Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification
Abstract
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 Xd=RdUd, the Euclidean ordering reduces to scores λdξi,d−Zi, where λd=√d sd(Rd)/𝔼 Rd and the Zi are independent standard Gaussian variables. This gives angular-dominated, intermediate, and radial-dominated entropy limits log 3, H∞(λ;F), and $0$. In hyperbolic space of curvature −κd2, the same master curve appears at λd,ℍ=√d τd A(sd), where τd=sd(Rd)/𝔼 Rd, sd=κd𝔼 Rd, and A(s)=scoth s. With a calibrated radial law and τd, and a monotone operating interval, entropy inversion identifies the scale-invariant target sd2; 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.
Related
- Constant mean curvature disks with circular boundary in the $3$-sphere and in hyperbolic space
- Hypersurfaces of constant higher order mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity
- Discontinuous phase transitions in the one-dimensional pair and triplet creation processes
- The asymptotic Plateau problem for Hypersurfaces of constant Hk curvature in hyperbolic space
- Volume Entropy and Scalar Curvature near a Hyperbolic Metric
BibTeX
@misc{deng2026farthestcell,
title = {{Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification}},
author = {Chongkun Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02362},
eprint = {2609.02362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02362},
}
RIS
TY - UNPB
TI - Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification
AU - Chongkun Deng
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02362
UR - https://arxiv.org/abs/2609.02362
AB - 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=√d sd(R_d)/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_∞(λ;F), and $0$. In hyperbolic space of curvature −κ_d^2, the same master curve appears at λ_d,ℍ=√d τ_d A(s_d), where τ_d=sd(R_d)/𝔼 R_d, s_d=κ_d𝔼 R_d, and A(s)=scoth 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.
ER -