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

Peter Flo, Luca Grossmann

arXiv:2608.25199v1cs.LGcs.AI

TL;DR

The paper asks whether Euclidean or hyperbolic prototype geometry better supports a tree-structured regularizer without distorting data likelihood. It compares the geometries in 150 regularized fits on WikiArt across dimensions, curvature, regularizer strength, and reference trees. Hyperbolic prototypes robustly improve local hierarchy preservation, while global tree-fidelity comparisons remain unstable.

  • Problem

    The paper tests whether choosing Euclidean R^d or the Poincare ball B^d_c affects how cheaply a tree-structured prototype regularizer can be satisfied without distorting class-conditional likelihood.

  • Method

    The study compares shared prototype-classifier models using Euclidean or hyperbolic latent manifolds across 150 seed-replicated regularized maximum-likelihood fits on WikiArt.

  • Results

    Hyperbolic prototypes improve local hierarchy preservation over matched Euclidean prototypes by +8.7 pp in sibling recall and +15.2 pp in cousin recall, while Euclidean prototypes match logistic regression on classification.

  • Takeaways & Limitations

    The results separate a robust local effect of latent geometry from an unstable global effect in this tree-regularized prototype model.

  • Takeaways & Limitations

    Global tree-fidelity conclusions depend on the reference tree, with the observed Spearman direction reversing between CLIP-derived and DINOv2-derived trees.

Abstract

from arXiv · show

We study a tree-structured regularizer over class-prototype layouts in a hierarchical-classification model and ask whether the choice of latent manifold for the prototypes (Euclidean R^d vs. the Poincare ball B^d_c) affects how well that regularizer can be satisfied without distorting the data likelihood. The two manifolds differ only in their volume growth: hyperbolic space grows exponentially with radius and embeds trees with provably lower distortion than R^d of matched dimension, so the structured regularizer should be cheaper to satisfy on B^d_c. Across 150 seed-replicated regularized maximum-likelihood fits spanning embedding dimension, curvature, and regularizer strength on WikiArt (27 styles, 81,446 paintings, frozen CLIP ViT-B/16 features), we find a single robust effect: Poincare 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, sign agreement 0.94), and the gap holds across three reference-tree definitions (hand-built lineage, CLIP-derived, and DINOv2-derived). On classification, Euclidean prototypes are tied with logistic regression on raw encoder features, indicating no detectable contribution from the latent geometry; only the hyperbolic fit improves on a k-NN encoder baseline for local retrieval. Global tree-fidelity comparisons are unstable across reference trees and we do not claim a winner. The results give an empirical separation, on a real hierarchical-classification problem, between two natural latent geometries for a class-structured regularizer.

1. Introduction

The paper asks whether hyperbolic or Euclidean prototype geometry makes a tree-structured regularizer cheaper to satisfy without distorting likelihood. Across 150 fits, hyperbolic prototypes improve local hierarchy preservation, while global fidelity remains inconclusive.

  • Motivation: A tree-structured regularizer balances class-likelihood fit against matching pairwise prototype distances to an externally specified class hierarchy.The likelihood pulls paintings toward style prototypes, while the regularizer constrains prototype geometry to respect tree shape.
  • Motivation: Hyperbolic space is expected to satisfy the regularizer more cheaply because its exponential volume growth embeds trees with lower distortion than matched-dimensional Euclidean space.The test concerns whether this geometric advantage appears in finite-sample hierarchical classification without distorting likelihood.
  • Study design: The study uses WikiArt with 27 styles and 81,446 paintings, frozen CLIP ViT-B/16 features, shared regularization, and 150 seed-replicated configurations.The sweep spans embedding dimension, curvature, regularizer strength, and three reference-tree variants.
  • Findings: Sibling recall improves by +8.7 pp and cousin recall by +15.2 pp for hyperbolic versus Euclidean prototypes, with sign agreement 0.94 and paired-t p < 10^-4.The effect persists for sibling sets derived from CLIP- and DINOv2-based empirical trees and across regularizer strengths.
  • Findings: Euclidean prototypes match logistic regression on classification, whereas only the hyperbolic fit improves local hierarchy retrieval over encoder-based baselines.The Euclidean fit adds no detectable structural value over the encoder, while global tree-fidelity comparisons do not establish a winner.

2. Data and reference trees

The study uses WikiArt-Refined, a 27-style, 81,446-painting corpus with imbalanced classes and multiple plausible reference trees. The trees include a lineage taxonomy, chronological grouping, and a flat null.

  • Dataset: WikiArt-Refined contains approximately 81,446 paintings labeled with one of 27 styles and uses the supplied 70/30 train/validation split.The dataset is heavily imbalanced, with a 133.3× ratio between Impressionism and Analytical Cubism.
  • Reference trees: The default reference tree follows standard art-historical lineage, while the chronological tree groups styles into six era buckets.The chronological buckets include Renaissance, Baroque–Rococo, nineteenth century, early twentieth century, modern post-war, and non-Western.
  • Reference trees: The flat tree makes every style a direct child of the root and serves as a deliberate null with constant off-diagonal pairwise distances.

