A Geometric Approach to Sample Compression

TL;DR

This paper introduces a geometric framework using hyperplane arrangements in hyperbolic space to address the Sample Compression Conjecture, providing new methods for compressing maximum concept classes and revealing limitations of embedding classes.

Contribution

It presents a novel geometric approach with hyperbolic and piecewise-linear arrangements to systematically compress finite maximum classes, extending Pachner moves to cubical complexes.

Findings

Hyperbolic hyperplane arrangements represent maximum classes.

PL arrangements can be d-swept to compress classes.

Some d-maximal classes cannot embed into larger maximum classes.

Abstract

The Sample Compression Conjecture of Littlestone & Warmuth has remained unsolved for over two decades. This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. Simple arrangements of hyperplanes in Hyperbolic space, and Piecewise-Linear hyperplane arrangements, are shown to represent maximum classes, generalizing the corresponding Euclidean result. A main result is that PL arrangements can be swept by a moving hyperplane to unlabeled d-compress any finite maximum class, forming a peeling scheme as conjectured by Kuzmin & Warmuth. A corollary is that some d-maximal classes cannot be embedded into any maximum class of VC dimension d+k, for any constant k. The construction of the PL sweeping involves Pachner moves on the one-inclusion graph, corresponding to moves of a hyperplane across the intersection of d other hyperplanes. This extends the well known Pachner moves for triangulations to cubical complexes.