Fréchet Means for Distributions of Persistence Diagrams - Turner, Katharine; Mileyko, Yuriy; Mukherjee, Sayan; Harer, John (Discrete & computational geometry, 2014)
Given a distribution ρ on persistence diagrams and observations X 1 , … , X n ∼ i i d ρ 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 ρ is a combination of Dirac masses ρ = 1 m ∑ i = 1 m δ 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 ρ . We illustrate the convergence of an empirical mean computed by the algorithm to a population mean by simulations from Gaussian random fields.