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

TL;DR

This paper investigates how bi-Lipschitz maps affect the geometric and topological properties of smooth manifolds, providing bounds and insights relevant to signal processing and machine learning applications.

Contribution

It characterizes the effects of bi-Lipschitz maps on manifold properties and establishes a lower bound on reach for linear bi-Lipschitz embeddings when the target dimension is less than or equal to the original.

Findings

Lower bound on reach for linear bi-Lipschitz embeddings

Relations between original and embedded manifold properties

Implications for signal processing and machine learning

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.