On the complexity of constrained VC-classes

Joel Ratsaby

doi:10.1016/j.dam.2007.05.041

On the complexity of constrained VC-classes

Joel Ratsaby

Research output: Contribution to journal › Article › peer-review

Abstract

Sauer's lemma is extended to classes H_N of binary-valued functions h on [n] = { 1, ..., n } which have a margin less than or equal to N on all x ∈ [n] with h (x) = 1, where the margin μ_h (x) of h at x ∈ [n] is defined as the largest non-negative integer a such that h is constant on the interval I_a (x) = [x - a, x + a] ⊆ [n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S ⊆ [n].

Original language	English
Pages (from-to)	903-910
Number of pages	8
Journal	Discrete Applied Mathematics
Volume	156
Issue number	6
DOIs	https://doi.org/10.1016/j.dam.2007.05.041
State	Published - 15 Mar 2008
Externally published	Yes

Keywords

Boolean functions
Integer partitions
Sauer's lemma
VC-dimension

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Applied Mathematics

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10.1016/j.dam.2007.05.041

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title = "On the complexity of constrained VC-classes",

abstract = "Sauer's lemma is extended to classes HN of binary-valued functions h on [n] = { 1, ..., n } which have a margin less than or equal to N on all x ∈ [n] with h (x) = 1, where the margin μh (x) of h at x ∈ [n] is defined as the largest non-negative integer a such that h is constant on the interval Ia (x) = [x - a, x + a] ⊆ [n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S ⊆ [n].",

keywords = "Boolean functions, Integer partitions, Sauer's lemma, VC-dimension",

author = "Joel Ratsaby",

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N2 - Sauer's lemma is extended to classes HN of binary-valued functions h on [n] = { 1, ..., n } which have a margin less than or equal to N on all x ∈ [n] with h (x) = 1, where the margin μh (x) of h at x ∈ [n] is defined as the largest non-negative integer a such that h is constant on the interval Ia (x) = [x - a, x + a] ⊆ [n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S ⊆ [n].

AB - Sauer's lemma is extended to classes HN of binary-valued functions h on [n] = { 1, ..., n } which have a margin less than or equal to N on all x ∈ [n] with h (x) = 1, where the margin μh (x) of h at x ∈ [n] is defined as the largest non-negative integer a such that h is constant on the interval Ia (x) = [x - a, x + a] ⊆ [n]. Estimates are obtained for the cardinality of classes of binary-valued functions with a margin of at least N on a positive sample S ⊆ [n].

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KW - VC-dimension

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On the complexity of constrained VC-classes

Abstract

Keywords

ASJC Scopus subject areas

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