Optimal terminal dimensionality reduction in Euclidean space

Abstract: Let $\varepsilon\in(0,1)$ and $X\subset\mathbb R<sup>d$ be arbitrary with $|X|$ having size $n&gt;1$. The Johnson-Lindenstrauss lemma states there exists $f:X\rightarrow\mathbb R<sup>m$ with $m = O(\varepsilon^{-2}\log n)$ such that $$\forall x\in X\ \forall y\in X, |x-y|_2 \le |f(x)-f(y)|_2 \le (1+\varepsilon)|x-y|_2 .$$ We show that a strictly stronger version of this statement holds, answering one of the main open questions of [MMMR18]: "$\forall y\in X$" in the above statement may be replaced with "$\forall y\in\mathbb R<sup>d$", so that $f$ not only preserves distances within $X$, but also distances to $X$ from the rest of space. Previously this stronger version was only known with the worse bound $m = O(\varepsilon^{-4}\log n)$. Our proof is via a tighter analysis of (a specific instantiation of) the embedding recipe of [MMMR18].