Statistical mirror symmetry
type: Item
instance of: scholarly article
title: Statistical mirror symmetry
zbMATH Open document ID: 1478.53022
author: Jun Zhang
author: Gabriel Khan
published in: Differential Geometry and its Applications
publication date: 2021-02-24T00:00:00Z
review text: Motivated by recent developments in information geometry, the authors define and investigate the notion of statistical mirror symmetry between a Hermitian manifold structure and an almost Kähler manifold structure on the tangent bundle \(TM\) of an affine manifold \(M\) endowed with a \(\mathcal{P}\)-geometry. Specifically, let \(M\) be a real, smooth, finite-dimensional manifold endowed with a Riemannian metric tensor \(g\), and a couple \((\nabla,\nabla^{*})\) of affine connection which are dually related with respect to \(g\) in the sense that \[ X\left(g(Y,Z)\right)=g\left(\nabla_{X}Y,Z\right) + g\left(Y,\nabla^{*}_{X}Z\right) \] holds for all vector fields \(X,Y,Z\) on \(M\). Then, it is not hard to show that \[ g\left(R^{\nabla}(Z,W)X,Y\right) + g\left(R^{\nabla^{*}}(Z,W)X,Y\right)=0 , \] where \(R^{\nabla}\) and \(R^{\nabla^{*}}\) denote, respectively, the curvature tensor of \(\nabla\) and \(\nabla^{*}\), and \[ \begin{split} g\left(T^{\nabla^{*}}(Z,X) - T^{\nabla}(Z,X),Y\right)&=\left(\nabla_{Z}g\right)(X,Y)- \left(\nabla_{X}g\right)(Z,Y)= \\ &=\left(\nabla_{X}^{*}g\right)(Z,Y) - \left(\nabla_{Z}^{*}g\right)(X,Y), \end{split} \] where \(T^{\nabla}\) and \(T^{\nabla^{*}}\) denote, respectively, the torsion tensors of \(\nabla\) and \(\nabla^{*}\). When both connection are torsion-less, the data \((M,g,\nabla,\nabla^{*})\) is referred to as a statistical manifold. The name is justified because this type of mathematical structure arises naturally in the context of statistical models where the manifold \(M\) is some manifold of parametrized probability distributions on some measure space (e.g., Gaussian measures on \(\mathbb{R}\)), the Riemannian metric tensor \(g\) is the Fisher-Rao metric tensor, and the couple \((\nabla,\nabla^{*})\) is defined either exploiting the family of \(\alpha\)-connections, or using a suitable extraction algorithm from a preferred relative entropy function, or divergence function, or contrast function. In this context, of particular…
reviewed by: Florio M. Ciaglia
Mathematics Subject Classification ID: 53B12
Mathematics Subject Classification ID: 53B35
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53D37
zbMATH DE Number: 7315171
zbMATH Keywords: information geometry
zbMATH Keywords: dual connections
zbMATH Keywords: statistical mirror symmetry
zbMATH Keywords: statistical manifolds admitting torsion
zbMATH Keywords: almost complex structure
zbMATH Keywords: quantum information geometry
Wikidata QID: Q115354719
MaRDI profile type: Publication
full work available at URL: j.difgeo.2020.101678
OpenAlex ID: W3082028296
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DOI: 10.1016/J.DIFGEO.2020.101678