… , we finally settle the optimality of the JL lemma. Furthermore, … upon the JL lemma by avoiding
the DJL lemma. Our main … ) via the DJL lemma combined with a union bound is optimal. …

… In other words, the JL lemma is optimal in the case where f must be linear. Our lower bound
is optimal: the identity map achieves the first term in the min, and the JL lemma provides the …

… JohnsonLindenstrauss transformations. The new bounds allow obtaining the well known Fast
Johnson-Lindenstrauss … , and is believed to be optimal. The possibility of obtaining such a …

… We also give a new elementary proof of the optimality of the JL lemma showing a lower
bound of Ω(log(1/δ)/ε 2 ) on the embedding dimension. Previously, Jayram and Woodruff [10] …

… The Johnson-Lindenstrauss transform is a dimensionality … In this article, we show that this
bound on k is optimal up to a … resolve the optimality of Johnson-Lindenstrauss transforms …

… In this section, we prove an optimal lower bound on the limit in Theorem 2 that matches the
upper bound from the previous section. The proof of … We begin with the following key lemma: …

… For a set X of N points in RD, the Johnson-Lindenstrauss lemma provides random linear
maps that approximately preserve all pairwise distances in X – up to multiplicative error (1±ǫ) …

… optimal up to the value of C, as was shown by Alon [3]; also see [14].) The Johnson–Lindenstrauss
lemma … In Section 5 we discuss a variant of the Johnson–Lindenstrauss lemma where …

… , is the Johnson-Lindenstrauss (JL) transform also optimal for … q, leaving the question of
optimality of the most important case of … that indeed the Johnson-Lindenstrauss bounds are best …

We give two different and simple constructions for dimensionality reduction in ℓ 2 via linear
mappings that are sparse: only an O(ε)-fraction of entries in each column of our embedding …