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Support Vector Machines in the Hilbert Geometry

Authors Aditya Acharya , Auguste H. Gezalyan , Julian Vanecek, David M. Mount , Sunil Arya

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Support vector machines
  • Hilbert geometry
  • linear classification
  • machine learning
  • LP-type problems

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Abstract

Support Vector Machines (SVMs) are a class of classification models in machine learning that are based on computing a maximum-margin separator between two sets of points. The SVM problem has been heavily studied for Euclidean geometry and for a number of kernels. In this paper, we consider the linear SVM problem in the Hilbert metric, a non-Euclidean geometry defined over a convex body. We present efficient algorithms for computing the SVM classifier for a set of n points in the Hilbert metric defined by convex polygons in the plane and convex polytopes in d-dimensional space. We also consider the problems in the related Funk distance.

Cite As Get BibTex

Aditya Acharya, Auguste H. Gezalyan, Julian Vanecek, David M. Mount, and Sunil Arya. Support Vector Machines in the Hilbert Geometry. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.3

Author Details

Aditya Acharya
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Auguste H. Gezalyan
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Julian Vanecek
  • Department of Computer Science, University of Maryland, College Park, MD, USA
David M. Mount
  • Department of Computer Science, University of Maryland, College Park, MD, USA
Sunil Arya
  • Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Hong Kong

Funding

  • Arya, Sunil: Research Grants Council of Hong Kong, China project number 16214721.

