Is the Geometry Doing the Work? An Operating-Point Audit of Hierarchy in Hyperbolic Vision-Language Models

Jul 16, 2026

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Whether a hyperbolic representation model uses its geometry cannot be inferred from curvature alone; it requires measuring the dimensionless operating point √cρ and checking whether radial and cone mechanisms are operational. Necessary-condition diagnostics audit MERU, HyCoCLIP, and PHyCLIP across released checkpoints and matched from-scratch interventions. All converged checkpoints remain near-Euclidean (H(u)≈1; none exceeds √cρ>1). Entailment cones are inactive or saturated, graded traversal fails under controlled readouts, and external parent–child ordering shows no shuffle-null radial signal. Gradient and a closed-form aperture identity reveal a low-curvature, wide-cone entailment shortcut.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages49

Reading Time2 h 3 min

Abstract

Introduction

Methodology and Operating-Point Diagnostics

Experimental Audit of Hyperbolic VLM Families

Geometry Clamping and Near-Euclidean Regime

Hierarchy Evidence and Containment Analysis

Gradient Diagnostics and Shortcut Identification

arXiv :2607 .05268v2 [ cs .CV] 13 Jul 2026 Is the Geometry Doing the Work? An Operating-Point Audit of Hierarchy in Hyperbolic Vision– Language Models Jaeyoung Kim MADI Eunseok Kim MADI Dongsuk Jang MADI [jykim@madidt. com](jykim@madidt. com) [eskim@madidt. com](eskim@madidt. com) [dsjang@madidt. com](dsjang@madidt. com) Abstract Whether a hyperbolic representation model uses its geometry cannot be inferred from curvature alone. It requires measuring the dimensionless operating point √cρ and testing whether the radial and cone mechanisms are operational. We develop necessary-condition diagnostics and audit three published hyperbolic vision–language families—MERU, HyCoCLIP, and PHyCLIP—across released checkpoints and matched from-scratch interventions. All audited converged checkpoints remain near-Euclidean (H(u) ≈ 1; none reaches √cρ > 1), and releasing the curvature floor changes c and norms without leaving the near-Euclidean regime or substantially degrading downstream performance. Entailment cones are inactive or saturated, and graded traversal fails under controlled readouts, including the models’ native distance metrics. External parent–child ordering shows no shuffle-controlled pair-specific radial signal at quantified sensitivity. The only pair-specific signal surviving these controls is a detectable but small residual on the GRIT box-to-full-caption relation, which remains nonoperative under the evaluated readouts. Under a cosine-plus-norm decomposition, taxonomy correlations show no detectable incremental norm contribution, while coarse-retrieval gains co-vary with box/compositional supervision without establishing an active radial mechanism. Gradient diagnostics expose a low-curvature, wide-cone shortcut in the entailment objective. A closed-form aperture identity places the saturation edge at √cρ ≤ 2K (K the models’ fixed aperture constant) . With the floor released, all trained relation-level parent means lie at or below this edge, leaving the parent cones fully or nearly saturated—clipping the aperture-widening gradient channel—and in the box-supervised families the trained image-parent coordinate settles at the edge across both scales. Entailment-off runs show no arrest and continue contracting. The shortcut is the dominant accelerator of collapse, not its sole cause: contrastive/alignment training alone also fails to sustain a nonlocal operating point. These audited formulations do not show an operative radial/cone mechanism under our diagnostics. We distill the audit into a five-number geometry report for hierarchy claims. 1 Introduction Vision-language models are increasingly expected to reason not only about visual similarity, but also about abstraction: an image of a “golden retriever” is also an image of a “dog”, an “animal”, and an “entity”. This has motivated a growing line of hyperbolic vision-language models, including MERU (Desai et al. , 2023), We used large language models as general-purpose assistive tools for code debugging, numerical cross-checking, organization of results, and manuscript polishing. All research questions, experimental design choices, statistical procedures, interpretations, and scientific claims are the authors’ responsibility. HyCoCLIP (Pal et al., 2025), and PHyCLIP (Yoshikawa & Matsubara, 2025), which replace Euclidean CLIPstyle representations (Radford et al., 2021) with negatively curved spaces. The motivation is compelling: hyperbolic spaces have exponential volume growth and are known to embed tree-like structures with low distortion (Bridson & Haefliger, 1999; Sarkar, 2011; Nickel & Kiela, 2017; Ganea et al., 2018) . In this view, general concepts should lie closer to the origin, specific concepts should move toward the boundary, andentailment cones should encode asymmetric abstraction relations. But do current hyperbolic vision-language models actually instantiate this hierarchy mechanism? This question is more subtle than asking whe

FAQs

What does the paper use to test whether hyperbolic geometry is actually being used?

It measures the dimensionless operating point √cρ and applies diagnostics to check whether radial and cone mechanisms are operational, rather than relying on curvature alone.

Which hyperbolic vision–language model families are audited in the study?

The paper audits MERU, HyCoCLIP, and PHyCLIP across multiple released checkpoints and matched from-scratch interventions.

What is the main conclusion about hierarchy mechanisms in the audited models?

The audited formulations do not show an operative radial/cone mechanism under the proposed diagnostics; entailment cones are inactive or saturated and hierarchy evidence is underdetermined.

Jul 16, 2026

Upload by Aria