Johnson-Lindenstrauss guarantees certain topological structure is preserved under random projections when project high dimensional deterministic vectors to low dimensional vectors. In this work, we try to understand how random matrix affect norms of random vectors. In particular we prove the distribution of the norm of random vector $X \in \mathbb{R}^n$, whose entries are i.i.d. random variables, is preserved by random projection $S:\mathbb{R}^n \to \mathbb{R}^m$. More precisely, \[ \frac{X^TS^TSX - mn}{\sqrt{\sigma^2 m^2n+2mn^2}} \xrightarrow[\quad m/n\to 0 \quad ]{ m,n\to \infty } \mathcal{N}(0,1) \] We also prove a concentration of the random norm transformed by either random projection or random embedding. Overall, our results showed random matrix has low distortion for the norm of random vectors with i.i.d. entries.
翻译:Johnson- Lindenstraus 保证在随机预测下保留某些表层结构。 当对低维矢量进行高维确定性矢量工程时, 我们试图了解随机矩阵是如何影响随机矢量规范的。 特别是我们证明了随机矢量 $X\ in\ mathbb{R ⁇ n$的分布, 其条目为 i. d. 随机变量, 由随机预测 $S:\mathbb{R ⁇ n\to\mathb{R ⁇ % m$ 。 更准确地说, \ [\ frac{ XQTS}TSX - mntsqrt\sigma2 m ⁇ 2n+2m% 2\\\\\\\xrightrow[\quad m/n\ to 0\quad]{ m, n\\ to\ infty}\ mathcal{N} {N}\\\) 我们还证明了随机预测改变随机规范的浓度。 总而言, 我们的随机矩阵显示随机矩阵对随机矢量输入的规律有低扭曲。 i. d. 。