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

2015/12/21 by Eftekhari, Armin, Wakin, Michael B.
#FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.1512.06906

Abstract

We study geometric and topological properties of the image of a smooth submanifold of ℝⁿ under a bi-Lipschitz map to 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 ≤ 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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