Optimality of the Johnson-Lindenstrauss Lemma

For any d, n ≥ 2 and 1/(min{n, d}) ^0.4999 <; ε <; 1, we show the existence of a set of n vectors X ⊂ ℝ ^d such that any embedding f : X → ℝ ^m satisfying ∀x, y ∈ X, (1-ε)∥x-y∥ ₂ ² ≤ ∥f(x)-f(y)∥ ₂ ² ≤ (1+ε)∥x-y∥ ₂ ² must have m = Ω(ε ^-2 lg n). This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of ε of interest, since there is always an isometric embedding into dimension min{d, n} (either the identity map, or projection onto span(X)). Previously such a lower bound was only known to hold against linear maps f, and not for such a wide range of parameters ε, n, d [LN16]. The best previously known lower bound for general f was m = Ω(ε ^-2 lg n/ lg(1/ε)) [Wel74], [Alo03], which is suboptimal for any ε = o(1).