What Happens to a Manifold Under a Bi-Lipschitz Map?

Abstract

We study geometric and topological properties of the image of a smooth submanifold of ${\mathbb{R}}^n$ under a bi-Lipschitz map to ${\mathbb{R}}^m$. In particular, we characterize how the dimension, diameter, volume, and reach of the embedded manifold relate to the original. Our main result establishes a lower bound on the reach of the embedded manifold in the case where $m\le n$ and the bi-Lipschitz map is linear. We discuss implications of this work in signal processing and machine learning, where bi-Lipschitz maps on low-dimensional manifolds have been constructed using randomized linear operators.