Turner, KatharineMileyko, YuriyMukherjee, SayanHarer, John2025-05-312025-05-310179-5376http://www.scopus.com/inward/record.url?scp=84904427571&partnerID=8YFLogxKhttps://hdl.handle.net/1885/733756239Given a distribution ρ on persistence diagrams and observations (Formula presented.) we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X1,...,Xn. If the underlying measure ρ is a combination of Dirac masses (Formula presented.) 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.SM and KT would like to acknowledge Shmuel Weinberger for discussions and insight. SM and KT would like to acknowledge E. Subag with help in obtaining persistence diagrams computed from random Gaussian fields and explaining the generative model. JH and YM are pleased to acknowledge the support from grants DTRA: HDTRA1-08-BRCWMD, DARPA: D12AP00001On, AFOSR: FA9550-10-1-0436, and NIH (Systems Biology): 5P50-GM081883. SM is pleased to acknowledge support from grants NIH (Systems Biology): 5P50-GM081883, AFOSR: FA9550-10-1-0436, and NSF CCF-1049290.27enAlexandrov spaceFréchet meanPersistence diagramPersistent homologyTopological data analysis201410.1007/s00454-014-9604-784904427571