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Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification — arXiv:2609.02362

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.02362
License (as archived): CC0-1.0
Archived: 2026-10-17T05:26:25.591Z
SHA-256 of the archived text: c58403b0284a20a70a2fe0c7acdbae918bb131dab516107bb0a58621ff073a4e
Stable URL of this page: https://metraipolis.com/documento/dc_4e4429c691ff

How to cite

Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification — arXiv:2609.02362 — arXiv (en), https://arxiv.org/abs/2609.02362 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_4e4429c691ff (Metraipolis archive dc_4e4429c691ff, sha256 c58403b0284a20a70a2fe0c7acdbae918bb131dab516107bb0a58621ff073a4e)

BibTeX

@misc{dc4e4429c691ff,
  title = {Farthest-cell triplet entropy: high-dimensional shell limits and hyperbolic curvature amplification — arXiv:2609.02362},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.02362},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_4e4429c691ff; sha256 c58403b0284a20a70a2fe0c7acdbae918bb131dab516107bb0a58621ff073a4e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.02362. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\lambda_d\xi_{i,d}-Z_i$, where $\lambda_d=\sqrt d\,\operatorname{sd}(R_d)/\mathbb{E} R_d$ and the $Z_i$ are independent standard Gaussian variables. This gives angular-dominated, intermediate, and radial-dominated entropy limits $\log 3$, $H_{\infty}(\lambda;F)$, and $0$. In hyperbolic space of curvature $-\kappa_d^2$, the same master curve appears at $\lambda_{d,\mathbb H}=\sqrt d\,\tau_d A(s_d)$, where $\tau_d=\operatorname{sd}(R_d)/\mathbb{E} R_d$, $s_d=\kappa_d\mathbb{E} R_d$, and $A(s)=s\coth s$. With a calibrated radial law and $\tau_d$, 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 $\kappa$ 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-03 in math.PR).
Categorie arXiv: Probability (math.PR) ; Metric Geometry (math.MG).
Pagina del lavoro: https://arxiv.org/abs/2609.02362 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T04:44:48.583Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-03?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).