Title: Stochastic topology and inference
Speaker: Sayan Mukherjee (Professor, Department of Statistical Science, Duke University)
Hosted by Elizabeth Meckes
Abstract: I will first talk about a problem at the interface of probability and topology and then an approach to using topology for statistical inference for shapes.
I) Given n points drawn from a uniform distribution on a manifold we place balls of size r around each point. The random object we consider is the union of the balls. We as n goes to infinity and r goes to zero about the limiting distribution of topological summaries such as Betti numbers and critical points. We provide scaling limits for this process.
II) We introduce the persistent homology transform (PHT), to model surfaces and shapes. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We show that the map from the space of simplicial complexes into the space spanned by the PHT is injective. This implies that we can use it to determine a metric on the space of piecewise linear shapes. We apply the PHT to study primate heel bones. We will also introduce a variation of the PHT called the Euler characteristic transform (ECT) and use it to predict disease free survival in Glioblastoma, we will see that the ECT can be used in the setting of a linear mixed model.