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arXiv · 1506.00165

Labeled compression schemes for extremal classes

Shay Moran·Manfred K. Warmuth
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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.

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Shay Moran, Manfred K. Warmuth. 2015-05-30. Labeled compression schemes for extremal classes. https://arxiv.org/abs/1506.00165

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