On Continuous Terminal Embeddings of Sets of Positive Reach
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| Format: | Preprint |
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2024
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| _version_ | 1866914902579347456 |
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| author | Brugiapaglia, Simone Chiclana, Rafael Hoheisel, Tim Iwen, Mark |
| author_facet | Brugiapaglia, Simone Chiclana, Rafael Hoheisel, Tim Iwen, Mark |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02812 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Continuous Terminal Embeddings of Sets of Positive Reach Brugiapaglia, Simone Chiclana, Rafael Hoheisel, Tim Iwen, Mark Optimization and Control 68R12, 47N10 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. |
| title | On Continuous Terminal Embeddings of Sets of Positive Reach |
| topic | Optimization and Control 68R12, 47N10 |
| url | https://arxiv.org/abs/2408.02812 |