Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Abstract

Persistence diagrams, an important summary in topological data analysis, consist of a set of ordered pairs, each with positive multiplicity. Persistence diagrams are obtained via Mobius inversion and may be compared using a one-parameter family of metrics called Wasserstein distances. In certain cases, Mobius inversion produces sets of ordered pairs which may have negative multiplicity. We call these virtual persistence diagrams. Divol and Lacombe recently showed that there is a Wasserstein distance for Radon measures on the half plane of ordered pairs that generalizes both the Wasserstein distance for persistence diagrams and the classical Wasserstein distance from optimal transport theory. Following this work, we define compatible Wasserstein distances for persistence diagrams and Radon measures on arbitrary metric spaces. We show that the 1-Wasserstein distance extends to virtual persistence diagrams and to signed measures. In addition, we characterize the Cauchy completion of persistence diagrams with respect to the Wasserstein distances. We also give a universal construction of a Banach space with a 1-Wasserstein norm. Persistence diagrams with the 1-Wasserstein distance isometrically embed into this Banach space.

arXiv comment33 pages, accepted to the Journal of Applied and Computational Topology

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• Homological approximations in persistence theory
• Graded persistence diagrams and persistence landscapes
• On the stability of multigraded Betti numbers and Hilbert functions
• Universality of persistence diagrams and the bottleneck and Wasserstein distances
• Exact weights, path metrics, and algebraic Wasserstein distances
• Basic Metric Geometry of the Bottleneck Distance
• Metric Geometry of Spaces of Persistence Diagrams
• Zig-Zag Modules: Cosheaves and K-Theory
• Topological and metric properties of spaces of generalized persistence diagrams
• Poincaré Duality for Generalized Persistence Diagrams of (co)Filtrations