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The Persistence Landscape and Some of Its Properties

  • Peter BubenikEmail author
Conference paper
  • 64 Downloads
Part of the Abel Symposia book series (ABEL, volume 15)

Abstract

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.

Notes

Acknowledgements

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. 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. The author would also like to thank Pawel Dlotko, Michael Kerber, and Oliver Vipond for helpful conversations, Leo Betthauser, Nikola Milicevic, and Alex Wagner for proofreading an earlier draft, and the Mathematisches Forschungsinstitut Oberwolfach (MFO) where some of this work was started.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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