Labeled compression schemes for extremal classes

TL;DR

This paper constructs a sample compression scheme for extremal classes, generalizing previous results for maximum classes, and explores open problems related to their combinatorial structure and unlabeled schemes.

Contribution

It introduces a compression scheme for extremal classes of size equal to their VC dimension, extending prior work beyond maximum classes.

Findings

Constructed a compression scheme for extremal classes matching their VC dimension.

Extended the concept of maximum classes to extremal classes using the Sandwich Theorem.

Identified open problems on the structure and unlabeled compression schemes for extremal classes.

Abstract

It is a long-standing open problem whether there always exists a compression scheme whose size is of the order of the Vapnik-Chervonienkis (VC) dimension $d$. Recently compression schemes of size exponential in $d$ have been found for any concept class of VC dimension $d$. Previously, compression schemes of size $d$ have been given for maximum classes, which are special concept classes whose size equals an upper bound due to Sauer-Shelah. We consider a generalization of maximum classes called extremal classes. Their definition is based on a powerful generalization of the Sauer-Shelah bound called the Sandwich Theorem, which has been studied in several areas of combinatorics and computer science. The key result of the paper is a construction of a sample compression scheme for extremal classes of size equal to their VC dimension. We also give a number of open problems concerning the combinatorial structure of extremal classes and the existence of unlabeled compression schemes for them.