We present recent results that show that for any portion of a compact manifold that admits a bi-Lipschitz parametrization by a Euclidean ball one may find a well-chosen set of eigenfunctions of the Laplacian that gives a bi-Lipschitz parametrization almost as good as the best possible. A similar, and in some respect stronger result holds by replacing eigenfunctions with heat kernels. These constructions are motivated by applications to the analysis of the geometry of data sets embedded in high-dimensional spaces, that are assumed to lie on, or close to, a low-dimensional manifold. This is joint work with P.W. Jones and R. Schul.
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