Optimal terminal dimensionality reduction in Euclidean space

TL;DR

This paper proves a stronger form of the Johnson-Lindenstrauss lemma, allowing distance preservation from points outside the set to the set, with improved embedding dimension bounds.

Contribution

It establishes that the JL embedding can preserve distances to the entire space, not just within the set, with a tighter bound on the embedding dimension.

Findings

Stronger JL lemma version proven with $m=O(rac{1}{\varepsilon^2}\log n)$

Distance preservation extends to all points in space, not just within the set

Improved bounds from previous $O(rac{1}{\varepsilon^4}\log n)$ result

Abstract

Let $\varepsilon\in(0,1)$ and $X\subset\mathbb R^d$ be arbitrary with $|X|$ having size $n>1$. The Johnson-Lindenstrauss lemma states there exists $f:X\rightarrow\mathbb R^m$ 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 "$\forall y\in\mathbb R^d$", 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].