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Towards a Canonical Divergence within Information Geometry
Publikationstyp
Journal Article
Date Issued
2021-06-22
Sprache
English
Author(s)
Ay, Nihat
Institut
Journal
Volume
4
Start Page
65
End Page
130
Citation
Information Geometry 4: 65-130 (2021-07)
Publisher DOI
Scopus ID
ArXiv ID
Publisher
Springer
In Riemannian geometry geodesics are integral curves of the Riemannian distance gradient. We extend this classical result to the framework of Information Geometry. In particular, we prove that the rays of level-sets defined by a pseudo-distance are generated by the sum of two tangent vectors. By relying on these vectors, we propose a novel definition of a canonical divergence and its dual function. We prove that the new divergence allows to recover a given dual structure (g,∇,∇*) of a dually convex set on a smooth manifold M. Additionally, we show that this divergence coincides with the canonical divergence proposed by Ay and Amari in the case of: (a) self-duality, (b) dual flatness, (c) statistical geometric analogue of the concept of symmetric spaces in Riemannian geometry. For a dually convex set, the case (c) leads to a further comparison of the new divergence with the one introduced by Henmi and Kobayashi.
Subjects
Mathematics - Differential Geometry
Mathematics - Differential Geometry