Gaussian distributions, Jacobi group, and Siegel-Jacobi space

Molitor, Mathieu

doi:10.1063/1.4903182

Cited by 11 publications

(16 citation statements)

References 73 publications

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“…For the convenience of the reader, the paper starts with a discussion on the relation between Kähler geometry and statistics (see Section 2). The material presented is mostly taken from [Mol14], except for Proposition 2.19 which seems to be new. We shall present the subject in a uniform way by using the concept of dually flat structure.…”

Section: Kähler Toric Manifold Dually Flat Spacementioning

confidence: 99%

“…The coordinate expression for the Fisher metric h F is the Hessian of the cumulant generating function: h F (θ) = Hess(ψ) = e θ . It follows from this and Proposition 2.17 that T P is Kähler isomorphic to C endowed with the Kähler metricg z (u, v) = e x Real(uv),where z, u, v ∈ C, z = x + iy, x, y ∈ R. The space of Kähler functions on T P = C is spanned by 1, e x , e 2 (to see this, use Proposition 2.25 in[Mol14]). Let ϕ : C → C be a diffeomorphism satisfying…”

mentioning

confidence: 94%

“…The objective of this paper is to "reshape" Delzant's correspondence in a more flexible way, in the Kähler setting, by replacing Delzant polytopes by dually flat manifolds (see below). Our motivation comes from the geometrization program of Quantum Mechanics that we pursued in previous works [Mol12,Mol13,Mol14,Mol15] 1 . Let us describe the objects involved.…”

Section: Introductionmentioning

confidence: 99%

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Kahler toric manifolds from dually flat spaces

Molitor

2021

Preprint

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We present a correspondence between real analytic Kähler toric manifolds and dually flat spaces, similar to Delzant correspondence in symplectic geometry. This correspondence gives rise to a lifting procedure: if f : M → M ′ is an affine isometric map between dually flat spaces and if N and N ′ are Kähler toric manifolds associated to M and M ′ , respectively, then there is an equivariant Kähler immersion N → N ′ . For example, we show that the Veronese and Segre embeddings are lifts of inclusion maps between appropriate statistical manifolds. We also discuss applications to Quantum Mechanics.

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Section: Kähler Toric Manifold Dually Flat Spacementioning

confidence: 99%

mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Kahler toric manifolds from dually flat spaces

Molitor

2021

Preprint

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“…Example 6.11 (Normal distributions [Mol14]). Let N be the set of Gaussian distributions, as defined in Example 6.4.…”

Section: Proof See [An00]mentioning

confidence: 99%

One-dimensional exponential families with constant Hessian scalar curvature

Molitor

2019

Preprint

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We give a complete classification of 1-dimensional exponential families E defined over a finite space Ω = {x 0 , ..., x m } whose Hessian scalar curvature is constant. We observe an interesting phenomenon: if E has constant Hessian scalar curvature, say λ, then λ = 2 k for some positive integer k ≤ m. We also discuss the central role played by the binomial distribution in this classification.

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“…As such, the metric has (NOAB). This metric is of independent interest, and for a more complete discussion, we refer the reader to the work of Molitor [26]. We will note a few of its curvature properties in passing.…”

Section: A Complete Complex Surface With (Noab) One Question Of Consi...mentioning

confidence: 99%

On the Kähler Geometry of Certain Optimal Transport Problems

Khan,

Zhang

2018

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Let X and Y be domains of R n equipped with probability measures µ and ν, respectively. We consider the problem of optimal transport from µ to ν with respect to a cost function c : X×Y → R. To ensure that the solution to this problem is smooth, it is necessary to make several assumptions about the structure of the domains and the cost function. In particular, Ma, Trudinger, and Wang [25] established regularity estimates when the domains are strongly relatively c-convex with respect to each other and the cost function has non-negative MTW tensor. For cost functions of the form c(x, y) = Ψ(x − y) for some convex function Ψ, we find an associated Kähler manifold whose orthogonal anti-bisectional curvature is proportional to the MTW tensor. We also show that relative c-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann [20].We provide several applications of this work. In particular, we find a complete Kähler surface with non-negative orthogonal anti-bisectional curvature that is not a Hermitian symmetric space or biholomorphic to C 2 . We also address a question in mathematical finance raised by Pal and Wong [32] on the regularity of pseudoarbitrages, or investment strategies which outperform the market.

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Gaussian distributions, Jacobi group, and Siegel-Jacobi space

Cited by 11 publications

References 73 publications

Kahler toric manifolds from dually flat spaces

Kahler toric manifolds from dually flat spaces

One-dimensional exponential families with constant Hessian scalar curvature

On the Kähler Geometry of Certain Optimal Transport Problems

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