Curvature Under Attack in hZACH-ViT: Gauge Symmetry, Boundary Saturation, and Adversarial Failure

Compact Vision Transformers are attractive for medical imaging in low-data and resource-constrained settings, but most existing variants assume that Euclidean latent geometry is sufficient for organizing image representations. We introduce hZACH-ViT, a family of curved-geometry extensions of ZACH-ViT, a compact zero-token Vision Transformer that removes positional embeddings and the class token and relies on global average pooling over patch representations. To isolate the role of geometry, we preserve the verified ZACH-ViT backbone and modify only the final representation space and prototype-based classifier head, enabling a controlled comparison between Euclidean, hyperbolic, and spherical latent geometries. We evaluate Poincaré, Klein, and spherical hZACH-ViT heads on seven MedMNIST datasets under an identical few-shot protocol with 50 samples per class and five random seeds. The completed benchmark contains 770 training runs spanning seven datasets, three non-Euclidean geometries, seven curvature magnitudes, and a Euclidean baseline. Across all seven datasets, the best non-Euclidean hZACH-ViT configuration improves over Euclidean ZACH-ViT, with an average gain of +0.021 in the dataset-specific primary metric and the largest improvement on OCTMNIST (+0.055 MacroF1). Fixed low-curvature configurations retain positive gains on the majority of datasets, and low curvature values (c = 0.1 or 0.2) account for six of the seven dataset-level winners. Rather than identifying a universally optimal manifold, our results establish geometry and curvature as dataset-dependent model-selection variables, with fixed low-curvature analyses confirming that gains persist beyond exhaustive per-dataset tuning.

Whether a hyperbolic representation model uses its geometry cannot be inferred from curvature alone: what matters is the dimensionless operating point $\sqrt{c}ρ$ and whether the radial and cone mechanisms are operational there. We develop necessary-condition diagnostics and audit three published hyperbolic vision-language families -- MERU, HyCoCLIP, and PHyCLIP -- across released checkpoints and matched interventions. All converged checkpoints remain near-Euclidean ($H(u)\approx1$; none reaches $\sqrt{c}ρ>1$), and releasing the curvature floor changes $c$ and norms without leaving this 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 surviving pair-specific signal, a statistically detectable but small residual on the GRIT box-to-full-caption relation, remains non-operative under the evaluated readouts. Taxonomy correlations show no detectable norm contribution beyond cosine, and 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 $\sqrt{c}ρ\le2K$: with the floor released, all trained relation-level parent means lie at or below this edge, leaving the parent cones fully or nearly saturated. Entailment-off runs pass the edge and continue contracting. The shortcut is the dominant accelerator of collapse, not its sole cause. 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.

Training billion-parameter Transformers is often brittle, with transient loss spikes and divergence that waste compute. Even though the recently developed Edge of Stability (EoS) theory provides a powerful tool to understand and control the stability of optimization methods via the (preconditioned) curvature, these curvature-controlling methods are not popular in large-scale Transformer training due to the complexity of curvature estimation. To this end, we first introduce a fast online estimator of the largest (preconditioned) Hessian eigenvalue (i.e., curvature) based on a warm-started variant for power iteration with Hessian-vector products. We show theoretically, and verify empirically, that the proposed method makes per-iteration curvature tracking feasible at billion parameter scale while being more accurate. Using this tool, we find that training instabilities coincide with surges in preconditioned curvature and that curvature grows with depth. Motivated by these observations, we propose architecture warm-up: progressively growing network depth to carefully control the preconditioned Hessian and stabilize training. Experiments on large Transformers validate that our approach enables efficient curvature tracking and reduces instabilities compared to existing state-of-the-art stabilization techniques without slowing down convergence.