Randomized large distortion dimension reduction

Abstract

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\). Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\)-dimensional affine subspaces of \({\mathbb {R}}^{n}\). We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\), $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\), which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\).