Is the Geometry Doing the Work? An Operating-Point Audit of Hierarchy in Hyperbolic Vision-Language Models
For researchers using hyperbolic embeddings for hierarchy, this work shows that published models fail to exploit the claimed geometry, undermining the validity of hierarchy claims based on these models.
The paper audits hyperbolic vision-language models (MERU, HyCoCLIP, PHyCLIP) and finds that their geometry is not actively used: the operating point remains near-Euclidean, entailment cones are inactive, and hierarchy evaluations are driven by angular distance or supervision rather than curvature. The authors identify a low-curvature shortcut that prevents the intended hyperbolic mechanism from being realized.
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.