The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry
Abstract
In this paper we study dually flat spaces arising from Delzant polytopes equipped with a symplectic potential together with their corresponding toric K\"ahler manifolds as their torifications.We introduce a dually flat structure and the associated Bregman divergence on the boundary from the viewpoint of toric K\"ahler geometry. We show a continuity and a generalized Pythagorean theorem for the divergence on the boundary. We also provide a characterization for a toric K\"ahler manifold to become a torification of a mixture family on a finite set.
Keywords
Cite
@article{arxiv.2305.08422,
title = {The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry},
author = {Hajime Fujita},
journal= {arXiv preprint arXiv:2305.08422},
year = {2023}
}
Comments
20 pages, 4 figures : Minor revisions on the format, title changed according to referee's suggestion, typos corrected, explanation added, to appear in Information Geometry : Mistake in the example in Section 6.1 corrected