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

arXiv: https://arxiv.org/abs/2608.25199 · HTML: https://mlsanity.com/paper/2608.25199/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 9 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)

## Abstract (arXiv)

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.