تقرير
On Continuous Terminal Embeddings of Sets of Positive Reach
| العنوان: | On Continuous Terminal Embeddings of Sets of Positive Reach |
|---|---|
| المؤلفون: | Brugiapaglia, Simone, Chiclana, Rafael, Hoheisel, Tim, Iwen, Mark |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Optimization and Control, 68R12, 47N10 |
| الوصف: | 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. Comment: 1 figure, 23 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2408.02812 |
| رقم الأكسشن: | edsarx.2408.02812 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |