Unlabeled sample compression schemes for oriented matroids

TL;DR

This paper constructs proper unlabeled sample compression schemes for classes of topes in oriented matroids, leveraging their combinatorial structure and solutions of oriented matroid programs, advancing understanding of VC-dimension bounds.

Contribution

It introduces proper unlabeled sample compression schemes for topes of oriented matroids, extending to affine OMs and complexes with corner peelings, based on oriented matroid program solutions.

Findings

Constructed proper unlabeled compression schemes for OMs

Extended results to affine OMs and complexes with corner peelings

Linked compression schemes to oriented matroid program solutions

Abstract

A long-standing sample compression conjecture asks to linearly bound the size of the optimal sample compression schemes by the Vapnik-Chervonenkis (VC) dimension of an arbitrary class. In this paper, we explore the rich metric and combinatorial structure of oriented matroids (OMs) to construct proper unlabeled sample compression schemes for the classes of topes of OMs bounded by their VC-dimension. The result extends to the topes of affine OMs, as well as to the topes of the complexes of OMs that possess a corner peeling. The main tool that we use are the solutions of certain oriented matroid programs.