  Consider a random matrix $H:\mathbb{R}^n\longrightarrow\mathbb{R}^m$. Let
$D\geq2$ 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 $1\leq
l\leq p$ and $x,y\in W_l$, \[
  \|x-y\|_2\leq\|Hx-Hy\|_2\leq D\|x-y\|_2. \] 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 $k,p,D$.
