Fitting manifolds to data.

The problems come in two flavors.

Extrinsic Flavor: Given a point cloud in R^N sampled from an unknown probability density, how can we decide whether that probability density is concentrated near a low-dimensional manifold M with reasonable geometry? If such an M exists, how can we find it? (Joint work with S. Mitter and H. Narayanan)

Intrinsic Flavor: How can we decide whether a given finite metric space is approximately isometric (in an appropriate sense) to an epsilon-net in a manifold M with reasonable geometry? If such an M exists, how can we find it?

(Joint work with S.Ivanov, Y. Kurylev, M. Lassas and H. Narayanan)