A remark on dimensionality reduction in discrete subgroups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909462550282240 |
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| author | Viera, Rodolfo |
| author_facet | Viera, Rodolfo |
| contents | In this short note, we prove a version of the Johnson-Lindenstrauss flattening Lemma for point sets taking values in discrete subgroups. More precisely, given $d,λ_0,N_0\in\mathbb{N}$ and $ε\in \left(0,\frac{1}{2}\right)$ suitably chosen, we show there exists a natural number $k=k(d,ε)=O\left(\frac{1}{ε^2}\log d\right)$, such that for every sufficiently large scaling factor $λ\in\mathbb{N}$ and any point set $\mathcal{D}\subset\fracλ{λ_0}\mathbb{Z}^d\cap B(0,λN_0)$ with cardinality $d$, there exists an embedding $F:\mathcal{D}\to\frac{1}{λ_0}\mathbb{Z}^k$, with distortion at most $\left(1+ε+\fracε{λλ_0}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A remark on dimensionality reduction in discrete subgroups Viera, Rodolfo Metric Geometry 46B85, 68R12 In this short note, we prove a version of the Johnson-Lindenstrauss flattening Lemma for point sets taking values in discrete subgroups. More precisely, given $d,λ_0,N_0\in\mathbb{N}$ and $ε\in \left(0,\frac{1}{2}\right)$ suitably chosen, we show there exists a natural number $k=k(d,ε)=O\left(\frac{1}{ε^2}\log d\right)$, such that for every sufficiently large scaling factor $λ\in\mathbb{N}$ and any point set $\mathcal{D}\subset\fracλ{λ_0}\mathbb{Z}^d\cap B(0,λN_0)$ with cardinality $d$, there exists an embedding $F:\mathcal{D}\to\frac{1}{λ_0}\mathbb{Z}^k$, with distortion at most $\left(1+ε+\fracε{λλ_0}\right)$. |
| title | A remark on dimensionality reduction in discrete subgroups |
| topic | Metric Geometry 46B85, 68R12 |
| url | https://arxiv.org/abs/2501.01396 |