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

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.

Improving Gradient Flow with Unrolled Highway Expectation Maximization

Integrating model-based machine learning methods into deep neural architectures allows one to leverage both the expressive power of deep neural nets and the ability of model-based methods to incorporate domain-specific knowledge. In particular, many works have employed the expectation maximization (EM) algorithm in the form of an unrolled layer-wise structure that is jointly trained with a backbone neural network. However, it is difficult to discriminatively train the backbone network by backpropagating through the EM iterations as they are prone to the vanishing gradient problem. To address this issue, we propose Highway Expectation Maximization Networks (HEMNet), which is comprised of unrolled iterations of the generalized EM (GEM) algorithm based on the Newton-Rahpson method. HEMNet features scaled skip connections, or highways, along the depths of the unrolled architecture, resulting in improved gradient flow during backpropagation while incurring negligible additional computation and memory costs compared to standard unrolled EM. Furthermore, HEMNet preserves the underlying EM procedure, thereby fully retaining the convergence properties of the original EM algorithm. We achieve significant improvement in performance on several semantic segmentation benchmarks and empirically show that HEMNet effectively alleviates gradient decay.