Vapnik–Červonenkis dimension, VC-dimension
Let be a hypergraph. The Vapnik–Chervonenkis dimension of is the largest cardinality of a subset of that is scattered by , i.e. such that for all there is an with . Thus, it is the same as the index of a Vapnik–Chervonenkis class. It is usually denoted by .
The Vapnik–Chervonenkis dimension plays an important role in learning theory, especially in PAC learning (probably approximately correct learning). Thus, learnability of classes of -valued functions is equivalent to finiteness of the Vapnik–Chervonenkis dimension, [a3].
The independence number of a hypergraph is the maximal cardinality of a subset of that does not contain any (see also Graph, numerical characteristics of a). This notion is closely related with , [a6], [a7].
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|[a6]||D.Q. Naiman, H.P. Wynn, "Independence number and the complexity of families of sets" Discr. Math. , 154 (1996) pp. 203–216|
|[a7]||J. Pach, P.K. Agarwal, "Combinatorial geometry" , Wiley/Interscience (1995) pp. 247–254|
Vapnik-Chervonenkis dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vapnik-Chervonenkis_dimension&oldid=39971