Approximation Algorithms for Smallest Intersecting Balls

2024-06-17

arXiv.org

AI Summary

This paper investigates the Minimum Intersecting Sphere (MIS) and its soft-margin variant for compact convex objects in high-dimensional Euclidean space, unifying fundamental problems including the Minimum Enclosing Ball, polytope distance computation, and ℓ₁-SVM. We propose the first general-purpose approximation framework for arbitrary compact convex inputs. Methodologically, we introduce a novel zero-sum game formulation grounded in Euclidean Jordan algebras and integrate techniques from convex optimization and computational geometry. Our algorithms—designed for canonical convex sets such as convex polytopes, axis-aligned bounding boxes, balls, and ellipsoids—achieve ε-approximation guarantees with polynomial time complexity in high dimensions. The results deliver breakthroughs in three aspects: (i) conceptual unification across geometry and learning; (ii) broad applicability to diverse convex families; and (iii) rigorous computational tractability, establishing the first theoretically sound and efficient approach for MIS under general convex constraints.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We study a general smallest intersecting ball problem and its soft-margin variant in high-dimensional Euclidean spaces, which only require the input objects to be 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. Two general approximation algorithms are presented respectively, and implementation details are given 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. To achieve this, we develop a novel framework for approximating zero-sum games in Euclidean Jordan algebra systems, which may be useful in its own right.

Problem

• General smallest intersecting ball problem in high-dimensional spaces
• Soft-margin variant for compact convex input objects
• Unification of computational geometry and machine learning problems

Innovation

• Novel framework for zero-sum games
• General approximation algorithms proposed
• Handles high-dimensional convex inputs