Information geometry of emergent symmetry quotients and their weak unfoldings

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 ελJ_-+λ²J₊/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=√n ε_n²: 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: Hol₀(∇^(a))=SO(2) for a=0, whereas Hol₀(∇^(a))=GL⁺(2,ℝ) for a≠0. The results separate the intrinsic geometry of the limiting quotient from the transverse geometry of its weak unfolding.

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BibTeX RIS

BibTeX

@misc{coatanhay2026information,
  title = {{Information geometry of emergent symmetry quotients and their weak unfoldings}},
  author = {Arnaud Coatanhay and Angélique Drémeau},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29260},
  eprint = {2609.29260},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29260},
}

RIS

TY  - UNPB
TI  - Information geometry of emergent symmetry quotients and their weak unfoldings
AU  - Arnaud Coatanhay
AU  - Angélique Drémeau
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.29260
UR  - https://arxiv.org/abs/2609.29260
AB  - 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 ελJ_-+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=√n ε_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: Hol_0(∇^(a))=SO(2) for a=0, whereas Hol_0(∇^(a))=GL^+(2,ℝ) for a≠0. The results separate the intrinsic geometry of the limiting quotient from the transverse geometry of its weak unfolding.
ER  -

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