[Paper Review] Probabilistic Fréchet Means and Statistics on Vineyards.

Elizabeth Munch, Paul Bendich+4 more|arXiv (Cornell University)|Jul 24, 2013

Topological and Geometric Data AnalysisComputer Science24 references7 citations

TL;DR

This paper proposes a probabilistic refinement of the Fréchet mean for persistence diagrams, defining it as a weighted sum of atomic measures over perturbed diagrams to ensure Holder continuity. The method enables stable statistical analysis on vineyards by resolving discontinuities in mean computation across continuously varying diagram sets.

ABSTRACT

In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [20], Mileyko and his collaborators made the first study of the properties of the Frechet mean in (Dp,Wp), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Frechet mean of a finite set of diagrams always exists, but is not necessarily unique. As an unfortunate consequence, one sees that the means of a continuously-varying set of diagrams do not themselves vary continuously, which presents obvious problems when trying to extend the Frechet mean definition to the realm of vineyards. We fix this problem by altering the original definition of Frechet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself the (Frechet mean) persistence diagram of a perturbation of the input diagrams. We show that this new definition defines a (Holder) continuous map, for each k, from (Dp) k → P (Dp), and we present several examples to show how it may become a useful statistic on vineyards.

Motivation & Objective

• To address the discontinuity issue in standard Fréchet means of persistence diagrams when applied to vineyards.
• To develop a stable, continuous statistical mean for sets of persistence diagrams under the p-th Wasserstein metric.
• To enable the use of mean-based statistics in dynamic settings like vineyards, where input diagrams vary continuously.
• To ensure the mean map is Holder continuous, facilitating robust statistical inference.

Proposed method

• Define the probabilistic Fréchet mean as a probability measure on the space of persistence diagrams.
• Construct the mean as a weighted sum of atomic measures, each corresponding to the Fréchet mean of a perturbed version of the input diagrams.
• Use perturbations of the input diagrams to generate multiple candidate means, ensuring diversity and stability.
• Prove that the resulting mean map is Holder continuous from (Dp)^k to P(Dp), the space of probability measures on Dp.
• Apply the method to vineyards, where input diagrams vary continuously over time.
• Demonstrate the continuity and stability of the mean under small variations in input diagrams through theoretical analysis and examples.

Experimental results

Research questions

• RQ1 Can a continuous mean be defined for persistence diagrams in the context of vineyards, where input diagrams vary smoothly over time?
• RQ2 How can the non-uniqueness and discontinuity of standard Fréchet means be resolved to enable reliable statistical analysis?
• RQ3 What properties does a probabilistic mean over perturbed diagrams exhibit, particularly in terms of continuity and stability?
• RQ4 To what extent does the proposed method preserve the topological information of the original diagrams during averaging?
• RQ5 Can the probabilistic mean be effectively used as a statistical summary in dynamic topological data analysis?

Key findings

• The proposed probabilistic Fréchet mean is a Holder continuous map from (Dp)^k to P(Dp), resolving the discontinuity problem of standard Fréchet means.
• The mean is defined as a weighted sum of atomic measures, each centered on the Fréchet mean of a perturbed input diagram set.
• The method ensures that small changes in input diagrams lead to small, controlled changes in the resulting mean measure.
• The approach enables stable statistical analysis on vineyards by ensuring continuity in the mean computation across time-varying diagram sequences.
• Theoretical and example-based results confirm that the probabilistic mean provides a robust and continuous alternative to classical Fréchet means.