Johnson-Lindenstrauss guarantees certain topological structure is preserved under random projections when embedding high dimensional deterministic vectors to low dimensional vectors. In this work, we try to understand how random projections affect norms of random vectors. In particular we prove the distribution of norm of random vectors $X \in \mathbb{R}^n$, whose entries are i.i.d. random variables, is preserved by random projection $S:\mathbb{R}^n \to \mathbb{R}^m$. More precisely, \[ \frac{X^TS^TSX - mn}{\sqrt{\sigma^2 m^2n+2mn^2}} \xrightarrow[\quad m/n\to 0 \quad ]{ m,n\to \infty } \mathcal{N}(0,1) \]

翻译：Johnson- Lindenstraus 保证在将高维确定性矢量嵌入低维矢量时,根据随机预测保留某些表层结构。 在这项工作中, 我们试图理解随机预测如何影响随机矢量的规范。 特别是, 我们证明随机矢量的常规分布 $X\ in\mathbb{R ⁇ n$, 其条目为 i. d. 随机变量, 随机预测保存在 $S:\mathbb{R ⁇ n\to\mathbb{R ⁇ @r ⁇ m$。 更准确地说, \\\ [\\ frac{X}X{TS}TSX - mn\sqrt\sigma2 m ⁇ 2n+2m%2\\\xrightrow[\quad m/n\to 0\qud]{ m, n\\\ to\ infty}\ m\mathcal{N} (0, 1\\]