Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability
Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability
Learning in Neural Networks: Theoretical Foundations
Learning in Neural Networks: Theoretical Foundations
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Consider a class $\mathcal{H}$ of binary functions h: X驴{驴驴驴1,驴+驴1} on an interval $X=[0, B]\subset \mbox{\rm IR}$ . Define the sample width of h on a finite subset (a sample) S驴驴驴X as 驴 S (h)驴=驴 min x驴驴驴S |驴 h (x)| where 驴 h (x)驴=驴h(x) max {a驴驴驴0: h(z)驴=驴h(x), x驴驴驴a驴驴驴z驴驴驴x驴+驴a}. Let $\mathbb{S}_\ell$ be the space of all samples in X of cardinality 驴 and consider sets of wide samples, i.e., hypersets which are defined as $A_{\beta, h} = \{S\in \mathbb{S}_\ell: \omega_{S}(h) \geq \beta\}. $ 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 $\{A_{\beta, h}: h\in\mathcal{H}\}$ , β驴驴0, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples $S\in\mathbb{S}_\ell$ of cardinality m. The estimate is $2\sum_{i=0}^{2\lfloor B/(2\beta)\rfloor}{m-\ell\choose i}$ .