  Let $\mathcal{M}$ be a compact $d$-dimensional submanifold of $\mathbb{R}^N$
with reach $\tau$ and volume $V_{\mathcal M}$. Fix $\epsilon \in (0,1)$. In
this paper we prove that a nonlinear function $f: \mathbb{R}^N \rightarrow
\mathbb{R}^{m}$ exists with $m \leq C \left(d / \epsilon^2 \right) \log
\left(\frac{\sqrt[d]{V_{\mathcal M}}}{\tau} \right)$ such that $$(1 - \epsilon)
\| {\bf x} - {\bf y} \|_2 \leq \left\| f({\bf x}) - f({\bf y}) \right\|_2 \leq
(1 + \epsilon) \| {\bf x} - {\bf y} \|_2$$ holds for all ${\bf x} \in
\mathcal{M}$ and ${\bf y} \in \mathbb{R}^N$. In effect, $f$ not only serves as
a bi-Lipschitz function from $\mathcal{M}$ into $\mathbb{R}^{m}$ with
bi-Lipschitz constants close to one, but also approximately preserves all
distances from points not in $\mathcal{M}$ to all points in $\mathcal{M}$ in
its image. Furthermore, the proof is constructive and yields an algorithm which
works well in practice. In particular, it is empirically demonstrated herein
that such nonlinear functions allow for more accurate compressive nearest
neighbor classification than standard linear Johnson-Lindenstrauss embeddings
do in practice.
