Fr\'echet Means for Distributions of Persistence diagrams

Abstract

Given a distribution $\rho$ on persistence diagrams and observations $X_1,...X_n \stackrel{iid}{\sim} \rho$ we introduce an algorithm in this paper that estimates a Fr\'echet mean from the set of diagrams $X_1,...X_n$. If the underlying measure $\rho$ is a combination of Dirac masses $\rho = \frac{1}{m} \sum_{i=1}^m \delta_{Z_i}$ then we prove the algorithm converges to a local minimum and a law of large numbers result for a Fr\'echet mean computed by the algorithm given observations drawn iid from $\rho$. We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.