Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport

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 measure space, and by observing that its metrics can be expressed as solutions of 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 study the 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 transport metrics, the existence of Fréchet means for any distribution of diagrams, and an exhaustive description of continuous linear representations of persistence diagrams. We also showcase the usefulness of this framework to study random persistence diagrams by providing several statistical results made meaningful thanks to this new formalism.

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

“…Ideas and preliminary results underlying persistent homology theory can be traced back to the 20th century, in particular in the works of Barannikov (1994) , Frosini (1992) , Robins (1999) . It started to know an important development in its modern form after the seminal works of Edelsbrunner et al (2002) and Zomorodian and Carlsson (2005) .…”

“…Topological data analysis ( tda ) is a recent field that emerged from various works in applied (algebraic) topology and computational geometry during the first decade of the century. Although one can trace back geometric approaches to data analysis quite far into the past, tda really started as a field with the pioneering works of Edelsbrunner et al (2002) and Zomorodian and Carlsson (2005) in persistent homology and was popularized in a landmark article in 2009 Carlsson (2009) . tda is mainly motivated by the idea that topology and geometry provide a powerful approach to infer robust qualitative, and sometimes quantitative, information about the structure of data [e.g., Chazal (2017 )].…”

A Framework for Differential Calculus on Persistence Barcodes

“…Properties of linear representations valued in Banach spaces such as continuity, lipschitzness and stochastic convergence are analyzed in [19,22]. Many vectorizations in the literature are linear representations, e.g., persistence images [1] and its variations [18,35,44], persistence silhouettes [13] and weighted Betti curves [47].…”

Tropical optimal transport and Wasserstein distances

“…where the infimum is taken over the set of all possible joint distributions (transport plans) π with marginals ρ 0 and ρ 1 , Π(ρ 0 , ρ 1 ). Here, the distanceW tr p depends the choice of p in the linear programming formulation (10). The following alternative gives an equivalent definition of the tropical Wassersteinp distances.…”

“…In Çelik et al [7], the Wasserstein distance between a probability distribution and an algebraic variety is minimized via transportation polytopes. In topological data analysis, where algebraic topology is leveraged to reduce the dimensionality of complex data spaces and extract shape features within the data, optimal transport theory has improved computational efficiency [20] and also has been used to study geometric aspects of algebraic topological invariants [10]. A prior transportation problem (distinct from the optimal transport setting) has been previously considered in tropical geometry by Richter-Gebert et al [41].…”