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Optimality of the Johnson-Lindenstrauss Lemma Larsen, Kasper Green

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For any integers $d, n \geq 2$ and $1/({\min\{n,d\}})^{0.4999} <\eps<1$, we show the existence of a set of $n$ vectors $X\subset \R^d$ such that any embedding $f:X\rightarrow \R^m$ satisfying $$ \forall x,y\in X,\ (1-\eps)\|x-y\|_2^2 \le \|f(x)-f(y)\|_2^2 \le (1+\eps)\|x-y\|_2^2 $$ must have $$ m = \Omega(\eps^{-2} \lg n). $$ This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL’84]. Furthermore, our lower bound holds for nearly the full range of $\eps$ 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 $\eps, n, d$ [LN’16]. The best previously known lower bound for general $f$ was $m = \Omega(\eps^{-2}\lg n/\lg(1/\eps))$ [W’74,A’03], which is suboptimal for any $\eps = o(1)$.

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