On Continuous Terminal Embeddings of Sets of Positive Reach

TL;DR

This paper proves the existence of H"{o}lder continuous embeddings of sets in Euclidean space with controlled distortion, depending on Gaussian width, applicable to finite, infinite, and manifold sets.

Contribution

It introduces new terminal embedding constructions with specific H"{o}lder regularity for various classes of sets, extending prior embedding results.

Findings

Embeddings exist with dimension depending on Gaussian width and distortion.

Finite sets admit locally rac{1}{2}-Hf6lder embeddings.

Infinite sets with positive reach admit rac{1}{4}-Hf6lder embeddings close to the set.

Abstract

In this paper we prove the existence of H\"{o}lder continuous terminal embeddings of any desired $X \subseteq \mathbb{R}^d$ into $\mathbb{R}^{m}$ with $m=\mathcal{O}(\varepsilon^{-2}\omega(S_X)^2)$, for arbitrarily small distortion $\varepsilon$, where $\omega(S_X)$ denotes the Gaussian width of the unit secants of $X$. More specifically, when $X$ is a finite set we provide terminal embeddings that are locally $\frac{1}{2}$-H\"{o}lder almost everywhere, and when $X$ is infinite with positive reach we give terminal embeddings that are locally $\frac{1}{4}$-H\"{o}lder everywhere sufficiently close to $X$ (i.e., within all tubes around $X$ of radius less than $X$'s reach). When $X$ is a compact $d$-dimensional submanifold of $\mathbb{R}^N$, an application of our main results provides terminal embeddings into $\tilde{\mathcal{O}}(d)$-dimensional space that are locally H\"{o}lder everywhere sufficiently close to the manifold.