachhöfer, Information geometry and sufficient statistics, arXiv:1207.6736.
Biography

Giovanni Pistone has been professor of Probability of the Politecnico di Torino to the year 2009 when he retired. Previously he was professor at the Università di Genova, where he served as Head of the Department of Mathematics. He obtained his Master degree from the Università di Torino in 1969, and the degree "docteur de 3me cycle" from the Universitè de Rennes (France) in 1975. Contributions to Probability and Mathematical Statistics cover various topics, e.g. Stochastic Partial Differential Equations, Industrial Statistics, Information Geometry, Algebraic Statistics. Currently he is affiliate of the de Castro Statistics Initiative of the Collegio Carlo Alberto, Moncalieri, Italy.

http://www.giannidiorestino.it