Tighter Bounds for Reconstruction from $\epsilon$-samples

TL;DR

This paper improves the bounds for reconstructing curves from epsilon-samples in various dimensions, showing tighter thresholds for guaranteed reconstruction and non-uniqueness, extending previous results.

Contribution

It establishes new lower bounds for successful curve reconstruction from epsilon-samples and demonstrates non-uniqueness at higher sampling thresholds across dimensions.

Findings

Reconstruction possible from 0.66-samples in any dimension using a modified NN-Crust algorithm.

Non-uniqueness of reconstruction occurs at 0.72-samples, extending to hypersurfaces in higher dimensions.

Previous bounds were 0.47 in 2D and 1/3 in higher dimensions for successful reconstruction.

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.