论文标题
降低维度的障碍
Barriers for Faster Dimensionality Reduction
论文作者
论文摘要
Johnson-Lindenstrauss的转换允许一个人嵌入$ n $点的数据集中的$ \ Mathbb {r}^d $中的$ \ \ \ \ \ \ \ \ \ m athbb {r}^m，$，同时保留任何点$（1 \ pm \ pm \ pm \ var pm \ var pm \ var pm \ varepsilon）$，前提是$ m = ph vareps $ = f vareps $ = fareps $ = f c.该转换发现了大量的算法应用程序，从而使算法加快了算法并以微小的准确性损失的价格减少记忆消耗。关于此类转换的中心研究，专注于开发快速嵌入算法，经典的例子是Ailon和Chazelle的快速JL变换。所有已知的此类算法的嵌入时间为$ω（d \ lg d）$，但没有下限排除清洁$ O（d）$嵌入时间。在这项工作中，我们为大型嵌入算法（包括最著名的上限）建立了第一个非平凡的下限（幅度$ω（M \ lg m）$）。
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 $m = Ω(\varepsilon^{-2} \lg n)$. 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 $Ω(d \lg d)$, 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 $Ω(m \lg m)$) for a large class of embedding algorithms, including in particular most known upper bounds.
