New Hyperbolic Entailment Cones Beyond the Capacity–Feasibility Trade-off

Entailment cones in hyperbolic space (Ganea et al., 2018b) represent the partial order of a hierarchy as containment between cones, and serve as a foundation of recent hyperbolic vision-language models. However, the cones' aperture cannot be defined everywhere; there is necessarily an infeasible ball region around the origin. The size of the infeasible ball trades off against the capacity (total aperture) of the cones, and at both ends the performance of the entailment cones degrades significantly. We first formalize this capacity–feasibility trade-off and show that this trade-off extends to any cones satisfying transitivity, i.e., the trade-off is an unavoidable problem in the original formulation. To overcome this trade-off, we then introduce $(\epsilon ,\delta )$-transitivity, which reformulates transitivity with two parameters: the radius of an infeasible region and an allowance for violations of transitivity. Building on this, we propose cones that can be defined everywhere except the origin, have at least the same capacity, and generalize the original entailment cones. On hierarchical word embeddings (WordNet, MCG, Hearst), the proposed method outperforms the original entailment cones in 10 of 12 settings, with gains of up to 27.6 points. On the hyperbolic vision-language model MERU, it preserves accuracy on zero-shot classification while yielding embeddings that better satisfy the entailment condition of the original formulation.