Tuesday, November 12, 2013, 4:00 p.m., B139 VV
Dejan Slepcev (Carnegie Mellon University)

Title: Regularity of minimizers of the average distance problem

Abstract: The average distance problem asks to find a good way to
approximate a high dimensional object, represented as a measure, by a one
dimensional object (measure with connected one-dimensional support). In
applications to data analysis the high dimensional object represents the
data points and the minimizer is known the (regularized) principal curve for
the data set. In mathematical community it was introduced by Buttazzo, Oudet
and Stepanov who also studied applications to network planning.
We will discuss recent examples that show that the even if the data measure
is smooth the minimizer may have corners. Nevertheless the curvature of the
minimizer can be considered as a measure. We will discuss a priori estimates
on the total curvature and ways to obtain information on topological
complexity of the minimizers. The talk is based on joint work with X.Y. Lu.