Labeled sample compression schemes for complexes of oriented matroids

TL;DR

This paper proves that complexes of oriented matroids have small labeled sample compression schemes proportional to their VC-dimension, advancing the understanding of sample compression in computational learning theory.

Contribution

It extends existing results by establishing proper labeled sample compression schemes for complexes of oriented matroids, linking combinatorial and metric graph theories.

Findings

Labeled sample compression schemes of size equal to VC-dimension for COMs.

Extension of previous results on ample classes and hyperplane arrangements.

Progress towards the sample compression conjecture.

Abstract

We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension $d$ admit a proper labeled sample compression scheme of size $d$. This considerably extends results of Moran and Warmuth on ample classes, of Ben-David and Litman on affine arrangements of hyperplanes, and of the authors on complexes of uniform oriented matroids, and is a step towards the sample compression conjecture -- one of the oldest open problems in computational learning theory. On the one hand, our approach exploits the rich combinatorial cell structure of COMs via oriented matroid theory. On the other hand, viewing tope graphs of COMs as partial cubes creates a fruitful link to metric graph theory.