Optimal terminal dimensionality reduction in Euclidean space

Abstract

Let ε ∈ (0, 1) and X ⊂ Rd be arbitrary with |X| having size n > 1. The Johnson-Lindenstrauss lemma states there exists f : X → Rm with m = O(ε−2 log n) such that ∀x ∈ X ∀y ∈ X, ∥x − y∥2 ≤ ∥ f (x) − f (y)∥2 ≤ (1 + ε)∥x − y∥2. We show that a strictly stronger version of this statement holds, answering one of the main open questions posed by Mahabadi et al. in STOC 2018: “∀y ∈ X” in the above statement may be replaced with “∀y ∈ Rd”, 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(ε−4 log n). Our proof is via a tighter analysis of (a specific instantiation of) the embedding recipe of Mahabadi et al.