Fréchet Means for Distributions of Persistence Diagrams

• Published in: Discrete & computational geometry Vol. 52; no. 1; pp. 44 - 70
• Main Authors: Turner, Katharine, Mileyko, Yuriy, Mukherjee, Sayan, Harer, John
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.07.2014; Springer Nature B.V
• Subjects: Algorithms; Combinatorics; Computation; Computational Mathematics and Numerical Analysis; Computer simulation; Convergence; Diagrams; Estimates; Gaussian; Geometry; Law; Mathematics; Mathematics and Statistics; Normal distribution; Texts; Persistence diagram; Persistent homology; Fréchet mean; Topological data analysis; Alexandrov space
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-014-9604-7

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.