A Geometric Condition for Uniqueness of Fréchet Means of Persistence Diagrams

TL;DR

This work addresses the non-uniqueness of Fréchet means in the space of persistence diagrams under the $2$-Wasserstein metric by introducing a geometric condition called flatness on groupings. It derives a variance expression for general groupings, proves that flat groupings yield a unique Fréchet mean, and establishes finite-sample convergence rates for both general and flat groupings, with interpretation within Alexandrov geometry. The authors also connect these ideas to practical computation by showing how truncating points near the diagonal can create flat groupings, enabling stable Fréchet-mean summaries for manifold-valued data and offering a principled way to approximate persistence diagrams via cropped diagrams. Together, these results provide theoretical guarantees and practical strategies for robust statistical summaries of persistent homology in complex geometries.

Abstract

The Fréchet 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 Fréchet 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; we prove that sets of persistence diagrams that exhibit flat groupings give rise to unique Fréchet means. We derive a finite sample convergence result for general groupings, which results in convergence for Fréchet means if the groupings are flat. We then interpret flat groupings in a recently-proposed general framework of Fréchet means in Alexandrov geometry. Finally, we show that for manifold-valued data, the persistence diagrams can be truncated to construct flat groupings.

Figures (4)

• Figure 1: Curvature is determined by the boundary. Consider three persistence diagrams: diagram $D_A$ with a single off-diagonal point $A$, diagram $D_B$ with a single off-diagonal point $B$, and the empty diagram $D_\emptyset$ with no off-diagonal point. The three edges of triangle $\triangle D_\emptyset D_AD_B$ are plotted with solid lines. For the comparison triangle, given $\mathrm{W}_2(D_A,D_\emptyset)=\|AA^\top\|$, $\mathrm{W}_2(D_A,D_B)=\|AB\|$, and $\angle D_\emptyset D_AD_B=\angle A^\top AB$, the length of the third edge is $\|A^\top B\|$. We see that $\|A^\top B\|>\|BB^\top\|=\mathrm{W}_2(D_B,D_\emptyset)$, indicating nonnegative Alexandrov curvature.
• Figure 2: An example of flat groupings. The off-diagonal points of $D_{\mathrm{red}},D_{\mathrm{blue}},D_{\mathrm{cyan}}$ are distributed as three clusters over the half-plane $\Omega$. Every dashed circle indicates a selection of the grouping. The Fréchet mean is given by $D_{\mathrm{black}}$.
• Figure 3: Counterexamples violating the conditions of flat groupings (Definition \ref{['def:flat_grouping']}). On the left panel, a grouping for two persistence diagrams $D_{\mathrm{red}},D_{\mathrm{black}}$ is depicted by solid lines. Four off-diagonal points form the corners of a square, hence the grouping violates conditions (i) and (ii). The Fréchet mean is not unique as the grouping depicted by dotted lines gives another Fréchet mean. On the right panel, a grouping is given by the solid lines. Let $\epsilon>0$ be a small positive number. The grouping satisfies condition (i) in Definition \ref{['def:flat_grouping']} because each selection has diameter $\lambda-\epsilon$. The dotted line parallel to the diagonal shows the distance between two selections is $\lambda+\epsilon$. Thus the grouping satisfies condition (ii) of Definition \ref{['def:flat_grouping']}. The dotted line orthogonal to the diagonal shows that the distances of off-diagonal points to the diagonal are sufficiently smaller than $\lambda$. Thus the grouping violates condition (iii) of Definition \ref{['def:flat_grouping']}. In this case, the mean of the grouping is not a Fréchet mean, since the optimal grouping here will match all off-diagonal points with the diagonal and the Fréchet mean is given by the intermediate points between off-diagonal points and the diagonal.
• Figure 4: Illustration of cropping. On the left panel, we have the original persistence diagrams (rotated $45^\circ$ clockwise) computed from 50 sample sets, each consisting of 1000 points from a torus. The shaded background represents the kernel density estimation for off-diagonal points from 50 persistence diagrams. On the right panel, we crop the off-diagonal points below the dotted line. The remaining points form two clusters, thus illustrating that the truncated persistence diagrams form a flat grouping.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Example 3
• Definition 4
• Remark 5
• Definition 6
• Theorem 7
• Theorem 8
• proof
• Definition 9