Information geometry of emergent symmetry quotients and their weak unfoldings

Arnaud Coatanay; Angélique Drémeau

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arXiv · 2609.29260

Information geometry of emergent symmetry quotients and their weak unfoldings

Arnaud Coatanay·Angélique Drémeau

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Abstract

A regular observed statistical model may converge to a limit in which a previously identifiable signed parameter becomes identifiable only modulo a reflection. We study the local information geometry of this transition. For a twice differentiable Hellinger embedding with an exact limiting reflection, the observed displacement is forced into the two-jet form $\varepsilonλJ_-+λ^2J_+/2$, up to higher-order terms. The mixed jet restores the sign away from the symmetric face, whereas the even jet is the first tangent inherited by the quotient. After nuisance elimination, a positive Gram determinant yields a nondegenerate cross-cap two-jet. The associated local asymptotic theory has three regimes governed by $τ_n=\sqrt n\,\varepsilon_n^2$: regular signed LAN, a critical curved Gaussian subexperiment, and a quotient regime with the $n^{-1/4}$ signed scale. We prove that the same parabolic critical experiment persists for predictive likelihoods along a single stationary dependent trajectory. The limiting quotient has a regular Fisher metric in the invariant coordinate, while its pullback degenerates in the signed coordinate. For a solvable CIR--OU benchmark motivated by coherent sea-clutter observations, we derive the quotient Fisher metric and curvature explicitly and show that the curvature is strictly negative. We also determine the restricted holonomy of the full Amari family: $\operatorname{Hol}_0(\nabla^{(a)})=SO(2)$ for $a=0$, whereas $\operatorname{Hol}_0(\nabla^{(a)})=GL^+(2,\mathbb R)$ for $a\neq0$. The results separate the intrinsic geometry of the limiting quotient from the transverse geometry of its weak unfolding.

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Arnaud Coatanay, Angélique Drémeau. 2026-09-24. Information geometry of emergent symmetry quotients and their weak unfoldings. https://arxiv.org/abs/2609.29260

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