Fréchet Means for Distributions of Persistence diagrams

2012/06/13 by Turner, Katharine, Mileyko, Yuriy, Mukherjee, Sayan +1 · 11 citations
#51 #62 #FOS: Mathematics #General Topology (math.GN) #Metric Geometry (math.MG) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1206.2790

Abstract

Given a distribution ρ on persistence diagrams and observations X₁,...X_n \stackreliid∼ ρ we introduce an algorithm in this paper that estimates a Fréchet mean from the set of diagrams X₁,...X_n. If the underlying measure ρ is a combination of Dirac masses ρ= (1)/(m) ∑_i=1^m δ_Zi 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.

Cited by

• Statistical Analysis of Network Collections Using Persistent Homology and Functional Data Analysis
• Continuous Edit Distance, Geodesics and Barycenters of Time-varying Persistence Diagrams
• Robust Global Fr'echet Regression via Weight Regularization
• Region-Aware Wasserstein Distances of Persistence Diagrams and Merge Trees
• A Mathematical Framework for Topological Causal Data Analysis
• Topology Aware Neural Interpolation of Scalar Fields
• Robust Barycenters of Persistence Diagrams
• Intrinsic Bottleneck Distance for Merge Trees
• Augmenting Bias Detection in LLMs Using Topological Data Analysis
• Spaces with Riemannian curvature bounds are universally infinitesimally Hilbertian
• Functional Connectivity Graph Neural Networks
• Path representations in multiparameter persistent homology

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