涌现对称商及其弱展开的信息几何
Information geometry of emergent symmetry quotients and their weak unfoldings
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中文总结 AI 辅助
本研究探讨统计模型极限中符号参数仅模反射可识别时的信息几何，揭示二射流结构、三种渐近区域及商度量性质，并显式计算CIR--OU基准的负曲率和Amari族的和乐群。
中文摘要 AI 辅助
一个正则的可观测统计模型可能收敛到一个极限，在该极限中，一个先前可识别的带符号参数仅能模反射地被识别。我们研究这一转变的局部信息几何。对于具有精确极限反射的二阶可微Hellinger嵌入，观测位移被迫呈现二射流形式\\(\varepsilon\lambda J_-+\lambda^2J_+/2\\)，直至高阶项。混合射流在对称面之外恢复符号，而偶射流是商继承的第一个切向量。在消除 nuisance 参数后，正 Gram 行列式产生非退化的交叉帽二射流。相关的局部渐近理论由\\(\tau_n=\sqrt n\\,\varepsilon_n^2\\) 控制，包含三个区域：正则带符号LAN、临界曲高斯子实验，以及具有\\(n^{-1/4}\\)带符号尺度的商区域。我们证明，沿单一平稳依赖轨迹的预测似然中，相同的抛物型临界实验持续存在。极限商在不变坐标下具有正则Fisher度量，而其拉回在带符号坐标下退化。对于由相干海杂波观测启发的可解CIR--OU基准，我们显式推导了商Fisher度量和曲率，并证明曲率严格为负。我们还确定了完整Amari族的限制和乐：对于\\(a=0\\)，\\(\operatorname{Hol}_0(\nabla^{(a)})=SO(2)\\)，而对于\\(a\neq0\\)，\\(\operatorname{Hol}_0(\nabla^{(a)})=GL^+(2,\mathbb R)\\)。这些结果将极限商的内在几何与其弱展开的横向几何区分开来。
英文摘要
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.
发表机构
- ENSTA（国立高等先进技术学院）
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