Some special vapnik-chervonenkis classes

✍ Scribed by R.S. Wenocur; R.M. Dudley

• Publisher: Elsevier Science
• Year: 1981
• Tongue: English
• Weight: 680 KB
• Volume: 33
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(81)90274-0

✦ Synopsis

For a class q of subsets of a set X, let V(v) be the smallest n such that no n-element set Fc X has all its subsets of the form ,4 nF, A E V. The condition V(%)C+QC has probabilistic implications. Ef any two-element wbset A of X satisfies both A n C = 8 arid A-c b for some C, DE. %, then V(q) = 2 if: furd otrly if 0 is linearly -ordered py in&s& Ef S is of the form %Zf(nFS1 Ci: CrE$, i =1,2 ., . . . , rl), where each 'gi is linearly ordkred by inclusiotr, then V(S) s n c 1. If #I is an (n -l)-dimensional afllne hyperplane in an n;dimensional vector-space of real functions on X, ad % is thte collection of all sets (x : f(x) > 0) for f in Ii; tlwi~ V(%) = n.