# A Continuous Mean for Finite Sets of Persistence Diagrams

Thursday, October 10, 2013 - 3:15pm - 4:05pm

Keller 3-180

Elizabeth Munch (University of Minnesota, Twin Cities)

Recent progress has shown that the abstract space of persistence diagrams is Polish, and found necessary and sufficient conditions for a set to be compact.

This also allowed for the definition of Frechet means, a construction which is possible for any metric space. Since the Frechet mean is a set, not an element, there were no guarantees that this set was non-empty, however Mileyko et al showed that for well behaved distributions, the Frechet mean is, in fact, non-empty.

However, there are simple counterexamples showing that the mean is non-unique, which creates issues when we are interested in continuous, time-varying systems. In order to solve this problem, we present a continuous mean which is a generalization of the Frechet mean.

This also allowed for the definition of Frechet means, a construction which is possible for any metric space. Since the Frechet mean is a set, not an element, there were no guarantees that this set was non-empty, however Mileyko et al showed that for well behaved distributions, the Frechet mean is, in fact, non-empty.

However, there are simple counterexamples showing that the mean is non-unique, which creates issues when we are interested in continuous, time-varying systems. In order to solve this problem, we present a continuous mean which is a generalization of the Frechet mean.

MSC Code:

26E60

Keywords: