Statistical manifolds with almost contact structures and its statistical submersions

Takano, Kazuhiko

doi:10.1007/s00022-006-0052-2

Cited by 53 publications

(110 citation statements)

References 12 publications

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“…The following is our definition of statistical manifold. [17]). Let (M, g) be a (pseudo) Riemannian manifold, ∇ a connection, and ∇ * the dual connection associated with g. If ∇ and ∇ * are torsion-free connections, then the triplet (M, g, ∇) is referred to as the statistical manifold.…”

Section: Information Geometric Characterization Of Parameter Mapsmentioning

confidence: 99%

“…Definition 4.5. (Almost complex structure and almost complex manifold, [19,17]). Let M be a manifold.…”

Section: Symplectic Information Geometric Characterization Of Parametmentioning

confidence: 99%

“…Definition 4.6. (Almost Hermite manifold, [19,17]). Let (M, g) be a (pseudo) Riemannian manifold, and J an almost complex structure.…”

Section: Symplectic Information Geometric Characterization Of Parametmentioning

confidence: 99%

See 2 more Smart Citations

Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Goto

Umeno

2018

Journal of Mathematical Physics

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Maps on a parameter space for expressing distribution functions are exactly derived from the Perron-Frobenius equations for a generalized Boole transform family. Here the generalized Boole transform family is a one-parameter family of maps where it is defined on a subset of the real line and its probability distribution function is the Cauchy distribution with some parameters. With this reduction, some relations between the statistical picture and the orbital one are shown. From the viewpoint of information geometry, the parameter space can be identified with a statistical manifold, and then it is shown that the derived maps can be characterized. Also, with an induced symplectic structure from a statistical structure, symplectic and information geometric aspects of the derived maps are discussed.

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Section: Information Geometric Characterization Of Parameter Mapsmentioning

confidence: 99%

“…Definition 4.5. (Almost complex structure and almost complex manifold, [19,17]). Let M be a manifold.…”

Section: Symplectic Information Geometric Characterization Of Parametmentioning

confidence: 99%

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Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Goto

Umeno

2018

Journal of Mathematical Physics

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“…It has applications in information geometry, which represents one of the main tools for machine learning and evolutionary biology. In 2004, K. Takano [2] defined and investigated Kähler-like statistical manifolds and their statistical submanifolds.…”

Section: Introductionmentioning

confidence: 99%

Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

et al. 2019

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We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ(2, 2). MSC: 53C05; 53C40where ∇ 0 is the Levi-Civita connection on M.Similar definitions can be considered for semi-Riemannian manifolds (see also [5]).A statistical structure is said to be of constant curvature ∈ R [6] if:for any vector fields X, Y, Z. The same equation holds for R * (X, Y)Z.A statistical structure of null constant curvature is called a Hessian structure. In [2,7], K. Takano considered a (semi-)Riemannian manifold M, g with an almost complex structure J, endowed with another tensor field J * of type (1, 1) satisfying:for vector fields X and Y on M, g . Then, M, g, J is called an almost Hermite-like manifold. It is easy to see that J * * = J, J * 2 = −I and g JX, J * Y = g (X, Y) . If J is parallel with respect to ∇, then M, g, ∇, J is called a Kähler-like statistical manifold [7]. One also has: g

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“…Meanwhile, there are scholars interested at the global properties of statistical manifolds themselves, rather than the applications of the theory of information geometry [21,24,39]. In this paper, we investigate statistical manifolds which are Einstein manifolds at the same time, that is, the following equation holds [10] Ric = −λg, (1.1) where λ is a constant, Ric is the Ricci curvature tensor and g is the Riemannian or pseudo-Riemannian metric.…”

Section: Introductionmentioning

confidence: 99%

Statistical Einstein manifolds of exponential families with group-invariant potential functions

Peng

Zhang

2019

Journal of Mathematical Analysis and Applications

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This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. A statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. With the Fisher information metric as a Riemannian metric, information geometry was developed to understand the intrinsic properties of statistical models, which play important roles in statistical inference, etc. Among all these models, exponential families is one of the most important kinds, whose geometric structures are fully determined by their potential functions. To classify statistical Einstein manifolds, we derive partial differential equations for potential functions of exponential families; special solutions of these equations are obtained through the ansatz method as well as group-invariant solutions via reductions using Lie point symmetries.

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Statistical manifolds with almost contact structures and its statistical submersions

Cited by 53 publications

References 12 publications

Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Maps on statistical manifolds exactly reduced from the Perron-Frobenius equations for solvable chaotic maps

Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

Statistical Einstein manifolds of exponential families with group-invariant potential functions

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