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Topological and metric properties of spaces of generalized persistence diagrams

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Abstract

Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics called Wasserstein distances. We study the topological and metric properties of these spaces. Some of our results are new even in the case of persistence diagrams on the half-plane. Under mild conditions, no persistence diagram has a compact neighborhood. If the underlying metric space is \(\sigma \)-compact then so is the space of persistence diagrams. However, under mild conditions, the space of persistence diagrams is not hemicompact and the space of functions from this space to a topological space is not metrizable. Spaces of persistence diagrams inherit completeness and separability from the underlying metric space. Some spaces of persistence diagrams inherit being path connected, being a length space, and being a geodesic space, but others do not. We give criteria for a set of persistence diagrams to be totally bounded and relatively compact. We also study the curvature and dimension of spaces of persistence diagrams and their embeddability into a Hilbert space. As an important technical step, which is of independent interest, we give necessary and sufficient conditions for the existence of optimal matchings of persistence diagrams.

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Acknowledgements

This research was partially supported by the Southeast Center for Mathematics and Biology, an NSF-Simons Research Center for Mathematics of Complex Biological Systems, under National Science Foundation Grant No. DMS- 1764406 and Simons Foundation Grant No. 594594. This material is based upon work supported by, or in part by, the Army Research Laboratory and the Army Research Office under contract/grant number W911NF-18-1-0307. The authors would like to thank Alex Elchesen and Alexander Dranishnikov for helpful discussions and comments. The authors would like to thank the referees whose careful reading led to many improvements in our paper.

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  1. Department of Mathematics, University of Florida, Gainesville, USA

    Peter Bubenik

  2. Department of Machine Learning, H. Lee Moffitt Cancer Center and Research Institute, Tampa, USA

    Iryna Hartsock

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  1. Peter Bubenik
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Correspondence to Peter Bubenik.

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Bubenik, P., Hartsock, I. Topological and metric properties of spaces of generalized persistence diagrams. J Appl. and Comput. Topology 8, 347–399 (2024). https://doi.org/10.1007/s41468-023-00157-2

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  • DOI: https://doi.org/10.1007/s41468-023-00157-2

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