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Statistical mirror symmetry

Authors :
Gabriel Khan
Jun Zhang
Source :
Differential Geometry and its Applications. 73:101678
Publication Year :
2020
Publisher :
Elsevier BV, 2020.

Abstract

In this paper, we investigate a duality between Hermitian and almost Kahler structures on the tangent manifold T M induced by pairs of conjugate connections on its base, affine Riemannian manifold M . In the context of information geometry, the classical theory of statistical manifold (which we call S -geometry) prescribes a parametrized family of probability distributions with a Fisher-Rao metric g and, using the Amari-Chensov tensor, a family of dualistic, torsion-free connections ∇ ( α ) , known as α-connections on M . Here we prescribe an alternative geometric framework (which we call P -geometry or partially flat geometry) by treating such parametrization as affine coordinates with respect to a flat connection ∇, and considering its g-conjugate connection ∇ ⁎ which is curvature-free but generally carries torsion. Under P -geometry, the triplet ( g , ∇ , ∇ ⁎ ) on M leads to a pair of complex and almost Kahler structures on T M , in “mirror correspondence” to each other. Such complex-to-symplectic correspondence is reminiscent of mirror symmetry in string theory. We discuss the statistical meaning of mirror correspondence in terms of reference duality and representation duality in (various generalizations of) contrast/divergence functions characterizing proximity of probability distributions within a parametric statistical model.

Details

ISSN :
09262245
Volume :
73
Database :
OpenAIRE
Journal :
Differential Geometry and its Applications
Accession number :
edsair.doi...........8ff5ca61d1ed034056c581421a824564
Full Text :
https://doi.org/10.1016/j.difgeo.2020.101678