It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in [BFN19] (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has 1 + ǫ average distortion for embedding into k-dimensional Euclidean space, where k = O(1/ǫ 2 ), and for more general q-norms of distortion, k = O(max{1/ǫ 2 , q/ǫ}), whereas tight lower bounds were established only for large values of q via reduction to the worst case.In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any 1 ≤ q ≤ O( log(2ǫ 2 n) ǫ ) and ǫ ≥ 1 √ n , where n is the size of the subset of Euclidean space. Our results also imply that the JL method is optimal for various distortion measures commonly used in practice, such as stress, energy and relative error. We prove that if any of these measures is bounded by ǫ then k = Ω(1/ǫ 2 ), for any ǫ ≥ 1

The seminal Fast Johnson-Lindenstrauss (Fast JL) transform by Ailon and Chazelle (SICOMP'09) embeds a set of n points in d-dimensional Euclidean space into optimal k = O(ε −2 ln n) dimensions, while preserving all pairwise distances to within a factor (1 ± ε). The Fast JL transform supports computing the embedding of a data point in O(d ln d + k ln 2 n) time, where the d ln d term comes from multiplication with a d × d Hadamard matrix and the k ln 2 n term comes from multiplication with a sparse k × d matrix. Despite the Fast JL transform being more than a decade old, it is one of the fastest dimensionality reduction techniques for many tradeoffs between ε, d and n.In this work, we give a surprising new analysis of the Fast JL transform, showing that the k ln 2 n term in the embedding time can be improved to (k ln 2 n)/α for an α = Ω(min{ε −1 ln(1/ε), ln n}). The improvement follows by using an even sparser matrix. We also complement our improved analysis with a lower bound showing that our new analysis is in fact tight.