Topological Consistency for Estimation of Density Level Sets

• Activity Number: 209
• Type: Invited
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Computing
• Author(s): Omer Bobrowski and Sayan Mukherjee and Jonathan Taylor
• Companies: Duke University and Duke University and Stanford University

Abstract

Let X_1,X_2,..., X_n be iid random variables in d-dimensional Euclidean space, generated by an unknown probability density function f. The level sets of the density function are of a considerable interest in many areas of statistics. Previous work focused on recovering the level sets of f, minimizing `local' measures such as the Hausdorff distance or the symmetric distance. We will focus on `global' features related to the topology of level sets. In particular, we are interested in the homology of these sets. Briefly, homology is an algebraic structure describing the topology of a set in terms of connected components and holes.

The main difficulty in recovering the homology is that even small perturbations to the estimated density function can generate a very large error. In this talk we discuss these problems, and present an estimator that overcomes these difficulties. We then show that this estimator is consistent, and discuss possible applications for clustering and topological manifold learning.

* This talk assumes no prior knowledge in algebraic topology.