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

Luca Grossmann, Peter Flo

Keywords: hyperbolic geometry, structured regularization, hierarchical classification, prototype networks

TL;DR: Hyperbolic-ball prototype classifiers on WikiArt's style hierarchy preserve local tree structure better than Euclidean ones on sibling/cousin recall, robust across reference-tree definitions, at a modest top-1 cost.

Abstract:

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.

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Submission Number: 164