3. Model and estimation

The model maps frozen CLIP features into either Euclidean or Poincare latent space, classifies by distance to learnable prototypes, and regularizes prototype distances toward a reference-tree metric. Optimization uses regularized maximum likelihood with geometry-specific prototype updates.

  • Model: A frozen CLIP ViT-B/16 encoder and two-layer MLP map each image into a latent manifold that is either Euclidean R^d or a Poincare ball B^d_c.The encoder and head architecture are shared across geometries; only the final transform and classifier metric change.
  • Model: The class-conditional likelihood is a softmax over distances from latent embeddings to learnable class prototypes.Each image is mapped to z, and prototype distances determine the class probabilities.
  • Regularization: The regularizer penalizes deviation between the pairwise prototype-distance matrix and the reference tree-distance matrix.Mean normalization makes the penalty invariant to the absolute distance scale and constrains relative prototype structure.
  • Estimation: Training minimizes L(θ, {p_k}) = LCE(θ, {p_k}) + R({p_k}) using stochastic gradient descent, with Riemannian Adam for hyperbolic prototypes.Each hyperbolic update is projected back onto the manifold.
  • Evaluation: The evaluation separates classification, global tree fidelity, and local tree preservation across default, CLIP-empirical, and DINOv2-empirical trees.Local metrics are sibling and cousin recall@k among nearest neighbors; global metrics include Spearman correlation and multiplicative distortion.

4. Results

Across the sweep, hyperbolic prototypes consistently preserve local tree structure better, while Euclidean prototypes lead classification and some global metrics; global tree-fidelity results remain unstable.

  • Dimension scaling: As dimension increases, Euclidean sibling recall@5 falls from 0.221 at d=2 to 0.142 at d=64, while hyperbolic recall stays nearly flat and opens a 5 pp gap.Top-1 saturates near d=16 for both geometries, with Euclidean ahead by 4–6 pp.
  • Local tree structure: Hyperbolic prototypes improve local structure over Euclidean ones, with mean gaps of +8.7 pp in sibling recall@5 and +15.2 pp in cousin recall.The paired comparisons show sign agreement 0.94 and paired-t p < 10^-4.
  • Local tree structure: At d=8, hyperbolic sibling recall@5 is 0.195 versus 0.149 for Euclidean on the default tree.The hyperbolic advantage persists when sibling and cousin sets are rebuilt from CLIP- or DINOv2-derived empirical trees.
  • Regularizer strength: The regularizer can raise tree-Spearman, but at λ=3 classification collapses to 46.9% for Euclidean and 37.4% for hyperbolic.The geometry ordering persists across the tested regularizer strengths: Euclidean leads top-1, while hyperbolic leads sibling recall.
  • Global tree fidelity: Global tree-fidelity conclusions depend on the metric and reference tree, so the experiments do not identify a geometry winner on that axis.Spearman effects reverse across CLIP- and DINOv2-derived empirical trees, while mean distortion favors Euclidean.

5. Discussion

The experiments separate local from global effects: hyperbolic prototypes consistently improve local hierarchy preservation, while global tree-fidelity comparisons remain unstable. Euclidean prototypes retain the classification advantage and match the raw-feature linear baseline.

  • Local hierarchy: The local-retrieval advantage persists when sibling sets are derived from CLIP- or DINOv2-based empirical trees.This robustness holds across the reference-tree definitions and the regularizer-strength sweep.
  • Classification: 64.1% top-1 from logistic regression on raw CLIP features is statistically tied with 64.3% for Euclidean prototypes.The hyperbolic fit underperforms the logistic baseline by 4–6 pp in the reported classification comparison.
  • Global tree fidelity: The global geometry comparison is unresolved: mean tree distortion favors Euclidean, while tree-Spearman is nonsignificant across the sweep at p = 0.68 and reverses sign across empirical trees.Accordingly, the paper does not claim a winner on global tree fidelity.
  • Reference-tree dependence: The training-tree ablation shows that regularization helps the metric anchored to the training tree, while non-default training trees collapse against the default evaluation tree.The geometry winner remains unchanged across training trees: Euclidean wins top-1 and hyperbolic wins sibling recall.
  • Extensions: A genuinely hyperbolic head is proposed as a follow-up to test whether the manifold can shape representations rather than only the decision boundary.The current experiments place only prototypes on the manifold.

Broader impact

The paper’s broader-impact concerns center on a Western, lineage-based taxonomy and severe dataset imbalance. These choices limit interpretation and may make retrieval less reliable for non-Western works or high-stakes attribution.

  • Taxonomy: The default hierarchy encodes a Western, lineage-based art-history canon, so every hierarchy-dependent metric inherits that taxonomy.The paper treats tree choice as configuration rather than a constant and evaluates variants as partial mitigation.
  • Dataset scope: WikiArt is heavily biased toward European painting, with Ukiyo-e the only style outside the European-and-American canon among 27 labels.East Asian ink-painting traditions spanning centuries are collapsed into that single label.
  • Deployment: A retrieval system trained on this data may under-rank non-Western works for ambiguous queries, and tighter hyperbolic neighborhoods could compound that effect.The paper therefore does not recommend hyperbolic style embeddings for attribution or authentication without expert human review.
  • Deployment: The reported 50–65% top-1 accuracy over 27 well-known styles is below the threshold suggested for serious provenance, insurance, or legal decisions.

