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arXiv · 2112.03656

Tighter Bounds for Reconstruction from $\epsilon$-samples

Håvard Bakke Bjerkevik
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Abstract

We show that reconstructing a curve in $\mathbb{R}^d$ for $d\geq 2$ from a $0.66$-sample is always possible using an algorithm similar to the classical NN-Crust algorithm. Previously, this was only known to be possible for $0.47$-samples in $\mathbb{R}^2$ and $\frac{1}{3}$-samples in $\mathbb{R}^d$ for $d\geq 3$. In addition, we show that there is not always a unique way to reconstruct a curve from a $0.72$-sample; this was previously only known for $1$-samples. We also extend this non-uniqueness result to hypersurfaces in all higher dimensions.

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Håvard Bakke Bjerkevik. 2021-12-07. Tighter Bounds for Reconstruction from $\epsilon$-samples. https://arxiv.org/abs/2112.03656

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