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

Armin Eftekhari; Michael B. Wakin

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arXiv · 1512.06906

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

Armin Eftekhari·Michael B. Wakin

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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.

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Armin Eftekhari, Michael B. Wakin. 2016-11-21. What Happens to a Manifold Under a Bi-Lipschitz Map?. https://arxiv.org/abs/1512.06906

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