Mathematics Colloquium, CSU
Friday, March 29, 2013
3PM–4PM in RT 1516
Reconstructing manifolds and functions
from noisy point samples
Vidit Nanda, University of Pennsylvania
Abstract. A classic problem in mathematics and computer science is that of manifold
learning. Given an unknown compact manifold embedded in Euclidean space and a
known uniform sample of points lying on or near that manifold, how much of the
manifold’s structure can be inferred from the points alone? We will survey some
fantastic work of Niyogi, Smale and Weinberger which provides explicit bounds on the
number of points required to reconstruct the manifold in question up to homotopy
type with high confidence. Next, we imagine the situation where an unknown function
between two such manifolds must be inferred from point samples lying near the domain
and range manifolds as well as the ability to evaluate the function at the domainsample. We will see that not only can such functions be reconstructed up to homotopy
from finite point samples with high confidence, but also that the entire process of
reconstruction is robust to certain models of sampling and evaluation noise.
* Refreshments at 2:30 PM in RT 1517