On the complexity of binary samples

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Consider a class H of binary functions h: X → {-1,+1} on an interval X = [0, B] ⊂ IR. Define the sample width of h on a finite subset (a sample) S ⊂ X as ωS (h) = min x∈Sh (x)| where ωh (x)=h(x) max {a ≥ 0: h(z) = h(x), x - a ≤ z ≤ x + a}. Let double-struck S sign be the space of all samples in X of ℓ and consider sets of wide samples, i.e., hypersets which are defined as Aβ, h} = {S ∈ double-struck S sign: ωS(h) ≥ β}. Through an application of the Sauer-Shelah result on the density of sets an upper estimate is obtained on the growth function (or trace) of the class {Aβ, h: h ∈ ℋ}, β > 0, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples S ∈ double-struk S sign of cardinality m. The estimate is 2∑i=02⌊ B/(2β)⌋ (m-ℓ i).

Original languageEnglish
Pages (from-to)55-65
Number of pages11
JournalAnnals of Mathematics and Artificial Intelligence
Issue number1
StatePublished - Jan 2008


  • Binary functions
  • Density of sets
  • VC-dimension


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