Approximation Algorithms for Smallest Intersecting Balls

TL;DR

This paper introduces novel approximation algorithms for the smallest intersecting ball problem and its soft-margin variant in high-dimensional spaces, unifying several fundamental problems in computational geometry and machine learning.

Contribution

It presents a new framework based on zero-sum games over symmetric cones and provides the first high-dimensional approximation algorithms for various convex inputs.

Findings

Algorithms efficiently solve large-scale instances.

First approximation methods for high-dimensional convex inputs.

Applicable to multiple geometric and machine learning problems.

Abstract

We study a general smallest intersecting ball problem and its soft-margin variant in high-dimensional Euclidean spaces for input objects that are compact and convex. These two problems link and unify a series of fundamental problems in computational geometry and machine learning, including smallest enclosing ball, polytope distance, intersection radius, $\ell_1$-loss support vector machine, $\ell_1$-loss support vector data description, and so on. Leveraging our novel framework for solving zero-sum games over symmetric cones, we propose general approximation algorithms for the two problems, where implementation details are presented for specific inputs of convex polytopes, reduced polytopes, axis-aligned bounding boxes, balls, and ellipsoids. For most of these inputs, our algorithms are the first results in high-dimensional spaces, and also the first approximation methods. Experimental results show that our algorithms can solve large-scale input instances efficiently.