Optimal Approximations Made Easy

Csikós, Mónika; Mustafa, Nabil H.

doi:10.48550/arxiv.2008.08970

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2022

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“…In this paper, we are interested in constructions where the parameters m and ε are independent of n. In contrast, since the family of all axis-parallel boxes has finite VC-dimension, one can construct (m, ε)-almost nets with m linear in n by sampling the points of P at random, see [8,6]. The dependence on n is unavoidable for points sampled at random.…”

Section: Introductionmentioning

confidence: 99%

Digital Almost Nets

Bukh

Chao

2022

Electron. J. Combin.

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Digital nets (in base 2) are the subsets of [0,1]d that contain exactly the expected number of points in every not-too-small dyadic box. We construct finite sets, which we call "almost nets", such that every such dyadic box contains almost the expected number of points from the set, but whose size is exponentially smaller than the one of nets. We also establish a lower bound on the size of such almost nets.

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Section: Introductionmentioning

confidence: 99%