The generalized Pythagorean theorem on the compactifications of certain dually flat spaces via toric geometry
arXiv:2305.08422 · doi:10.1007/s41884-023-00123-y
Abstract
In this paper we study dually flat spaces arising from Delzant polytopes equipped with a symplectic potential together with their corresponding toric Kähler manifolds as their torifications.We introduce a dually flat structure and the associated Bregman divergence on the boundary from the viewpoint of toric Kähler geometry. We show a continuity and a generalized Pythagorean theorem for the divergence on the boundary. We also provide a characterization for a toric Kähler manifold to become a torification of a mixture family on a finite set.
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