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

We study geometric and topological properties of the image of a smooth submanifold of $$\mathbb {R}^{n}$$Rn under a bi-Lipschitz map to $$\mathbb {R}^{m}$$Rm. 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 $$m \le n$$m≤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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