Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2403.03969 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913256554102784 |
|---|---|
| author | Chiclana, Rafael Iwen, Mark A. Roach, Mark Philip |
| author_facet | Chiclana, Rafael Iwen, Mark A. Roach, Mark Philip |
| contents | The celebrated Johnson-Lindenstrauss lemma states that for all $\varepsilon \in (0,1)$ and finite sets $X \subseteq \mathbb{R}^N$ with $n>1$ elements, there exists a matrix $Φ\in \mathbb{R}^{m \times N}$ with $m=\mathcal{O}(\varepsilon^{-2}\log n)$ such that \[ (1 - \varepsilon) \|x-y\|_2 \leq \|Φx-Φy\|_2 \leq (1+\varepsilon)\| x- y\|_2 \quad \forall\, x, y \in X.\] Herein we consider terminal embedding results which have recently been introduced in the computer science literature as stronger extensions of the Johnson-Lindenstrauss lemma for finite sets. After a short survey of this relatively recent line of work, we extend the theory of terminal embeddings to hold for arbitrary (e.g., infinite) subsets $X \subseteq \mathbb{R}^N$, and then specialize our generalized results to the case where $X$ is a low-dimensional compact submanifold of $\mathbb{R}^N$. In particular, we prove the following generalization of the Johnson-Lindenstrauss lemma: For all $\varepsilon \in (0,1)$ and $X\subseteq\mathbb{R}^N$, there exists a terminal embedding $f: \mathbb{R}^N \longrightarrow \mathbb{R}^{m}$ such that $$(1 - \varepsilon) \| x - y \|_2 \leq \left\| f(x) - f(y) \right\|_2 \leq (1 + \varepsilon) \| x - y \|_2 \quad \forall \, x \in X ~{\rm and}~ \forall \, y \in \mathbb{R}^N.$$ Crucially, we show that the dimension $m$ of the range of $f$ above is optimal up to multiplicative constants, satisfying $m=\mathcal{O}(\varepsilon^{-2} ω^2(S_X))$, where $ω(S_X)$ is the Gaussian width of the set of unit secants of $X$, $S_X=\overline{\{(x-y)/\|x-y\|_2 \colon x \neq y \in X\}}$. Furthermore, our proofs are constructive and yield algorithms for computing a general class of terminal embeddings $f$, an instance of which is demonstrated herein to allow for more accurate compressive nearest neighbor classification than standard linear Johnson-Lindenstrauss embeddings do in practice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03969 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Outer Bi-Lipschitz Extensions of Linear Johnson-Lindenstrauss Embeddings of Subsets of $\mathbb{R}^N$ Chiclana, Rafael Iwen, Mark A. Roach, Mark Philip Metric Geometry Data Structures and Algorithms Numerical Analysis 51F30, 65D18, 68R12 The celebrated Johnson-Lindenstrauss lemma states that for all $\varepsilon \in (0,1)$ and finite sets $X \subseteq \mathbb{R}^N$ with $n>1$ elements, there exists a matrix $Φ\in \mathbb{R}^{m \times N}$ with $m=\mathcal{O}(\varepsilon^{-2}\log n)$ such that \[ (1 - \varepsilon) \|x-y\|_2 \leq \|Φx-Φy\|_2 \leq (1+\varepsilon)\| x- y\|_2 \quad \forall\, x, y \in X.\] Herein we consider terminal embedding results which have recently been introduced in the computer science literature as stronger extensions of the Johnson-Lindenstrauss lemma for finite sets. After a short survey of this relatively recent line of work, we extend the theory of terminal embeddings to hold for arbitrary (e.g., infinite) subsets $X \subseteq \mathbb{R}^N$, and then specialize our generalized results to the case where $X$ is a low-dimensional compact submanifold of $\mathbb{R}^N$. In particular, we prove the following generalization of the Johnson-Lindenstrauss lemma: For all $\varepsilon \in (0,1)$ and $X\subseteq\mathbb{R}^N$, there exists a terminal embedding $f: \mathbb{R}^N \longrightarrow \mathbb{R}^{m}$ such that $$(1 - \varepsilon) \| x - y \|_2 \leq \left\| f(x) - f(y) \right\|_2 \leq (1 + \varepsilon) \| x - y \|_2 \quad \forall \, x \in X ~{\rm and}~ \forall \, y \in \mathbb{R}^N.$$ Crucially, we show that the dimension $m$ of the range of $f$ above is optimal up to multiplicative constants, satisfying $m=\mathcal{O}(\varepsilon^{-2} ω^2(S_X))$, where $ω(S_X)$ is the Gaussian width of the set of unit secants of $X$, $S_X=\overline{\{(x-y)/\|x-y\|_2 \colon x \neq y \in X\}}$. Furthermore, our proofs are constructive and yield algorithms for computing a general class of terminal embeddings $f$, an instance of which is demonstrated herein to allow for more accurate compressive nearest neighbor classification than standard linear Johnson-Lindenstrauss embeddings do in practice. |
| title | On Outer Bi-Lipschitz Extensions of Linear Johnson-Lindenstrauss Embeddings of Subsets of $\mathbb{R}^N$ |
| topic | Metric Geometry Data Structures and Algorithms Numerical Analysis 51F30, 65D18, 68R12 |
| url | https://arxiv.org/abs/2403.03969 |