THE EGUCHI CONSTRUCTION ARISES FROM BI-LAGRANGIAN STRUCTURES
Abstract
We propose a new geometric interpretation of Eguchi's construction, which associates to a statistical divergence, an essential ingredient of Information Geometry, a statistical structure on a differentiable manifold. Our approach is based on bi-Lagrangian geometry. Building on the work of O. E. Barndorff-Nielsen and P. E. Jupp [1], we show that a statistical divergence naturally induces a bi-Lagrangian structure, hence a para-Kähler structure. Within this framework, the Riemannian metric g D and the pair of dual connections (∇ D , * ∇ D ) of Eguchi's construction are induced by the para-Kähler structure: g D is the restriction to the diagonal of the para-Kähler metric G of signature (d, d), while ∇ D and * ∇ D are obtained, through the two natural projections, from the Bott connections of the two transverse Lagrangian foliations, which are themselves the restrictions of the Levi-Civita connection of G (the Hess, Künneth or bi-Lagrangian connection).
This perspective shows that the statistical structure associated with a divergence is not merely the formal outcome of successive differentiations, but the manifestation of an underlying bi-Lagrangian geometry. Our approach thus provides an intrinsic and unified understanding of Eguchi's construction, highlights the connections between information geometry, symplectic geometry and para-Kähler geometry, and opens new research perspectives at the interface of these fields.
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BibTeX
@misc{gnandi2026eguchi,
title = {{THE EGUCHI CONSTRUCTION ARISES FROM BI-LAGRANGIAN STRUCTURES}},
author = {Emmanuel Gnandi and Stéphane Puechmorel},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32539},
eprint = {2610.32539},
archivePrefix = {arXiv},
url = {https://enac.hal.science/hal-05834597v1},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - THE EGUCHI CONSTRUCTION ARISES FROM BI-LAGRANGIAN STRUCTURES
AU - Emmanuel Gnandi
AU - Stéphane Puechmorel
PY - 2026
DA - 2026/10/22
PB - HAL
DO - 10.48550/arxiv.2610.32539
UR - https://enac.hal.science/hal-05834597v1
AB - We propose a new geometric interpretation of Eguchi's construction, which associates to a statistical divergence, an essential ingredient of Information Geometry, a statistical structure on a differentiable manifold. Our approach is based on bi-Lagrangian geometry. Building on the work of O. E. Barndorff-Nielsen and P. E. Jupp [1], we show that a statistical divergence naturally induces a bi-Lagrangian structure, hence a para-Kähler structure. Within this framework, the Riemannian metric g D and the pair of dual connections (∇ D , * ∇ D ) of Eguchi's construction are induced by the para-Kähler structure: g D is the restriction to the diagonal of the para-Kähler metric G of signature (d, d), while ∇ D and * ∇ D are obtained, through the two natural projections, from the Bott connections of the two transverse Lagrangian foliations, which are themselves the restrictions of the Levi-Civita connection of G (the Hess, Künneth or bi-Lagrangian connection). This perspective shows that the statistical structure associated with a divergence is not merely the formal outcome of successive differentiations, but the manifestation of an underlying bi-Lagrangian geometry. Our approach thus provides an intrinsic and unified understanding of Eguchi's construction, highlights the connections between information geometry, symplectic geometry and para-Kähler geometry, and opens new research perspectives at the interface of these fields.
ER -