Hyperbolic Latent Geometry for Tree-Structured Prototype Networks: A Local-vs-Global Trade-off

We ask how the choice of latent manifold for class prototypes (Euclidean $\mathbb{R}^d$ vs.\ the Poincar\'e ball $\mathbb{B}^d_c$) affects fitting a tree-structured regularizer on prototype layouts in a hierarchical-classification model, since hyperbolic space embeds trees with provably lower distortion than $\mathbb{R}^d$ at matched dimension. Across 150 seed-replicated regularized maximum-likelihood fits on WikiArt (27 styles, 81{,}446 paintings, frozen CLIP ViT-B/16 features) spanning embedding dimension, curvature, and regularizer strength, Poincar\'e prototypes preserve the topology of the nearest-neighbor graph in latent space substantially better than matched Euclidean prototypes (sibling recall@5 $+8.7$\,pp, cousin recall $+15.2$\,pp; paired-$t$ $p<10^{-4}$), and the gap holds across hand-built, CLIP-derived, and DINOv2-derived reference trees. Calibrated against logistic-regression and $k$-NN baselines on raw encoder features, only the hyperbolic fit improves on the encoder for local retrieval; the Euclidean fit ties logistic regression on classification but adds no detectable structural value over it. Global tree-fidelity comparisons are unstable across reference trees and we do not claim a winner there. The results separate two natural latent geometries for a class-structured regularizer on a real hierarchical-classification problem.