From Hessian to Weitzenböck: manifolds with torsion-carrying connections

Zhang, Jun; Khan, Gabriel

doi:10.1007/s41884-019-00018-x

Cited by 8 publications

(4 citation statements)

References 12 publications

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“…An almost dual basis {θ k } k=1,...,n 2 −1 of {L k } k=1,...,n 2 −1 is then easily seen to be associated with a choice of basis {ω k } k=0,...,n 2 −1 for B sa (H) in such a way that ω 0 = I and Tr(ω k ) = 0. Indeed, setting θ k := dl k with l k (ρ) := Tr(ρω k ) (61) the expectation-value function associated with σ k , a direct computation shows that…”

Section: Fundingmentioning

confidence: 99%

“…Moreover, it is worth noting how the investigation of the appearance of torsion also in Classical Information Geometry is recently gaining interest (cf. [40,61,62,63]). In particular, the present work may be thought of as complementing some theoretical aspects already discussed in [61] (especially in connection with the explicit construction of what is called the g-biorthogonal frame), and providing a natural extension to the case of Quantum Information Geometry.…”

Section: Introductionmentioning

confidence: 99%

“…[40,61,62,63]). In particular, the present work may be thought of as complementing some theoretical aspects already discussed in [61] (especially in connection with the explicit construction of what is called the g-biorthogonal frame), and providing a natural extension to the case of Quantum Information Geometry.…”

Section: Introductionmentioning

confidence: 99%

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G-dual teleparallel connections in Information Geometry

Ciaglia¹,

Cosmo²,

Ibort³

et al. 2022

Preprint

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Given a real, finite-dimensional, smooth parallelizable Riemannian manifold (N , G) endowed with a teleparallel connection ∇ determined by a choice of a global basis of vector fields on N , we show that the G-dual connection ∇ * of ∇ in the sense of Information Geometry must be the teleparallel connection determined by the basis of G-gradient vector fields associated with a basis of differential one-forms which is (almost) dual to the basis of vector fields determining ∇. We call any such pair (∇, ∇ * ) a G-dual teleparallel pair. Then, after defining a covariant (0, 3) tensor T uniquely determined by (N , G, ∇, ∇ * ), we show that T being symmetric in the first two entries is equivalent to ∇ being torsion-free, that T being symmetric in the first and third entry is equivalent to ∇ * being torsion free, and that T being symmetric in the second and third entries is equivalent to the basis vectors determining ∇ (∇ * ) being parallel-transported by ∇ * (∇). Therefore, G-dual teleparallel pairs provide a generalization of the notion of Statistical Manifolds usually employed in Information Geometry, and we present explicit examples of G-dual teleparallel pairs arising both in the context of both Classical and Quantum Information Geometry.

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Section: Fundingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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G-dual teleparallel connections in Information Geometry

Ciaglia¹,

Cosmo²,

Ibort³

et al. 2022

Preprint

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“…Geometrically, the standard model (denoted the

-model in this paper) uses a pair of affine connections that are torsion-free, though in general, they are not curvature-free. An alternative, “partially flat” model (denoted the

-model in this paper) was recently investigated in [ 3 ], leading to the notion of “statistical mirror symmetry” [ 4 ]. Under the

-model, the affine connections ∇ and

are allowed to carry torsion, but are both curvature-free.…”

Section: Introductionmentioning

confidence: 99%

λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

Zhang

Wong

2022

Entropy

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This paper systematically presents the λ-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the λ-deformed case: λ-convexity, λ-conjugation, λ-biorthogonality, λ-logarithmic divergence, λ-exponential and λ-mixture families, etc. In particular, λ-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical λ-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the λ-exponential family, in turn, coincides with the λ-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, λ-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.

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New Geometry of Parametric Statistical Models

Zhang

Khan

2019

Lecture Notes in Computer Science

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From Hessian to Weitzenböck: manifolds with torsion-carrying connections

Cited by 8 publications

References 12 publications

G-dual teleparallel connections in Information Geometry

G-dual teleparallel connections in Information Geometry

λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

New Geometry of Parametric Statistical Models

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