The Johnson-Lindenstrauss transform allows one to embed a dataset of $n$ points in $\mathbb{R}^d$ into $\mathbb{R}^m,$ while preserving the pairwise distance between any pair of points up to a factor $(1 \pm \varepsilon)$, provided that $m = \Omega(\varepsilon^{-2} \lg n)$. The transform has found an overwhelming number of algorithmic applications, allowing to speed up algorithms and reducing memory consumption at the price of a small loss in accuracy. A central line of research on such transforms, focus on developing fast embedding algorithms, with the classic example being the Fast JL transform by Ailon and Chazelle. All known such algorithms have an embedding time of $\Omega(d \lg d)$, but no lower bounds rule out a clean $O(d)$ embedding time. In this work, we establish the first non-trivial lower bounds (of magnitude $\Omega(m \lg m)$) for a large class of embedding algorithms, including in particular most known upper bounds.

翻译：Johnson- Lindensstraus 变换允许将一组美元点数的数据集嵌入 $mathbb{R ⁇ d$, 以美元为单位, 将一组美元点数的数据集嵌入 $mathbb{R ⁇ d$, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以快速嵌入算法为主, 典型的例子是 Ailon 和 Chazelle 的快速 JL 变换。 所有已知的算法都以美元为单位, 以美元为单位, 但没有下限排除一个干净的 $(d) 美元嵌入时间 。 在这项工作中, 我们建立了第一个非三边底底线, 包括已知的大型嵌入级的 。