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

2026-07-06

AI Summary

It remains unclear whether existing hyperbolic vision–language models genuinely leverage hyperbolic geometry for hierarchical modeling. This work proposes a diagnostic framework centered on the dimensionless operating point $\sqrt{c}\rho$, integrating entailment cone activation detection, radial depth sensitivity analysis, and angle–curvature disentanglement experiments to systematically audit models such as MERU, HyCoCLIP, and PHyCLIP. The study reveals that these models predominantly operate in near-Euclidean regimes, with their purported hierarchical capabilities stemming not from hyperbolic geometric mechanisms but rather from angular distances or composite supervisory signals. The root cause lies in curvature collapse induced by low curvature and overly wide cone configurations, which create shortcut pathways. This work is the first to demonstrate that the geometric machinery of hyperbolic models remains largely inactive and provides a reproducible evaluation paradigm.

Abstract

Whether a hyperbolic representation model uses its geometry cannot be read off its curvature parameter: what matters is the dimensionless operating point $\sqrt{c}ρ$ and whether the radial and cone machinery is active there. We develop a battery of necessary-condition diagnostics and audit three published hyperbolic vision-language families -- MERU, HyCoCLIP, and PHyCLIP -- across released checkpoints and controlled interventions on a fixed GRIT snapshot, identifying three failure modes. First, curvature is not an active resource: the operating point stays near-Euclidean ($H(u)\approx 1$; no audited converged checkpoint reaches $\sqrt{c}ρ>1$), and releasing the curvature floor moves curvature and norms but keeps the operating point near-Euclidean, without substantial downstream degradation. Second, the cone and traversal machinery is measured inoperative: entailment cones are inactive, saturated, or misaligned, and graded traversal fails under controlled readouts, while directed radial depth is a bounded non-detection above shuffle-null controls at quantified sensitivity; the one surviving native-relation residual remains non-operative. Third, hierarchy-looking evaluations are underdetermined: taxonomy correlations are carried by angular distance, and coarse-retrieval gains track box/compositional supervision, not curvature. A mechanistic account explains why: the entailment objective admits a low-curvature, wide-cone shortcut, and a parameter-free aperture identity (cones saturate iff $\sqrt{c}ρ\le 2K$) locates the edge where every entailment-trained unclamped run settles; entailment-off runs show no arrest there. The shortcut is the dominant accelerator of collapse, not its sole cause. These formulations, as released, do not instantiate the radial/cone mechanism their geometry motivates; we distill the audit into a five-number geometry report for future hierarchy claims.

Problem

• hyperbolic geometry
• vision-language models
• hierarchy
• entailment cones
• operating point

Innovation

• hyperbolic geometry
• operating point
• entailment cones
• hierarchy audit
• vision-language models