The Fast Johnson-Lindenstrauss Transform is Even Faster
(2204.01800)Abstract
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 $(1 \pm \varepsilon)$. 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 $d \times d$ Hadamard matrix and the $k \ln2 n$ term comes from multiplication with a sparse $k \times d$ 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 $(k \ln2 n)/\alpha$ for an $\alpha = \Omega(\min{\varepsilon{-1}\ln(1/\varepsilon), \ln n})$. 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.
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