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

The Johnson-Lindenstrauss lemma is optimal for linear dimensionality reduction