On Continuous Terminal Embeddings of Sets of Positive Reach

Abstract

In this paper we prove the existence of Hölder continuous terminal embeddings of any desired $X \subseteq \mathbb{R}^d$ into $\mathbb{R}^{m}$ with $m=\mathcal{O}(\varepsilon^{-2}ω(S_X)^2)$, for arbitrarily small distortion $\varepsilon$, where $ω(S_X)$ 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 $\frac{1}{2}$-Hölder almost everywhere, and when $X$ is infinite with positive reach we give terminal embeddings that are locally $\frac{1}{4}$-Hölder 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 Hölder everywhere sufficiently close to the manifold.