# Minimax Rates for Hyperbolic Hierarchical Learning

arXiv: https://arxiv.org/abs/2601.20047 · HTML: https://mlsanity.com/paper/2601.20047/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 23 references that are in the map.
- First posted on arXiv 27 Jan 2026.

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

- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [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 Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Comparing Euclidean and Hyperbolic Embeddings on the WordNet Nouns Hypernymy Graph](https://mlsanity.com/paper/2109.07488/) (2021)
- [Why are hyperbolic neural networks effective? A study on hierarchical representation capability](https://mlsanity.com/paper/2402.02478/) (2024)

## Abstract (arXiv)

We prove an exponential separation in sample complexity between Euclidean and hyperbolic representations for learning on hierarchical data under standard Lipschitz regularization. For depth-R hierarchies with branching factor m, we first establish a geometric obstruction for Euclidean space: any bounded-radius embedding forces volumetric collapse, mapping exponentially many tree-distant points to nearby locations. This necessitates Lipschitz constants scaling as exp(Omega(R)) to realize even simple hierarchical targets, yielding exponential sample complexity under capacity control. We then show this obstruction vanishes in hyperbolic space: constant-distortion hyperbolic embeddings admit O(1)-Lipschitz realizability, enabling learning with n = O(mR log m) samples. A matching Omega(mR log m) lower bound via Fano's inequality establishes that hyperbolic representations achieve the information-theoretic optimum. We also show a geometry-independent bottleneck: any rank-k prediction space captures only O(k) canonical hierarchical contrasts.