[Paper Review] Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport

Vincent Divol, Théo Lacombe|arXiv (Cornell University)|Jan 10, 2019

Topological and Geometric Data AnalysisComputer Science18 references13 citations

TL;DR

This paper introduces a formalism based on optimal partial transport to generalize persistence diagrams as Radon measures on the upper half-plane, enabling a unified geometric and topological analysis of their space. It establishes convergence criteria, characterizes Fréchet means, and proves stability of expected diagrams, extending Wasserstein metrics to continuous measures for statistical applications in topological data analysis.

ABSTRACT

Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a space of discrete measures, and by observing that its metrics can be expressed as optimal partial transport problems, we introduce a generalization of persistence diagrams, namely Radon measures supported on the upper half plane. Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g.\ persistence surfaces) but also as limits for laws of large numbers on persistence diagrams or as expectations of probability distributions on the persistence diagrams space. We explore topological properties of this new space, which will also hold for the closed subspace of persistence diagrams. New results include a characterization of convergence with respect to Wasserstein metrics, a geometric description of barycenters (Fr\'echet means) for any distribution of diagrams, and an exhaustive description of continuous linear representations of persistence diagrams. We also showcase the strength of this framework to study random persistence diagrams by providing several statistical results made meaningful thanks to this new formalism.

Motivation & Objective

• To formalize the space of persistence diagrams as a subset of Radon measures on the upper half-plane, enabling continuous representations and statistical limits.
• To extend Wasserstein-type metrics (dp) from discrete diagrams to general Radon measures, ensuring consistency for statistical convergence and expectations.
• To provide a geometric and topological characterization of convergence, barycenters (Fréchet means), and linear representations in the context of optimal transport.
• To establish stability results for expected persistence diagrams under perturbations of the underlying data distribution.
• To unify theoretical frameworks in topological data analysis with optimal transport theory for improved statistical inference and machine learning applications.

Proposed method

• Represents persistence diagrams as discrete Radon measures on the upper half-plane Ω = {(t₁, t₂) ∈ ℝ² : t₂ > t₁}, generalizing to continuous measures.
• Reformulates the Wasserstein and bottleneck distances as optimal partial transport problems between measures, using the diagonal ∂Ω as a matching boundary.
• Introduces the VM topology (a stronger topology than vague convergence) to ensure convergence of measures in the space of Radon measures.
• Defines the OTp metric on Radon measures as a generalization of dp, enabling convergence analysis and statistical inference.
• Applies Bochner integration to define the linear expectation of a random persistence measure, ensuring consistency with measure-theoretic convergence.
• Uses optimal transport duality and coupling arguments to prove stability of Fréchet means and expected diagrams under distributional perturbations.

Experimental results

Research questions

• RQ1 How can persistence diagrams be generalized to continuous measures in a way that preserves metric and topological structure?
• RQ2 What is the correct topology and convergence criterion for sequences of persistence diagrams and their continuous counterparts?
• RQ3 How can Fréchet means (barycenters) of probability distributions on persistence diagrams be characterized geometrically and existentially?
• RQ4 What is the stability of the expected persistence diagram under perturbations of the underlying data-generating distribution?
• RQ5 Can optimal transport metrics be consistently extended from discrete diagrams to Radon measures to support statistical inference in TDA?

Key findings

• Convergence of persistence diagrams in the OTp metric is equivalent to weak convergence of their associated Radon measures under the VM topology.
• Fréchet means (barycenters) of any probability distribution on the space of persistence diagrams exist and are characterized as solutions to optimal transport problems on the space of Radon measures.
• The expected persistence diagram of a random process is a Radon measure, and its OTp distance to another expected diagram is bounded by the p-Wasserstein distance between the underlying data distributions.
• For i.i.d. samples from a point process in ℝᵈ, the OTp distance between expected diagrams decays as n · Wₚ₋ₖ(ξ, ξ′)ᵖ⁻ᵏ for p > k + d, with k > d, showing strong convergence under sampling.
• The bottleneck distance between expected diagrams is bounded by the Wasserstein distance between the underlying point process laws, yielding a stability result in the limit.
• The framework enables continuity of standard TDA representations (e.g., persistence surfaces, Betti curves) by characterizing them as continuous linear functionals on the space of Radon measures.