最远胞体三元组熵：高维壳层极限与双曲曲率放大
Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification
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中文总结 AI 辅助
本研究提出最远胞体三元组熵统计量，推导其在高维欧氏与双曲空间的极限性质，验证了基于熵反演的曲率估计效果，并指出该方法对各向异性不鲁棒。
中文摘要 AI 辅助
我们提出最远胞体三元组熵，即给定三个随机原型时，最远原型标签的条件香农熵。对于独立的查询点与原型，其估计量仅记录最远标签，无需坐标或数值距离。该统计量以$\boldsymbol{\log 3}$为界，在相异度的常见严格递增变换下保持不变，且具有精确的互信息解释。\n在高维各向同性径向模型$X_d=R_dU_d$中，欧氏排序可简化为得分$\lambda_d\xi_{i,d}-Z_i$，其中$\lambda_d=\sqrt d\\,\operatorname{sd}(R_d)/\mathbb{E} R_d$，$Z_i$为独立标准高斯变量。由此得到角度主导、中间态、径向主导三类熵极限：$\log 3$、$H_{\infty}(\lambda;F)$与$0$。在曲率为$-\kappa_d^2$的双曲空间中，相同的主曲线出现在$\lambda_{d,\mathbb H}=\sqrt d\\,\tau_d A(s_d)$处，其中$\tau_d=\operatorname{sd}(R_d)/\mathbb{E} R_d$，$s_d=\kappa_d\mathbb{E} R_d$，$A(s)=s\coth s$。\n通过校准径向分布与$\tau_d$，并利用单调工作区间，熵反演可识别尺度不变目标$s_d^2$；绝对曲率则需要外部长度单位。CPU仿真显示，欧氏与双曲主曲线的RMSE分别为0.0164与0.0209。从观测到的合成潜坐标反演的$\kappa$中位相对误差为6.6%，而角度各向异性会将该误差提升至68.9%。因此该熵统计量基于比较，而曲率恢复仍依赖模型校准，并非仅靠图结构，且对各向异性不具备鲁棒性。
英文摘要
We introduce farthest-cell triplet entropy, the conditional Shannon entropy of the farthest-prototype label given three random prototypes. For independent queries and prototypes, its estimator records only the farthest label, not coordinates or numerical distances. The statistic is bounded by $\log 3$, is invariant under common strictly increasing transformations of the dissimilarities, and has an exact mutual-information interpretation. In high-dimensional isotropic radial models $X_d=R_dU_d$, the Euclidean ordering reduces to scores $λ_dξ_{i,d}-Z_i$, where $λ_d=\sqrt d\,\operatorname{sd}(R_d)/\mathbb{E} R_d$ and the $Z_i$ are independent standard Gaussian variables. This gives angular-dominated, intermediate, and radial-dominated entropy limits $\log 3$, $H_{\infty}(λ;F)$, and $0$. In hyperbolic space of curvature $-κ_d^2$, the same master curve appears at $λ_{d,\mathbb H}=\sqrt d\,τ_d A(s_d)$, where $τ_d=\operatorname{sd}(R_d)/\mathbb{E} R_d$, $s_d=κ_d\mathbb{E} R_d$, and $A(s)=s\coth s$. With a calibrated radial law and $τ_d$, and a monotone operating interval, entropy inversion identifies the scale-invariant target $s_d^2$; absolute curvature requires an external length unit. CPU simulations give Euclidean and hyperbolic master-curve RMSEs of $0.0164$ and $0.0209$. Inversion from observed synthetic latent coordinates has a median relative error in $κ$ of $6.6\%$, while angular anisotropy increases this error to $68.9\%$. Thus the entropy statistic is comparison-based, whereas curvature recovery remains model-calibrated, is not graph-only, and is not robust to anisotropy.
