Fréchet 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 Fréchet 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 Fréchet 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.

References in corpus (3)

• Random geometric complexes
• Semiconcave functions in Alexandrov's geometry
• Limit theorems for Betti numbers of random simplicial complexes

Cited by in corpus (4)

• A persistence landscapes toolbox for topological statistics
• Principal Component Analysis of Persistent Homology Rank Functions with case studies of Spatial Point Patterns, Sphere Packing and Colloids
• Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces
• Nonparametric Risk Assessment and Density Estimation for Persistence Landscapes