Keynote Speakers
Invited talks will be scheduled with 3 Keynote speakers that will be invited for plenary sessions:
 Yann OLLIVIER (ParisSud Univ.): “Informationgeometric 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, ParisSud University, France Informationgeometric 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 (informationgeometric 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 KullbackLeibler 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 

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 ParisSud 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 


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 1form α such that Dα 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, nonassociative 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; (D_{X} g)(Y, Z) = (D_{Y} 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;(D_{X} g)(Y, Z) = (D_{Y} g)(X, Z) [Del]. For a Codazzi structure (D, g) we can define a new torsionfree connection D' by Xg(Y, Z) = g(D_{X}Y, Z) + g(Y, D'_{X}Z). Then we have D' = 2∇  D where ∇ is the LeviCivita 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 nondegenerate 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 

Biography 
Emeritus Professor of Yamaguchi University 


Giovanni Pistone, Collegio Carlo Alberto, Italy Nonparametric Information Geometry 

Abstract 
The differentialgeometric 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 [15] 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 [78] 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 [910]. 
References 

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. 
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