Information geometry of the space of probability measures and barycenter maps

2021 ◽  
Vol 34 (2) ◽  
pp. 231-253
Author(s):  
Mitsuhiro Itoh ◽  
Hiroyasu Satoh
Keyword(s):  
Information Geometry ◽  
Hadamard Manifold ◽  
Probability Measures ◽  
Ideal Boundary ◽  
Recent Developments ◽  
Fisher Metric ◽  
The Ideal

In this article, we present recent developments of information geometry, namely, geometry of the Fisher metric, dualistic structures, and divergences on the space of probability measures, particularly the theory of geodesics of the Fisher metric. Moreover, we consider several facts concerning the barycenter of probability measures on the ideal boundary of a Hadamard manifold from a viewpoint of the information geometry.

Information geometry of Busemann-barycenter for probability measures

2015 ◽  
Vol 26 (06) ◽  
pp. 1541007 ◽  
Author(s):  
Mitsuhiro Itoh ◽  
Hiroyasu Satoh
Keyword(s):  
Natural Extension ◽  
Information Geometry ◽  
Acta Math ◽  
Space Structure ◽  
Hadamard Manifold ◽  
Probability Measures ◽  
Positive Density ◽  
Ideal Boundary ◽  
Push Forward ◽  
The Ideal

Using Busemann function of an Hadamard manifold X we define the barycenter map from the space 𝒫+(∂X, dθ) of probability measures having positive density on the ideal boundary ∂X to X. The space 𝒫+(∂X, dθ) admits geometrically a fiber space structure over X from Fisher information geometry. Following the arguments in [E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math.157 (1986) 23–48; G. Besson, G. Courtois and S. Gallot, Entropies et rigidités des espaces localement symétriques de coubure strictement négative, Geom. Funct. Anal.5 (1995) 731–799; Minimal entropy and Mostow's rigidity theorems, Ergodic Theory Dynam. Systems16 (1996) 623–649], we exhibit that under certain geometrical hypotheses a homeomorphism Φ of the ideal boundary ∂X induces, by the aid of push-forward, an isometry of X whose extension is Φ.

Time-preserving conjugacies of geodesic flows

1992 ◽  
Vol 12 (1) ◽  
pp. 67-74 ◽  
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Sufficient Conditions ◽  
Universal Covering ◽  
Probability Measures ◽  
Geodesic Flows ◽  
Ideal Boundary ◽  
Negatively Curved ◽  
Unit Tangent Bundle ◽  
Borel Probability ◽  
Necessary And Sufficient ◽  
The Ideal

AbstractIn this note we study Borel-probability measures on the unit tangent bundle ofa compact negatively curved manifold M that are invariant under the geodesic flow. We interpret the entropy of such a measure as a Hausdorff dimension with respect to a natural family of distances on the ideal boundary of the universal covering of M. This in term yields necessary and sufficient conditions for the existence of time preserving conjugacies of geodesic flows.

scholarly journals The ideal boundaries and global geometric properties of complete open surfaces

1990 ◽  
Vol 120 ◽  
pp. 181-204 ◽  
Author(s):  
Takashi Shioya
Keyword(s):  
Equivalence Relation ◽  
Focal Point ◽  
Asymptotic Relation ◽  
Total Curvature ◽  
Equivalence Classes ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Ideal Boundary ◽  
Geometric Properties ◽  
The Ideal

In this paper we study the ideal boundaries of surfaces admitting total curvature as a continuation of [Sy2] and [Sy3]. The ideal boundary of an Hadamard manifold is defined to be the equivalence classes of rays. This equivalence relation is the asymptotic relation of rays, defined by Busemann [Bu]. The asymptotic relation is not symmetric in general. However in Hadamard manifolds this becomes symmetric. Here it is essential that the manifolds are focal point free.

scholarly journals A Riemann on the ideal boundary of a Riemann surface

1956 ◽  
Vol 32 (6) ◽  
pp. 409-411 ◽  
Author(s):  
Shin'ichi Mori ◽  
Minoru Ota
Keyword(s):  
Riemann Surface ◽  
Ideal Boundary ◽  
The Ideal

The parabolicity of Brelot's harmonic spaces

1995 ◽  
Vol 59 (1) ◽  
pp. 20-27
Author(s):  
Hideo Imai
Keyword(s):  
Harmonic Function ◽  
Ideal Boundary ◽  
Minimal Growth ◽  
Positive Harmonic Function ◽  
The Ideal

AbstractThe parabolicity of Brelot's harmonic spaces is characterized by the fact that every positive harmonic function is of minimal growth at the ideal boundary.

scholarly journals Prospects for the development of fungal vaccines.

1997 ◽  
Vol 10 (4) ◽  
pp. 585-596 ◽  
Author(s):  
G S Deepe
Keyword(s):  
Vaccine Development ◽  
Fungal Infections ◽  
Low Cost ◽  
Mode Of Delivery ◽  
Cost Effective ◽  
Fungal Diseases ◽  
Haemophilus Influenzae Type ◽  
Medical Problems ◽  
Recent Developments ◽  
The Ideal

In an era that emphasizes the term "cost-effective," vaccines are the ideal solution to preventing disease at a relatively low cost to society. Much of the previous emphasis has been on childhood scourges such as measles, mumps, rubella, poliomyelitis, and Haemophilus influenzae type b. The concept of vaccines for fungal diseases has had less impact because of the perceived limited problem. However, fungal diseases have become increasingly appreciated as serious medical problems that require recognition and aggressive management. The escalation in the incidence and prevalence of infection has prompted a renewed interest in vaccine development. Herein, I discuss the most recent developments in the search for vaccines to combat fungal infections. Investigators have discovered several inert substances from various fungi that can mediate protection in animal models. The next challenge will be to find the suitable mode of delivery for these immunogens.

scholarly journals Carnot metrics, dynamics and local rigidity

2021 ◽  
pp. 1-51
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Symmetric Spaces ◽  
Holonomy Group ◽  
Tangent Cones ◽  
Ideal Boundary ◽  
Uniform Lattice ◽  
New Techniques ◽  
Carathéodory Metric ◽  
Local Rigidity ◽  
Lattice Actions ◽  
The Ideal

Abstract This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

scholarly journals Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

2016 ◽  
Author(s):  
Frédéric Barbaresco
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Statistical Physics ◽  
Geometric Theory ◽  
Adjoint Action ◽  
Information Geometry ◽  
Affine Group ◽  
Exponential Families ◽  
Natural Exponential Families ◽  
Fisher Metric

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 moment space, defining physical observables like energy, heat, and moment 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. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. 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. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of Information Geometry for the action of an affine Group for exponentiel families, and provide some illustration of use cases for multivariate Gaussian densities. Information Geometry is presented in the context of seminal work of Fréchet and his Clairait-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

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