A Geometric Condition for Uniqueness of Fréchet Means of Persistence Diagrams

The Fr\'echet mean is an important statistical summary and measure of centrality of data; it has been defined and studied for persistent homology captured by persistence diagrams. However, the complicated geometry of the space of persistence diagrams implies that the Fr\'echet mean for a given set of persistence diagrams is not necessarily unique, which prohibits theoretical guarantees for empirical means with respect to population means. In this paper, we derive a variance expression for a set of persistence diagrams exhibiting a multi-matching between the persistence points known as a grouping. Moreover, we propose a condition for groupings, which we refer to as flatness: sets of persistence diagrams that exhibit flat groupings give rise to unique Fr\'echet means. We derive a finite sample convergence result for general groupings, which results in convergence for Fr\'echet means if the groupings are flat. Finally, we interpret flat groupings in a recently-proposed general framework of Fr\'echet means in Alexandrov geometry. Together with recent results from Alexandrov geometry, this allows for the first derivation of a finite sample convergence rate for sets of persistence diagrams and lays the ground for viability of the Fr\'echet mean as a practical statistical summary of persistent homology.