It is a long-standing open problem whether there 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.
CITATION STYLE
Moran, S., & Warmuth, M. K. (2016). Labeled compression schemes for extremal classes. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9925 LNAI, pp. 34–49). Springer Verlag. https://doi.org/10.1007/978-3-319-46379-7_3
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