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

We study geometric and topological properties of the image of a smooth submanifold of Runder a bi-Lipschitz map to R. In particular, we characterize how the dimension, diameter,volume, and reach of the embedded manifold relate to the original. Our main result establishesa lower bound on the reach of the embedded manifold in the case where m ≤ n and the bi-Lipschitz map is linear. We discuss implications of this work in signal processing and machinelearning, where bi-Lipschitz maps on low-dimensional manifolds have been constructed usingrandomized linear operators.