Vapnik-Chervonenkis dimension

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Vapnik–Červonenkis dimension, VC-dimension

Let $H = (V_H,E_H)$ be a hypergraph. The Vapnik–Chervonenkis dimension of $H$ is the largest cardinality of a subset $F$ of $V_H$ that is scattered by $E_H$, i.e. such that for all $A \subseteq F$ there is an $E \in E_H$ with $A = F \cap E$. Thus, it is the same as the index of a Vapnik–Chervonenkis class. It is usually denoted by $\mathrm{VC}(H)$.

Computing the Vapnik–Chervonenkis dimension is $\mathcal{NP}$-hard (cf. also $\mathcal{NP}$) for many classes of hypergraphs, [a1], [a2].

The Vapnik–Chervonenkis dimension plays an important role in learning theory, especially in probably approximately correct (PAC) learning. Thus, learnability of classes of $\{0,1\}$-valued functions is equivalent to finiteness of the Vapnik–Chervonenkis dimension, [a3].

For the role of the Vapnik–Chervonenkis dimension in neural networks, see, e.g., [a4], [a5].

The independence number of a hypergraph $H$ is the maximal cardinality of a subset $A$ of $V_H$ that does not contain any $E \in E_H$ (see also Graph, numerical characteristics of a). This notion is closely related with $\mathrm{VC}(H)$, [a6], [a7].


[a1] E. Kranakis, D. Krizanc, B. Ruf, J. Urrutia, G. Wöginger, "The VC-dimension of set systems defined by graphs" Discr. Appl. Math. , 77 : 3 (1997) pp. 237–257
[a2] C.H. Papadimitriou, M. Yannakakis, "On limited nondeterminism and the complexity of VC-dimension" J. Comput. Syst. Sci. , 53 : 2 (1996) pp. 161–170
[a3] S. Ben-David, N. Cesa-Bianchi, D. Haussler, P.M. Long, "Characterizations of learnability of $\{0,\ldots,n\}$-valued functions" J. Comput. Syst. Sci. , 50 : 1 (1995) pp. 74–86 DOI 10.1006/jcss.1995.1008 Zbl 0827.68095
[a4] S.B. Holden, "Neural networks and the VC-dimension" J.G. McWhirter (ed.) , Mathematics in Signal Processing , III , Oxford Univ. Press (1994) pp. 73–84
[a5] W. Maass, "Perspectives of current research about the complexity of learning on neural nets" V. Roychowdhury (ed.) et al. (ed.) , Theoretical Advances in Neural Computation and Learning , Kluwer Acad. Publ. (1994) pp. 295–336
[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
How to Cite This Entry:
Vapnik-Chervonenkis dimension. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article