B. Class distribution and tree-distance matrices

The hyperbolic curvature sweep exposes a trade-off between classification and local hierarchy retrieval. Higher curvature improves sibling recall while reducing top-1 accuracy within the tested configurations.

  • Curvature sweep: Higher curvature exchanges 2–3 pp of top-1 accuracy for 4–5 pp of sibling recall within the hyperbolic family.Higher c pulls prototypes closer to the Poincare-ball boundary, where the metric is most curved.
  • Curvature sweep: At d=8, curvature c=0.3 optimizes top-1 at 59.8%, while c=3.0 optimizes sibling recall at 0.230.

E. Training-tree ablation, full panel set

Training against a reference tree different from the evaluation tree provides little useful hierarchical signal, while the dataset’s per-style counts are highly imbalanced. The tree matrices compare three alternative hierarchy structures over the same styles.

  • Training-tree ablation: Tree-Spearman against the default tree drops to approximately 0.14 for both non-default training trees.Top-1 returns to the λ=0 baseline because the mismatched regularizer no longer competes with the likelihood.
  • Training-tree ablation: Euclidean wins top-1 and hyperbolic wins sibling recall for every training tree tested.
  • Class distribution: Impressionism has 13,060 examples, while Analytical Cubism, Action Painting, and Synthetic Cubism each have fewer than 250.The imbalance ratio between the largest and smallest classes is 133.3×.
  • Tree-distance matrices: The three tree-distance matrices represent default lineage, chronological era grouping, and a flat null, with styles ordered identically across panels.

F. Empirical reference trees and DINOv2 cross-encoder

Empirical reference trees reveal that global prototype-tree alignment depends on the encoder, while hyperbolic prototypes retain their local-hierarchy advantage across reference definitions.

  • Reference-tree relationships: The hand-built default tree is an outlier: its pairwise-distance Spearman correlations are 0.08 with the CLIP-empirical tree and 0.12 with the DINOv2-empirical tree, versus 0.50 between empirical trees.The encoders therefore capture similar structure that only weakly matches the hand-built lineage.
  • Local hierarchy: Hyperbolic prototypes lead in sibling recall@5 on the default, CLIP-empirical, and DINOv2-empirical trees at d=8: Eu 0.149/0.356/0.322 versus Hy 0.195/0.416/0.380.The reference-tree sensitivity test rebuilds sibling and cousin sets from each empirical tree’s binary linkage before rerunning recall.
  • Global tree fidelity: Against the DINOv2-empirical tree, Euclidean prototypes lead at every d ≥4, reaching ρEu − ρHy = +0.06 at d=64 (0.413 vs. 0.357).This reverses the global-Spearman direction observed against the CLIP-empirical tree.
  • Regularizer strength: Mean tree distortion decreases monotonically with λ in both geometries, paralleling the rise in tree-Spearman.The full regularizer-strength sweep reports this trend across the two latent geometries.
  • Training-tree ablation: Tree distortion against the default tree is uniformly higher when training against a non-default tree.This result comes from the full four-panel training-tree ablation.

G. Class-imbalance robustness

Inverse-frequency class weighting changes absolute classification and balanced-accuracy values but preserves the geometry gap across every reported axis.

  • Class-imbalance robustness: The geometry gap is preserved on every axis after inverse-frequency class-weighted cross-entropy reruns of the d=8 winners.The robustness check uses three seeds and addresses WikiArt’s severe class imbalance.
  • Class-imbalance robustness: Top-1 accuracy drops by roughly 6 pp for both geometries: Eu 64.3 →58.3% and Hy 59.8 →53.4%.Balanced accuracy rises sharply under weighting: Eu 56.4 →65.6% and Hy 44.5 →62.1%.

I. Reproducibility

The paper documents a 150-row sweep, scripts for reproducing training and evaluation, and figures comparing hand-built with empirical reference trees across encoders.

  • Reproducibility: The full sweep CSV records training hyperparameters and every evaluation metric for 150 configurations.The reproducibility materials identify the complete configuration-level record.
  • Reproducibility: The pipeline provides scripts for three sweep phases, baseline evaluation, significance tests, and figure regeneration.The listed commands run phases with approximately 14, 10, and 3 minutes on MPS, followed by baselines, significance, and figure generation.
  • Reproducibility: Code, sweep configurations, and exact metric implementations are available in the project repository.The repository URL is supplied in the paper materials.
  • Reference-tree alignment: Phase 4 aligns prototypes against hand-built default and data-driven empirical reference trees.The comparison is explicitly organized around default versus empirical references.
  • Reference-tree alignment: The reference-tree figures compare prototype-tree Spearman and sibling recall across hand-built, CLIP-empirical, and DINOv2-empirical constructions.The DINOv2 figure reports global-Spearman and sibling-recall comparisons across dimensions, while the confusion-matrix figure orders styles by default-tree traversal.
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