## Abstract

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 ω_{S} (h) = min _{x∈S} |ω_{h} (x)| where ω_{h} (x)=h(x) max {a ≥ 0: h(z) = h(x), x - a ≤ z ≤ x + a}. Let double-struck S sign_{ℓ} be the space of all samples in X of ℓ and consider sets of wide samples, i.e., hypersets which are defined as A_{β}, h} = {S ∈ double-struck S sign_{ℓ}: ω_{S}(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 {A_{β}, 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_{ℓ} of cardinality m. The estimate is 2∑_{i=0}^{2⌊ B/(2β)⌋ (m-ℓ i).}

Original language | English |
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Pages (from-to) | 55-65 |

Number of pages | 11 |

Journal | Annals of Mathematics and Artificial Intelligence |

Volume | 52 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2008 |

## Keywords

- Binary functions
- Density of sets
- VC-dimension