Frechet Means for Distributions of Persistence diagrams

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 Fr'echet 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 Fr'echet 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.

💡 Research Summary

The paper addresses the fundamental problem of defining and computing a statistical mean for persistence diagrams, which are multiset representations of topological features extracted from data. By treating persistence diagrams as points in a metric space equipped with the p‑Wasserstein distance, the authors adopt the Fréchet functional
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