# Classifiers in High Dimensional Hilbert Metrics

arXiv: https://arxiv.org/abs/2601.13410 · HTML: https://mlsanity.com/paper/2601.13410/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 19 Jan 2026.

## Builds on (most cited first, 16 of 16)

- [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)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Embedding Text in Hyperbolic Spaces](https://mlsanity.com/paper/1806.04313/) (2018)
- [Sinkhorn Divergences for Unbalanced Optimal Transport](https://mlsanity.com/paper/1910.12958/) (2019)
- [The Numerical Stability of Hyperbolic Representation Learning](https://mlsanity.com/paper/2211.00181/) (2022)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Tropical Attention: Neural Algorithmic Reasoning for Combinatorial Algorithms](https://mlsanity.com/paper/2505.17190/) (2025)
- [Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning](https://mlsanity.com/paper/2302.09738/) (2023)
- [Non-linear Embeddings in Hilbert Simplex Geometry](https://mlsanity.com/paper/2203.11434/) (2022)
- [Voronoi Diagrams in the Hilbert Metric](https://mlsanity.com/paper/2112.03056/) (2021)
- [Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian family](https://mlsanity.com/paper/2508.14369/) (2025)
- [Economical Convex Coverings and Applications](https://mlsanity.com/paper/2303.08349/) (2023)

## Abstract (arXiv)

Classifying points in high dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address classifying points in the d-dimensional Hilbert polygonal metric. The Hilbert metric is a generalization of the Cayley-Klein hyperbolic distance to arbitrary convex bodies and has a diverse range of applications in machine learning and convex geometry. We first present an efficient LP-based algorithm in the metric for the large-margin SVM problem. Our algorithm runs in time polynomial to the number of points, bounding facets, and dimension. This is a significant improvement on previous works, which either provide no theoretical guarantees on running time, or suffer from exponential runtime. We also consider the closely related Funk metric. We also present efficient algorithms for the soft-margin SVM problem and for nearest neighbor-based classification in the Hilbert metric.