Lower Bounds on the Low-Distortion Embedding Dimension of Submanifolds of $\mathbb{R}^n$

TL;DR

This paper establishes lower bounds on the minimal dimension needed for low-distortion embeddings of smooth submanifolds in Euclidean space, showing that random linear maps are nearly optimal for manifold dimension reduction.

Contribution

It provides new lower bounds on embedding dimension based on manifold properties, confirming the near-optimality of random linear maps for manifold embeddings.

Findings

Lower bounds depend on reach, volume, and dimension of the manifold.

Random matrices achieve near-optimal embedding dimensions.

Results imply limitations on nonlinear measurement maps with RIP.

Abstract

Let $\mathcal{M}$ be a smooth submanifold of $\mathbb{R}^n$ equipped with the Euclidean (chordal) metric. This note considers the smallest dimension $m$ for which there exists a bi-Lipschitz function $f: \mathcal{M} \mapsto \mathbb{R}^m$ with bi-Lipschitz constants close to one. The main result bounds the embedding dimension $m$ below in terms of the bi-Lipschitz constants of $f$ and the reach, volume, diameter, and dimension of $\mathcal{M}$. This new lower bound is applied to show that prior upper bounds by Eftekhari and Wakin (arXiv:1306.4748) on the minimal low-distortion embedding dimension of such manifolds using random matrices achieve near-optimal dependence on both reach and volume. This supports random linear maps as being nearly as efficient as the best possible nonlinear maps at reducing the ambient dimension for manifold data. In the process of proving our main result, we also prove similar results concerning the impossibility of achieving better nonlinear measurement maps with the Restricted Isometry Property (RIP) in compressive sensing applications.