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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

It remains unclear whether hyperbolic prototypes let a tree-structured regularizer fit class-prototype distances with less likelihood distortion than Euclidean prototypes on real hierarchical-classification problems. Comparing both geometries under the same regularizer, the study finds stronger local tree preservation with hyperbolic prototypes.

  • Problem

    It remains unclear whether hyperbolic prototypes let a tree-structured regularizer fit class-prototype distances with less likelihood distortion than Euclidean prototypes on real hierarchical-classification problems.

    Source: §1 · p. 1

  • Method

    The study compares Euclidean and Poincare prototype classifiers under the same tree-distance regularizer, varying only output manifold on WikiArt.

    Source: §1 · p. 1

  • Results

    Across paired configurations, hyperbolic prototypes gained +8.7 pp sibling recall and +15.2 pp cousin recall over matched Euclidean prototypes, robust across three reference trees.

    Source: §1 · p. 2, §5 · p. 6

  • Takeaways & Limitations

    Only hyperbolic fits improve on the encoder for local hierarchy preservation, while Euclidean fits match it on classification.

    Source: §1 · p. 2, §5 · p. 6

    Global tree-fidelity results identify no consistent winner: global metrics disagree, and the Spearman direction changes with the reference tree.

    Source: §4.6 · p. 5

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 tests whether hyperbolic prototypes make a tree-distance regularizer easier to satisfy than Euclidean prototypes without distorting class likelihood. Hyperbolic prototypes show robust gains in local hierarchy retrieval, but the results do not establish a geometry winner on global tree fidelity.

  • On WikiArt-Refined, frozen CLIP features feed a prototype classifier with a distance-based softmax and a regularizer pulling pairwise prototype distances toward tree distances.The corpus has 27 styles and 81,446 paintings; the classifier-output manifold is the sole design choice changed between Euclidean and hyperbolic runs. The study sweeps 150 seed-replicated configurations over dimension, curvature, regularizer strength, and reference-tree variants.Source: §1 · pp. 1–2
  • Because trees have exponential leaf growth while Euclidean-ball volume grows polynomially, the paper hypothesizes that hyperbolic space can satisfy the tree regularizer with less distortion of class likelihood.The Poincare ball also has exponential volume growth, matching a tree’s growth.Source: §1 · p. 1
  • Sibling recall@5 was 0.195 vs 0.149 at d=8; across 18 paired seeds, hyperbolic gains were +8.7 pp for sibling and +15.2 pp for cousin recall (sign agreement 0.94; p < 10^-4).The local-recall advantage persists across regularizer strengths and when sibling sets come from CLIP- or DINOv2-derived empirical trees.Source: §1 · p. 2
  • Raw-CLIP logistic regression’s 64.1% top-1 was statistically tied with Euclidean’s 64.3%, while k-NN sibling recall@5 (0.160) tied Euclidean (0.149) but trailed hyperbolic (0.195) by 3.5 pp.Hyperbolic prototypes were the only trained models to improve encoder local hierarchy; Euclidean prototypes added no detectable structural value over the encoder.Source: §1 · p. 2
  • Global tree-fidelity results do not identify a winner: although mean distortion against the default tree favors Euclidean, prototype-tree Spearman is not significant and the small effect flips across empirical trees.The full-sweep Spearman test has p = 0.68; the effect changes sign between CLIP- and DINOv2-derived reference trees.Source: §1 · p. 2
  • Against prior few-shot findings (Khrulkov et al., 2020), this study confirms the neighborhood/top-1 asymmetry on a hand-built taxonomy, adds linear and k-NN calibration, and makes tree-distance penalties explicit and tunable.Hyperbolic prototype embeddings and tree-distance penalties have prior precedents (Nickel & Kiela, 2017; Ganea et al., 2018; Chami et al., 2020).Source: §1 · p. 2

2. Data and reference trees

The study uses an imbalanced 27-style WikiArt corpus and evaluates prototype structure against three art-historical reference trees plus CLIP- and DINOv2-derived trees. It uses class-bias-free logits and reports balanced accuracy alongside top-1.

2. Data and reference trees

  • WikiArt-Refined contains ∼81,446 paintings across 27 styles, with a 133.3× class-size ratio motivating class-bias-free logits and balanced-accuracy reporting.The supplied train/validation split is 70/30.Source: §2 · p. 2
  • Evaluation uses lineage, chronological, and flat null trees, plus CLIP- and DINOv2-derived trees; tree distance is the unweighted shortest-path edge count.The empirical trees are built by agglomerative clustering of class-mean encoder features and used only for evaluation.Source: §2 · p. 2

3. Model and estimation

The model compares Euclidean and Poincaré latent prototypes under a scale-invariant tree-distance penalty, fits them by regularized maximum likelihood, and evaluates classification, global tree fidelity, and local tree preservation.

