@inproceedings{yang-etal-2026-hyperbolicity,
title = "A Hyperbolicity Atlas of Large Language Model Hidden States",
author = "Yang, Zhichao and
Hu, Yuanze and
Li, Gen and
Yu, Qingchen and
Duan, Shiying and
Wang, Xinyu and
Qiu, Ye and
Liu, Zeming and
Chen, Guangxu and
Fan, Zhaoxin",
editor = "Lee, Hung-Yi and
Dev, Sunipa and
Li, Jessy and
Elliott, Desmond",
booktitle = "Findings of the {A}ssociation for {C}omputational {L}inguistics: {EMNLP} 2026",
month = oct,
year = "2026",
address = "Budapest, Hungary",
publisher = "Association for Computational Linguistics",
url = "https://aclanthology.org/2026.findings-emnlp.2003/",
pages = "42692--42701",
ISBN = "979-8-89176-462-0",
abstract = "Hidden states of large language models (LLMs) are typically analyzed as flat Euclidean vectors through probing, anisotropy, and intrinsic-dimension lenses. Yet much of what these models process (taxonomies, program structure, multi-step derivations) is hierarchical, and whether hidden-state distances reflect such hierarchy remains untested. We present the first systematic study of Gromov hyperbolicity (GH), a distance-based measure of tree-likeness, in prompt-token hidden states of open-weight LLMs. Across model families, scales, layer depths, and domains spanning math, code, commonsense, and truthfulness benchmarks, we construct a hyperbolicity atlas charting the normalized statistic $\delta_{\mathrm{rel}}$ along each axis. Four findings emerge: (i) depth dominates, with middle layers forming a high-$\delta_{\mathrm{rel}}$ plateau and final layers dropping toward tree-like structure; (ii) scale is weak and non-monotonic; (iii) matched-scale families occupy distinct regimes, with specialization gaps exceeding scale effects; and (iv) domains interact with specialization, e.g., HumanEval shifts from the highest- to the lowest-$\delta_{\mathrm{rel}}$ domain between base and coder checkpoints. GH thus serves as a lightweight diagnostic that localizes where hierarchical distance structure emerges and which comparisons warrant deeper analysis."
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<abstract>Hidden states of large language models (LLMs) are typically analyzed as flat Euclidean vectors through probing, anisotropy, and intrinsic-dimension lenses. Yet much of what these models process (taxonomies, program structure, multi-step derivations) is hierarchical, and whether hidden-state distances reflect such hierarchy remains untested. We present the first systematic study of Gromov hyperbolicity (GH), a distance-based measure of tree-likeness, in prompt-token hidden states of open-weight LLMs. Across model families, scales, layer depths, and domains spanning math, code, commonsense, and truthfulness benchmarks, we construct a hyperbolicity atlas charting the normalized statistic δ_rel along each axis. Four findings emerge: (i) depth dominates, with middle layers forming a high-δ_rel plateau and final layers dropping toward tree-like structure; (ii) scale is weak and non-monotonic; (iii) matched-scale families occupy distinct regimes, with specialization gaps exceeding scale effects; and (iv) domains interact with specialization, e.g., HumanEval shifts from the highest- to the lowest-δ_rel domain between base and coder checkpoints. GH thus serves as a lightweight diagnostic that localizes where hierarchical distance structure emerges and which comparisons warrant deeper analysis.</abstract>
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%0 Conference Proceedings
%T A Hyperbolicity Atlas of Large Language Model Hidden States
%A Yang, Zhichao
%A Hu, Yuanze
%A Li, Gen
%A Yu, Qingchen
%A Duan, Shiying
%A Wang, Xinyu
%A Qiu, Ye
%A Liu, Zeming
%A Chen, Guangxu
%A Fan, Zhaoxin
%Y Lee, Hung-Yi
%Y Dev, Sunipa
%Y Li, Jessy
%Y Elliott, Desmond
%S Findings of the Association for Computational Linguistics: EMNLP 2026
%D 2026
%8 October
%I Association for Computational Linguistics
%C Budapest, Hungary
%@ 979-8-89176-462-0
%F yang-etal-2026-hyperbolicity
%X Hidden states of large language models (LLMs) are typically analyzed as flat Euclidean vectors through probing, anisotropy, and intrinsic-dimension lenses. Yet much of what these models process (taxonomies, program structure, multi-step derivations) is hierarchical, and whether hidden-state distances reflect such hierarchy remains untested. We present the first systematic study of Gromov hyperbolicity (GH), a distance-based measure of tree-likeness, in prompt-token hidden states of open-weight LLMs. Across model families, scales, layer depths, and domains spanning math, code, commonsense, and truthfulness benchmarks, we construct a hyperbolicity atlas charting the normalized statistic δ_rel along each axis. Four findings emerge: (i) depth dominates, with middle layers forming a high-δ_rel plateau and final layers dropping toward tree-like structure; (ii) scale is weak and non-monotonic; (iii) matched-scale families occupy distinct regimes, with specialization gaps exceeding scale effects; and (iv) domains interact with specialization, e.g., HumanEval shifts from the highest- to the lowest-δ_rel domain between base and coder checkpoints. GH thus serves as a lightweight diagnostic that localizes where hierarchical distance structure emerges and which comparisons warrant deeper analysis.
%U https://aclanthology.org/2026.findings-emnlp.2003/
%P 42692-42701
Markdown (Informal)
[A Hyperbolicity Atlas of Large Language Model Hidden States](https://aclanthology.org/2026.findings-emnlp.2003/) (Yang et al., Findings 2026)
ACL
- Zhichao Yang, Yuanze Hu, Gen Li, Qingchen Yu, Shiying Duan, Xinyu Wang, Ye Qiu, Zeming Liu, Guangxu Chen, and Zhaoxin Fan. 2026. A Hyperbolicity Atlas of Large Language Model Hidden States. In Findings of the Association for Computational Linguistics: EMNLP 2026, pages 42692–42701, Budapest, Hungary. Association for Computational Linguistics.