Terminal Embeddings in Sublinear Time
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929449900965888 |
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| author | Cherapanamjeri, Yeshwanth Nelson, Jelani |
| author_facet | Cherapanamjeri, Yeshwanth Nelson, Jelani |
| contents | Recently (Elkin, Filtser, Neiman 2017) introduced the concept of a {\it terminal embedding} from one metric space $(X,d_X)$ to another $(Y,d_Y)$ with a set of designated terminals $T\subset X$. Such an embedding $f$ is said to have distortion $ρ\ge 1$ if $ρ$ is the smallest value such that there exists a constant $C>0$ satisfying
\begin{equation*}
\forall x\in T\ \forall q\in X,\ C d_X(x, q) \le d_Y(f(x), f(q)) \le C ρd_X(x, q) .
\end{equation*}
When $X,Y$ are both Euclidean metrics with $Y$ being $m$-dimensional, recently (Narayanan, Nelson 2019), following work of (Mahabadi, Makarychev, Makarychev, Razenshteyn 2018), showed that distortion $1+ε$ is achievable via such a terminal embedding with $m = O(ε^{-2}\log n)$ for $n := |T|$. This generalizes the Johnson-Lindenstrauss lemma, which only preserves distances within $T$ and not to $T$ from the rest of space. The downside of prior work is that evaluating their embedding on some $q\in \mathbb{R}^d$ required solving a semidefinite program with $Θ(n)$ constraints in~$m$ variables and thus required some superlinear $\mathrm{poly}(n)$ runtime. Our main contribution in this work is to give a new data structure for computing terminal embeddings. We show how to pre-process $T$ to obtain an almost linear-space data structure that supports computing the terminal embedding image of any $q\in\mathbb{R}^d$ in sublinear time $O^* (n^{1-Θ(ε^2)} + d)$. To accomplish this, we leverage tools developed in the context of approximate nearest neighbor search. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_08691 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Terminal Embeddings in Sublinear Time Cherapanamjeri, Yeshwanth Nelson, Jelani Data Structures and Algorithms Computational Geometry Machine Learning Recently (Elkin, Filtser, Neiman 2017) introduced the concept of a {\it terminal embedding} from one metric space $(X,d_X)$ to another $(Y,d_Y)$ with a set of designated terminals $T\subset X$. Such an embedding $f$ is said to have distortion $ρ\ge 1$ if $ρ$ is the smallest value such that there exists a constant $C>0$ satisfying \begin{equation*} \forall x\in T\ \forall q\in X,\ C d_X(x, q) \le d_Y(f(x), f(q)) \le C ρd_X(x, q) . \end{equation*} When $X,Y$ are both Euclidean metrics with $Y$ being $m$-dimensional, recently (Narayanan, Nelson 2019), following work of (Mahabadi, Makarychev, Makarychev, Razenshteyn 2018), showed that distortion $1+ε$ is achievable via such a terminal embedding with $m = O(ε^{-2}\log n)$ for $n := |T|$. This generalizes the Johnson-Lindenstrauss lemma, which only preserves distances within $T$ and not to $T$ from the rest of space. The downside of prior work is that evaluating their embedding on some $q\in \mathbb{R}^d$ required solving a semidefinite program with $Θ(n)$ constraints in~$m$ variables and thus required some superlinear $\mathrm{poly}(n)$ runtime. Our main contribution in this work is to give a new data structure for computing terminal embeddings. We show how to pre-process $T$ to obtain an almost linear-space data structure that supports computing the terminal embedding image of any $q\in\mathbb{R}^d$ in sublinear time $O^* (n^{1-Θ(ε^2)} + d)$. To accomplish this, we leverage tools developed in the context of approximate nearest neighbor search. |
| title | Terminal Embeddings in Sublinear Time |
| topic | Data Structures and Algorithms Computational Geometry Machine Learning |
| url | https://arxiv.org/abs/2110.08691 |