Ample completions of OMs and CUOMs

TL;DR

This paper demonstrates that oriented matroids and complexes of uniform oriented matroids can be completed to ample partial cubes with the same VC-dimension, supporting the sample compression conjecture in learning theory.

Contribution

It proves the existence of such completions for OMs and CUOMs, advancing understanding of their structure and implications for the sample compression conjecture.

Findings

OMs and CUOMs can be completed to ample partial cubes

These completions preserve VC-dimension

Supports the sample compression conjecture

Abstract

This paper considers completions of COMs (complexes oriented matroids) to ample partial cubes of the same VC-dimension. We show that these exist for OMs (oriented matroids) and CUOMs (complexes of uniform oriented matroids). This implies that OMs and CUOMs satisfy the sample compression conjecture -- one of the central open questions of learning theory. We conjecture that every COM can be completed to an ample partial cube without increasing the VC-dimension.