2021
DOI: 10.1017/jsl.2020.55
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Thicket Density

Abstract: We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s -rank.

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Cited by 2 publications
(2 citation statements)
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“…Proof. First, create a sequence (c i ) i∈N with c i ∈ X for all i ∈ N such that, for all I ∈ N ≤d , (1) f (c i ) : i ∈ I is linearly independent.…”
Section: Vc-dimension and Littlestone Dimension Of Zero Setsmentioning
confidence: 99%
See 1 more Smart Citation
“…Proof. First, create a sequence (c i ) i∈N with c i ∈ X for all i ∈ N such that, for all I ∈ N ≤d , (1) f (c i ) : i ∈ I is linearly independent.…”
Section: Vc-dimension and Littlestone Dimension Of Zero Setsmentioning
confidence: 99%
“…Bhaskar has studied a shatter function related to Littlestone dimension (called "thicket dimension" in [1]). Bhaskar showed that the size of the set system that well-labels the leaves of the tree of depth n (this is a natural notion for the shatter function for Littlestone dimension) is bounded above by n ≤d , where d is the Littlestone dimension.…”
Section: Introductionmentioning
confidence: 99%