Likelihood model

  • A frozen CLIP ViT-B/16 encoder feeds a two-layer GELU-and-dropout MLP, which maps images to learnable class prototypes scored by a softmax over manifold distance.Source: §3 · p. 2
  • Geometry changes only the head’s final transform and classifier metric; the backbone, MLP width, dropout, batch size, optimizer, and schedule are shared.The final transform is identity for Euclidean space and the exponential map for the Poincaré ball.Source: §3 · p. 2

Tree-structured regularizer

  • The tree penalty matches mean-normalized prototype and reference-tree pairwise distances, constraining relative layout rather than absolute scale; λ = 0 recovers the cross-entropy baseline.It is a penalty, not a normalized Bayesian prior, so the estimates are regularized maximum-likelihood estimates rather than MAP estimates.Source: §3 · pp. 2–3

Estimation

  • Regularized negative log-likelihood is minimized using Adam for the head and Riemannian Adam for hyperbolic prototypes, with the penalty evaluated over 351 style pairs per step.The study ran 150 fits; headline configurations were replicated across three seeds, and paired-t comparisons used 18 dimension-and-seed pairs.Source: §3 · p. 3

Distortion bound

  • Sala et al. (2018) give distortion bounds of approximately 1/d for the Poincaré ball versus a constant floor for Euclidean space at fixed dimension.The empirical question is whether this asymptotic result appears at modest dimension on a real dataset under a classification likelihood and finite-sample fitting.Source: §3 · p. 3

Evaluation

  • Evaluation covers classification accuracy, global tree fidelity via distance-matrix correlation and distortion, and local sibling/cousin recall@k against three reference trees.Local recall uses k ∈ {5, 10}; the trees are the default, CLIP-empirical, and DINOv2-empirical references.Source: §3 · p. 3

4. Results

Across the 150-run sweep, the geometries split along a local-versus-global axis: hyperbolic prototypes better preserve local tree structure, while global tree-fidelity comparisons are unstable across metrics and reference trees. Against raw-CLIP baselines, only the hyperbolic fit adds local-structural value over the encoder.

4.2. Local tree structure

  • Hyperbolic prototypes gain +8.7 pp on sibling recall@5 and +15.2 pp on cousin recall, with sign agreement 0.94 and paired-t p < 10^-4.The sibling-recall gap widens as Euclidean recall falls from 0.221 at d=2 to 0.142 at d=64 while hyperbolic recall stays near 0.19. The lead persists across hand-built, CLIP-derived, and DINOv2-derived trees; at d=8, the CLIP-tree gap is +6.0 pp versus +4.6 pp on the default tree. Hyperbolic prototypes also settle near the Poincare-ball boundary, consistent with the exponential-volume regime (Nickel & Kiela, 2017).Source: §4.2 · pp. 3–4, Figure 2 · p. 5, Figure 3 · p. 6

4.3. Calibration baselines

  • Against raw-CLIP baselines, only hyperbolic improves sibling recall over k-NN-5 (0.188 vs. 0.160), while Euclidean recall is lower (0.136).Best-top-1 Euclidean reaches 65.9% top-1, 1.8 pp above logistic regression, but has 55.4% balanced accuracy versus 61.1%; best-top-1 hyperbolic reaches 60.6% top-1, 2.9 pp below k-NN. At d=8, hyperbolic sibling recall@5 is 0.195 on the default tree and 0.416 on the empirical tree, versus 0.160 and 0.366 for k-NN.Source: §4.3 · p. 4, Figure 4 · p. 6

4.1. Headline comparison

  • Best configurations split: Euclidean leads top-1 by +5.3 pp, top-5 by +2.4 pp, and global-tree Spearman by +0.108; hyperbolic leads sibling recall@5 by +0.052 and cousin recall@5 by +0.047.The seed bands do not overlap on any of the five metrics.Source: §4.1 · p. 3, Table 1 · p. 5

4.4. Regularizer strength

  • The Euclidean top-1 lead and hyperbolic sibling-recall lead persist at every tested regularizer strength λ.At λ=0.1, Euclidean d=8 top-1 rises 0.7 pp to 65.0% and tree-Spearman rises from 0.292 to 0.345, while hyperbolic gains a fraction of a point on top-1 and 2 pp on sibling recall. At λ=3, both reach tree-Spearman ≈0.7 but classification collapses to 46.9% for Euclidean and 37.4% for hyperbolic.Source: §4.4 · pp. 4–5

