Optimality of the Johnson-Lindenstrauss Lemma

TL;DR

This paper proves that the Johnson-Lindenstrauss lemma's dimension bound is essentially optimal for all relevant parameters, establishing a tight lower bound matching the known upper bound.

Contribution

It extends the lower bound to all embeddings, not just linear maps, for a broad range of parameters, confirming the optimality of the JL lemma.

Findings

Lower bound matches JL upper bound for embedding dimension

Lower bound applies to all embeddings, not just linear

Bound holds for a wide range of epsilon, n, d

Abstract

For any integers $d, n \geq 2$ and $1/({\min\{n,d\}})^{0.4999} < \varepsilon<1$, we show the existence of a set of $n$ vectors $X\subset \mathbb{R}^d$ such that any embedding $f:X\rightarrow \mathbb{R}^m$ satisfying $$ \forall x,y\in X,\ (1-\varepsilon)\|x-y\|_2^2\le \|f(x)-f(y)\|_2^2 \le (1+\varepsilon)\|x-y\|_2^2 $$ must have $$ m = \Omega(\varepsilon^{-2} \lg n). $$ 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 $\varepsilon$ of interest, since there is always an isometric embedding into dimension $\min\{d, n\}$ (either the identity map, or projection onto $\mathop{span}(X)$). Previously such a lower bound was only known to hold against linear maps $f$, and not for such a wide range of parameters $\varepsilon, n, d$ [LN16]. The best previously known lower bound for general $f$ was $m = \Omega(\varepsilon^{-2}\lg n/\lg(1/\varepsilon))$ [Wel74, Lev83, Alo03], which is suboptimal for any $\varepsilon = o(1)$.