The Johnson-Lindenstrauss lemma is optimal for linear dimensionality reduction

Abstract: For any $n&gt;1$ and $0&lt;\varepsilon&lt;1/2$, we show the existence of an $n<sup>{O(1)}$-point subset $X$ of $\mathbb{R}<sup>n$ such that any linear map from $(X,\ell_2)$ to $\ell_2<sup>m$ with distortion at most $1+\varepsilon$ must have $$m = \Omega(\min{n, \varepsilon^{-2}\log n})$$. Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma, improving the previous lower bound of Alon by a $\log(1/\varepsilon)$ factor.