4.5. Training-tree and class-imbalance robustness

  • Across training trees, Euclidean wins top-1 and hyperbolic wins sibling recall; inverse-frequency class weighting preserves the geometry gap on every reported metric.The top-1 gap remains 5 pp for Euclidean and the sibling-recall gap 4 pp for hyperbolic; DINOv2-based empirical-tree construction reverses the small global-Spearman gap but preserves the local-recall lead.Source: §4.5 · p. 5

4.6. Global tree fidelity is unstable

  • Global tree-fidelity results do not establish a winner: default-tree Spearman has p = 0.68 and sign agreement 0.50, while mean tree distortion favors Euclidean (+0.040, p < 10^-4).Spearman direction reverses with the reference tree: hyperbolic leads on CLIP-derived trees, while Euclidean leads on DINOv2-derived trees. Across four reference-tree constructions, only the instability of the comparison is consistent.Source: §4 · p. 3, §4.6 · p. 5

5. Discussion

Hyperbolic prototypes preserve local hierarchy better than Euclidean prototypes, while classification and global tree-fidelity results do not establish a general geometric winner. The findings are limited to frozen-encoder WikiArt experiments and motivate tests with hyperbolic heads, alternative reference trees, and deeper hierarchies.

5. Discussion

  • +8.7 pp sibling and +15.2 pp cousin recall over the dimension sweep favor hyperbolic prototypes (paired-t p < 10−4), with the advantage holding across empirical reference trees.At d=8, sibling recall@5 is 0.195 for hyperbolic, 0.149 for Euclidean, and 0.160 for k-NN-5 on raw CLIP features; the hyperbolic fit improves over the encoder by 3.5 pp.Source: §5 · p. 6
  • 64.1% top-1 from logistic regression on raw CLIP features is matched by the Euclidean fit at d=8 (64.3%), while hyperbolic underperforms by 4–6 pp.The classification gap does not show meaningful classification work beyond what a linear head on the encoder features supports.Source: §5 · p. 6
  • Global tree-fidelity comparisons do not establish a winner: mean tree distortion favors Euclidean against the default tree, while global-Spearman gaps reverse between CLIP- and DINOv2-derived trees.Class-center/tree Spearman is not significantly different from zero across the sweep (p = 0.68); sibling and cousin recall favor hyperbolic on every tested reference tree.Source: §5 · p. 6
  • The results may not generalize beyond medium-scale, Western-canon-heavy WikiArt trained with frozen CLIP-ViT-B/16 features or to a deeper hierarchy; only prototypes, not the head MLP, live on the manifold.The study uses a 27-leaf taxonomy, and the local advantage may not survive encoder fine-tuning.Source: §5 · p. 6
  • Three proposed follow-ups are a genuinely hyperbolic head, reference trees built in hyperbolic space or by direct tree-learning, and replication on a deeper hierarchy.A hyperbolic head would test whether the classification gap closes; deeper-hierarchy tests could use iNaturalist with WordNet or finer WikiArt subdivisions.Source: §5 · p. 7

Broader impact · A. Default style hierarchy · B. Class distribution and tree-distance matrices

The paper warns that its hierarchy-based results inherit a Western art-history canon and that WikiArt’s class imbalance risks disadvantaging non-Western works. It cautions against using hyperbolic style embeddings for consequential attribution or authentication without expert review.

Broader impact

  • The hand-built default reference tree follows standard art-history references, and hierarchy-based metrics implicitly endorse its Western lineage canon.The paper says tree-variant experiments only partially mitigate this dependence and recommends treating the tree as a configurable choice in museums or classrooms.Source: Broader impact · p. 7, A. Default style hierarchy · p. 9
  • WikiArt’s 27 styles include only one non-European-and-American label, Ukiyo-e, while centuries-spanning East Asian ink traditions are collapsed into it.The paper warns that retrieval trained on this corpus will under-rank non-Western works for ambiguous queries.Source: Broader impact · p. 7
  • Hyperbolic geometry could compound this bias by tightening local neighborhoods and making them more confident, so the authors advise expert review for attribution or authentication.They do not recommend deployment for these tasks without human review.Source: Broader impact · p. 7
  • The system achieves 65% top-1 accuracy over 27 well-known styles, far below the threshold any serious provenance, insurance, or legal decision should require.The system is a small 27-way classifier on cached features.Source: Broader impact · pp. 7–8

C. Curvature within the hyperbolic family · D. Regularizer-strength sweep, full panel set

Within the hyperbolic family, curvature trades top-1 accuracy against sibling recall, with different optimal curvature values for the two metrics at d=8.

C. Curvature within the hyperbolic family

E. Training-tree ablation, full panel set

Training against a non-evaluation tree provides little hierarchical signal: tree-Spearman falls to about 0.14, while top-1 returns to the λ=0 baseline. The supplied figures also show the reference-tree distance matrices and substantial style-count imbalance.

