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On the Vapnik-Chervonenkis dimension of products of intervals in $\mathbb{R}^d$

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arxiv 2104.07136 v1 pith:RHLOZHAP submitted 2021-04-14 math.MG cs.LGmath.COstat.ML

Alirio Gómez Gómez , Pedro L. Kaufmann This is my paper · ORCID
classification math.MGcs.LGmath.COstat.ML
keywords vapnik-chervonenkisdimensionintervalsmathbbproductsballscertainclasses
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abstract

We study combinatorial complexity of certain classes of products of intervals in $\mathbb{R}^d$, from the point of view of Vapnik-Chervonenkis geometry. As a consequence of the obtained results, we conclude that the Vapnik-Chervonenkis dimension of the set of balls in $\ell_\infty^d$ -- which denotes $\R^d$ equipped with the sup norm -- equals $\lfloor (3d+1)/2\rfloor$.

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  1. VC-dimension of generalized progressions in some nonabelian groups

    math.GR 2025-05 conditional novelty 8.0 of 10

    Generalized progressions in the integer Heisenberg group have VC-dimension at most 267, and in the free group on k generators at most 3k-1.

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