Technical Report CS0838

Authors: S. Ben-David and L. Gurvits
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Abstract: Vapnik and Chervonekis proposed in [VC71] a combinatorial notion of dimension that reflects the `combinatorial complexity' of families of sets. In the two decades that have passed since that paper, this notion - the Vapnik-Chervonekis dimension (VC-dimension) - has been discovered to be of primal importance in quite a wide variety of topics in both pure mathematics and theoretical computer science. The VC-dimension of a family of sets (sometimes called a class) is defined as the maximal cardinality of a set that this class `shatters'. If a class shatters arbitrarily large finite sets ten its VC-dimension is defined to be \infty. In this paper we turn out attention to classes with infinite VC-dimension, a realm thrown into one big bag by the usual VC-dimension. We identify 3 levels of combinatorial complexity of classes with infinite VC-dimension. We show that these levels fall under the set-theoretic definition of \sigma-ideals (in particular, each of them is closed under countable unions), and that they are all distinct. Maybe the most surprising contribution of this work is an intimate relation, that we demonstrate, between the VC-dimension of a class of subsets of the natural numbers and the Lebesgue measure of the set of reals defined when these subsets are viewed as binary representations of real numbers.
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