References

  1. A. Abdelkader and D. M. Mount. Economical Delone sets for approximating convex bodies. In Proc. 16th Scand. Workshop Algorithm Theory, pages 4:1-4:12, 2018. URL: https://doi.org/10.4230/LIPIcs.SWAT.2018.4.
  2. A. Abdelkader and D. M. Mount. Convex approximation and the Hilbert geometry. In Proc. SIAM Symp. Simplicity in Algorithms (SOSA24), pages 286-298, 2024. URL: https://doi.org/10.1137/1.9781611977936.26.
  3. S. Arya, G. D. da Fonseca, and D. M. Mount. Economical convex coverings and applications. In Proc. 34th Annu. ACM-SIAM Sympos. Discrete Algorithms, pages 1834-1861, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch70.
  4. A. Ben-Hur, C. S. Ong, S. Sonnenburg, B. Schölkopf, and G. Rätsch. Support vector machines and kernels for computational biology. PLoS Comput. Biology, 4(10):e1000173, 2008. URL: https://doi.org/10.1371/journal.pcbi.1000173.
  5. C. M. Bishop. Pattern Recogn. and Mach. Learn. Springer, 2016. URL: https://doi.org/10.1007/978-0-387-45528-0.
  6. M. Bumpus, C. Dai, A. H. Gezalyan, S. Munoz, R. Santhoshkumar, S. Ye, and D. M. Mount. Software and analysis for dynamic Voronoi diagrams in the Hilbert metric, 2023. URL: https://arxiv.org/abs/2304.02745.
  7. C. J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Min. Knowl. Discov., 2(2):121-167, 1998. URL: https://doi.org/10.1023/A:1009715923555.
  8. O. Chapelle, P. Haffner, and V. N. Vapnik. Support vector machines for histogram-based image classification. IEEE Trans. Neural Netw., 10:1055-1064, 1999. URL: https://doi.org/10.1109/72.788646.
  9. B. Chazelle and J. Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension. Journal of Algorithms, 21:579-597, 1996. URL: https://doi.org/10.1006/jagm.1996.0060.
  10. H. Cho, B. DeMeo, J. Peng, and B. Berger. Large-margin classification in hyperbolic space. In Proc. 22nd Internat. Conf. Artif. Intell. and Stat., pages 1832-1840. PMLR, 2019. URL: https://doi.org/10.48550/arXiv.1806.00437.
  11. K. L. Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small. J. Assoc. Comput. Mach., 42:488-499, 1995. URL: https://doi.org/10.1145/201019.201036.
  12. C. Cortes and V. Vapnik. Support-vector networks. Mach. Learn., 20:273-297, 1995. URL: https://doi.org/10.1007/BF00994018.
  13. N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, 2000. Google Scholar
  14. S. Das, S. R. Dev, and Sarvottamananda. A worst-case optimal algorithm to compute the Minkowski sum of convex polytopes. Discrete Appl. Math., 350:44-61, 2024. URL: https://doi.org/10.1016/j.dam.2024.02.004.
  15. D. DeCoste and B. Schölkopf. Training invariant support vector machines. Mach. Learn., 46:161-190, 2002. URL: https://doi.org/10.1023/A:1012454411458.
  16. G. M. Foody and A. Mathur. A relative evaluation of multiclass image classification by support vector machines. IEEE Trans. Geosci. Remote Sensing, 42:1335-1343, 2004. URL: https://doi.org/10.1109/TGRS.2004.827257.
  17. A. H. Gezalyan, S. Kim, C. Lopez, D. Skora, Z. Stefankovic, and D. M. Mount. Delaunay triangulations in the Hilbert metric. In Proc. 19th Scand. Workshop Algorithm Theory, pages 25:1-25:17, 2024. URL: https://doi.org/10.4230/LIPIcs.SWAT.2024.25.
  18. A. H. Gezalyan and D. M. Mount. Voronoi diagrams in the Hilbert metric. In Proc. 39th Internat. Sympos. Comput. Geom., pages 35:1-35:16, 2023. URL: https://doi.org/10.4230/LIPIcs.SoCG.2023.35.
  19. D. Hilbert. Ueber die gerade Linie als kürzeste Verbindung zweier Punkte. Math. Annalen, 46:91-96, 1895. URL: https://doi.org/10.1007/BF02096204.
  20. T. Hofmann, B. Schölkopf, and A. J. Smola. Kernel methods in machine learning. The Annals of Statistics, 36:1171-1220, 2008. URL: https://doi.org/10.1214/009053607000000677.
  21. S. Huang, N. Cai, P. P. Pacheco, S. Narrandes, Y. Wang, and W. Xu. Applications of support vector machine (SVM) learning in cancer genomics. Cancer Genomics & Proteomics, 15:41-51, 2018. URL: https://doi.org/10.21873/cgp.20063.
  22. T. Joachims. Learning to Classify Text Using Support Vector Machines. Springer, 2002. URL: https://doi.org/10.1007/978-1-4615-0907-3.
  23. S. Krig. Computer Vision Metrics: Survey, Taxonomy, and Analysis. Springer nature, 2014. Google Scholar
  24. F. Nielsen and L. Shao. On balls in a Hilbert polygonal geometry (multimedia contribution). In Proc. 33rd Internat. Sympos. Comput. Geom., pages 67:1-67:4, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.67.
  25. F. Nielsen and K. Sun. Clustering in Hilbert’s projective geometry: The case studies of the probability simplex and the elliptope of correlation matrices. In Frank Nielsen, editor, Geometric Structures of Information, pages 297-331. Springer Internat. Pub., 2019. URL: https://doi.org/10.1007/978-3-030-02520-5_11.
  26. A. Papadopoulos and M. Troyanov. From Funk to Hilbert geometry. In Handbook of Hilbert geometry, volume 22 of IRMA Lectures in Mathematics and Theoretical Physics, pages 33-68. European Mathematical Society Publishing House, 2014. URL: https://doi.org/10.4171/147-1/2.
  27. A. Papadopoulos and M. Troyanov. Handbook of Hilbert geometry, volume 22 of IRMA Lectures in Mathematics and Theoretical Physics. European Mathematical Society Publishing House, 2014. URL: https://doi.org/10.4171/147.
  28. S. J. Russell and P. Norvig. Artificial Intelligence: A Modern Approach. Pearson, 4th edition, 2021. Google Scholar
  29. R. Seidel. Low dimensional linear programming and convex hulls made easy. Discrete Comput. Geom., 6:423-434, 1991. URL: https://doi.org/10.1007/BF02574699.
  30. M. Sharir and E. Welzl. A combinatorial bound for linear programming and related problems. In Proc. 9th Internat. Sympos. on Theoret. Aspects of Comp. Sci., pages 567-579, 1992. URL: https://doi.org/10.1007/3-540-55210-3_213.
  31. I. Steinwart and A. Christmann. Support Vector Machines. Springer, 2008. URL: https://doi.org/10.1007/978-0-387-77242-4.
  32. R. Szeliski. Computer Vision: Algorithms and Applications. Springer, 2022. URL: https://doi.org/10.1007/978-3-030-34372-9.
  33. S. Tong and D. Koller. Support vector machine active learning with applications to text classification. J. Mach. Learn. Res., 2:45-66, 2001. URL: https://doi.org/10.1162/153244302760185243.
  34. V. N. Vapnik and A. Y. Chervonenkis. On a class of algorithms of learning pattern recognition. Automation and Remote Control, 25(6):937-945, 1964. (In Russian). URL: http://mi.mathnet.ru/at11678.
  35. C. Vernicos. On the Hilbert geometry of convex polytopes. In Handbook of Hilbert geometry, volume 22 of IRMA Lectures in Mathematics and Theoretical Physics, pages 111-126. European Mathematical Society Publishing House, 2014. URL: https://doi.org/10.48550/arXiv.1406.0733.
  36. Constantin Vernicos and Cormac Walsh. Flag-approximability of convex bodies and volume growth of Hilbert geometries, 2018. URL: https://arxiv.org/abs/1809.09471.
  37. H. Wendland. Scattered Data Approximation. Cambridge University Press, 2004. URL: https://doi.org/10.1017/CBO9780511617539.
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