E. Training-tree ablation, full panel set

  • Training on either non-default tree drops tree-Spearman against the default tree to ≈0.14 and returns top-1 to the λ=0 baseline.The regularizer no longer competes with the likelihood when the training tree differs from the evaluation tree.Source: E. Training-tree ablation, full panel… · p. 9
  • Impressionism has 13,060 examples, while the three named Cubist and Action Painting styles each have fewer than 250, producing a 133.3× extrema imbalance.The counts are for WikiArt-Refined.Source: Figure 7 · p. 10
  • Figure 8 shows distance matrices for the default lineage, chronological era grouping, and flat-null reference hierarchies.Styles appear in the same row and column order across panels.Source: Figure 8 · p. 10

F. Empirical reference trees and DINOv2 cross-encoder

Hyperbolic prototypes retain a sibling-recall@5 lead across the default and both empirical reference trees, while global tree-fidelity comparisons depend on the reference tree. The hand-built lineage tree is an outlier relative to the encoder-derived trees.

F. Empirical reference trees and DINOv2 cross-encoder

  • Pairwise-distance Spearman is 0.08 for default vs. CLIP-empirical, 0.12 for default vs. DINOv2-empirical, and 0.50 for CLIP-empirical vs. DINOv2-empirical trees.The hand-built lineage tree is the outlier; the two encoders capture more similar structures.Source: F. Empirical reference trees and… · p. 10
  • At d=8, sibling recall@5 favors Hy over Eu on default (0.195 vs. 0.149), CLIP-empirical (0.416 vs. 0.356), and DINOv2-empirical (0.380 vs. 0.322) trees.Sibling and cousin sets were rebuilt from each empirical tree’s binary linkage; the hyperbolic lead was preserved on every reference tree.Source: F. Empirical reference trees and… · p. 10
  • At d=64, against DINOv2-empirical, global Spearman is 0.413 for Eu vs. 0.357 for Hy (ρEu − ρHy = +0.06).Euclidean prototypes lead at every d ≥4 against this tree, qualifying the global-fidelity comparison.Source: F. Empirical reference trees and… · p. 10
  • Mean tree distortion decreases monotonically with λ in both geometries, mirroring the rise in tree-Spearman.Figure 9 shows the full four-panel regularizer-strength sweep; the body figure omits its third panel for legibility.Source: Figure 9 · p. 11
  • Tree distortion against the default tree is uniformly higher when training against a non-default tree.Figure 10 presents the full four-panel training-tree ablation; the body figure omits its third panel for legibility.Source: Figure 10 · p. 11

G. Class-imbalance robustness · H. Confusion structure

Inverse-frequency weighting improves balanced accuracy while lowering top-1 accuracy for both geometries, but preserves their reported geometry gaps. The balanced-accuracy gap narrows, with Euclidean still ahead.

G. Class-imbalance robustness

  • Top-1 accuracy shifts from 64.3 to 58.3% for Euclidean and 59.8 to 53.4% for hyperbolic prototypes under class weighting.Balanced accuracy rises from 56.4 to 65.6% for Euclidean and 44.5 to 62.1% for hyperbolic prototypes.Source: G. Class-imbalance robustness · p. 11
  • The balanced-accuracy gap narrows from 11.9 to 3.5 percentage points under weighting, but remains in Euclidean’s favour.The authors suggest hyperbolic’s main default-setup classification weakness was disproportionately on rare classes.Source: G. Class-imbalance robustness · p. 11
  • Despite weighting, top-1 accuracy remains 5 percentage points higher for Euclidean, while sibling recall remains 4 points higher for hyperbolic.Class-center and tree Spearman correlations shift by less than 0.01.Source: G. Class-imbalance robustness · p. 11
  • WikiArt is heavily imbalanced: Impressionism has 133× more training examples than Action Painting.The default sweep uses unweighted cross-entropy; the weighted rerun covers the d=8 winners of each geometry with three seeds.Source: G. Class-imbalance robustness · p. 11

I. Reproducibility

The section provides a 150-row sweep record and scripts for rerunning experiments, baselines, significance tests, and figures; code, configurations, and exact metric implementations are available in the project repository.

  • A full-sweep CSV records training hyperparameters and every evaluation metric for all 150 configurations.Source: §I · p. 11
  • Run sweep phases 1–3 on MPS for 90, 48, and 18 configurations in approximately 14, 10, and 3 minutes, respectively.Additional scripts run logistic-regression and kNN baselines, paired-t and sign tests, and figure generation.Source: §I · p. 11
  • Code, sweep configurations, and exact metric implementations are available in the project's GitHub repository.Source: §I · p. 11

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