The Johnson Lindenstrauss lemma is optimal for linear dimensionality reduction
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 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.
ICALP