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 $\rho\ge 1$ if $\rho$ 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 \rho d_X(x, q) . \end{equation*} In the case that $X,Y$ are both Euclidean metrics with $Y$ being $m$-dimensional, recently (Narayanan, Nelson 2019), following work of (Mahabadi, Makarychev, Makarychev, Razenshteyn 2018), showed that distortion $1+\epsilon$ is achievable via such a terminal embedding with $m = O(\epsilon^{-2}\log n)$ 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 is that evaluating the embedding on some $q\in \mathbb{R}^d$ required solving a semidefinite program with $\Theta(n)$ 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 $n^{1-\Theta(\epsilon^2)+o(1)} + dn^{o(1)}$. To accomplish this, we leverage tools developed in the context of approximate nearest neighbor search.

翻译：最近 (Elkin, filtser, Neiman 2017) 引入了一个概念, 从一个公吨空间$( X, d_ X) 到另外的$( Y, d_ Y), 其中有一套指定终端$ T\ subset X$ 。 据说这样的嵌入美元扭曲了$\ rho\ ge 1美元, 如果$是最小值, 以至于存在一个恒定的 $>0 美元满足 =gin{ =quation\\\ restal} =xxxxx, c=xxxx, c=xxxxx) 至另外的 美元终端 。 如果 $X, Y 和 Euclidean 等量度都存在一个恒定值, 美元为美元, 最近( Narson 2019), 根據主數( Mahabd, Makarychel - 美元, Makary = maxim dalmax) 的數據數數數數數據數據數數數數數數數數數數數數數數數數數數數是數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數數