The Fast Johnson-Lindenstrauss Transform is Even Faster

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(varepsilon-2 ln n) dimensions, while preserving all pairwise distances to within a factor (1 ± varepsilon). The Fast JL transform supports computing the embedding of a data point in O(d ln d +k ln2 n) time, where the d ln d term comes from multiplication with a d × d Hadamard matrix and the k ln2 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 varepsilon, d and n. In this work, we give a surprising new analysis of the Fast JL transform, showing that the k ln2 n term in the embedding time can be improved to (k ln2 n)/α for an α = Omega(minvarepsilon-1ln(1/varepsilon), 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.