Fréchet Means for Distributions of Persistence Diagrams

Abstract

Given a distribution $$\rho $$ ? on persistence diagrams and observations $$X_{1},\ldots ,X_{n} \mathop {\sim }\limits ^{iid} \rho $$ X 1 , ? , X n ~ i i d ? we introduce an algorithm in this paper that estimates a Frechet mean from the set of diagrams $$X_{1},\ldots ,X_{n}$$ X 1 , ? , X n . If the underlying measure $$\rho $$ ? is a combination of Dirac masses $$\rho = \frac{1}{m} \sum _{i=1}^{m} \delta _{Z_{i}}$$ ? = 1 m ? i = 1 m ? Z i then we prove the algorithm converges to a local minimum and a law of large numbers result for a Frechet 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.