# Unlabeled Compression Schemes for Maximum Classes

Dima Kuzmin, Manfred K. Warmuth.

Year: 2007, Volume: 8, Issue: 70, Pages: 2047−2081

#### Abstract

Maximum concept classes of VC dimension *d*
over *n* domain points have size *n* C ≤*d*, and this is an upper bound on the size of any concept class
of VC dimension *d* over *n* points.
We give a compression scheme for any maximum class that
represents each concept by a subset of up
to *d* unlabeled domain points
and has the property that for
any sample of a concept in the class,
the representative of exactly
one of the concepts consistent with the sample
is a subset of the domain of the sample.
This allows us to compress any sample of a concept
in the class to a subset of up to *d* unlabeled sample points such that
this subset represents a concept consistent with the
entire original sample.
Unlike the previously known compression scheme
for maximum classes (Floyd and Warmuth, 1995) which compresses
to labeled subsets of the sample of size equal *d*,
our new scheme is tight in the sense that the number of possible
unlabeled compression sets of size at most *d* equals the
number of concepts in the class.