Optimality of the Johnson-Lindenstrauss Lemma (1609.02094v2)

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 $\varepsilon = o(1)$.