Abstract
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