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.

翻译：$mathcal{M} $Let $mathcal{R{M} 美元是 $mathbb{R ⁇ n$ 的平滑的底部, 配有 Euclidean (chodal) 度量值。 本说明考虑的是存在双- Lipschitz 函数的最小维度 $f:\ mathcal{M}\ mpsto \ mathcal{M} {M} \ maptsto \ mathcalbb{R ⁇ m$, 和 bi- Lipschitz 常量值接近于 1 的 mathbb{R{M} 。 主要结果将以下嵌入维维维维维维维维 。 支持随机线性地图, 几乎是减少 $\\ mathcal developal developmental impressiveal importiveal import asiveal import (Regresulate the mapal main.