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Classifiers in High Dimensional Hilbert Metrics

Aditya Acharya, Auguste H. Gezalyan, David M. Mount

TL;DR

This work studies binary classification in Hilbert geometry induced by a polytopal domain, formulating a scalable large-margin SVM via LP-feasibility that achieves a polynomial dependence on the dimension $d$, the number of bounding hyperplanes $m$, the dataset size $n$, input bit-length $B$, and a tolerance $\varepsilon$. By leveraging Birkhoff's formulation, it expresses Hilbert and Funk balls as polytopes and reduces margin optimization to a sequence of LPs, with a constructive method to build Hilbert balls in $O(ndm^{2})$ time. The paper extends the framework to the Funk metric, introduces a soft-margin variant with boundary-weighted penalties, and proposes a nearest-neighbor classifier through an isometric embedding into a normed space, broadening applicability while maintaining polynomial-time guarantees. Collectively, these contributions provide principled, geometry-aware, scalable classification tools for high-dimensional Hilbert spaces, with practical relevance to convex geometry and applications where boundary sensitivity is important.

Abstract

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.

Classifiers in High Dimensional Hilbert Metrics

TL;DR

This work studies binary classification in Hilbert geometry induced by a polytopal domain, formulating a scalable large-margin SVM via LP-feasibility that achieves a polynomial dependence on the dimension , the number of bounding hyperplanes , the dataset size , input bit-length , and a tolerance . By leveraging Birkhoff's formulation, it expresses Hilbert and Funk balls as polytopes and reduces margin optimization to a sequence of LPs, with a constructive method to build Hilbert balls in time. The paper extends the framework to the Funk metric, introduces a soft-margin variant with boundary-weighted penalties, and proposes a nearest-neighbor classifier through an isometric embedding into a normed space, broadening applicability while maintaining polynomial-time guarantees. Collectively, these contributions provide principled, geometry-aware, scalable classification tools for high-dimensional Hilbert spaces, with practical relevance to convex geometry and applications where boundary sensitivity is important.

Abstract

Classifying points in high dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address classifying points in the -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.
Paper Structure (14 sections, 5 theorems, 27 equations, 4 figures)

This paper contains 14 sections, 5 theorems, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contributions
  4. Mathematical Preliminaries
  5. Hilbert Geometry
  6. Birkhoff's Formulation and Metric Balls
  7. Hilbert SVM
  8. Overall Algorithm
  9. LP-feasibility
  10. Constructing Hilbert Balls
  11. Complexity Analysis
  12. Extensions
  13. A Soft-Margin Classifier
  14. A Nearest Neighbor-based Classifier

Key Result

Theorem 1

Consider a polytope $\Omega$ in $\mathbb{R}^d$ defined by $m$ hyperplanes, and two linearly separable point sets $P^+$ and $P^-$ of total size $n$ contained within $\Omega$'s interior. Assume further that all of the defining coordinates and coefficients are representable as $B$-bit rational numbers.

Figures (4)

  • Figure 1: The (a) forward Funk, (b) reverse Funk, (c) and Hilbert distances between $p$ and $q$ in $\Omega$.
  • Figure 2: Defining balls and their bounding hyperplanes for (a) Funk, (b) Reverse Funk, and (c) Hilbert.
  • Figure 3: Intuitive description of the dual: K separates $\mathcal{B}(r)^+$ and $\mathcal{B}(r)^-$. $K_B$ is a conical combination of $S_1$ and $S_2$ while lying strictly above $K$.
  • Figure 4: Constructing $B^H_\Omega(p_i,r)$.

Theorems & Definitions (8)

  • Theorem 1
  • Proposition 2: Birkhoff's Formulation
  • Lemma 3: Funk Balls
  • Lemma 4: Hilbert Balls
  • Lemma 5
  • Remark 6: Intuitive Description
  • Remark 7: LP for Funk
  • Remark 8: Funk Balls