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Classifiers in High Dimensional Hilbert Metrics

arXiv 2601.13410 ↗First posted 19 Jan 2026cs.CG cs.LGcited by 0

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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.

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References · 16

  1. 2017Poincar'e Embeddings for Learning Hierarchical Representationscited by 684
  2. 2018Hyperbolic Neural Networkscited by 402
  3. 2019Hyperbolic Graph Convolutional Neural Networkscited by 402
  4. 2021Hyperbolic Deep Neural Networks: A Surveycited by 139
  5. 2019Learning with Fenchel-Young Lossescited by 119
  6. 2018Tropical Geometry of Deep Neural Networkscited by 95
  7. 2018Embedding Text in Hyperbolic Spacescited by 76
  8. 2019Sinkhorn Divergences for Unbalanced Optimal Transportcited by 55
  9. 2022The Numerical Stability of Hyperbolic Representation Learningcited by 47
  10. 2018Large-Margin Classification in Hyperbolic Spacecited by 41
  11. 2025Tropical Attention: Neural Algorithmic Reasoning for Combinatorial Algorithmscited by 12
  12. 2023Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learningcited by 10
4 more
  1. 2022Non-linear Embeddings in Hilbert Simplex Geometrycited by 6
  2. 2021Voronoi Diagrams in the Hilbert Metriccited by 4
  3. 2025Hilbert geometry of the symmetric positive-definite bicone: Application to the geometry of the extended Gaussian familycited by 2
  4. 2023Economical Convex Coverings and Applicationscited by 1

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