A stable cardinality distance for topological classification

Abstract

This work incorporates topological features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are multisets summarizing the connectedness and holes of given data. A new distance on the space of persistence diagrams generates relevant input features for a classification algorithm for materials science data. This distance measures the similarity of persistence diagrams using the cost of matching points and a regularization term corresponding to cardinality differences between diagrams. Establishing stability properties of this distance provides theoretical justification for the use of the distance in comparisons of such diagrams. The classification scheme succeeds in determining the crystal structure of materials on noisy and sparse data retrieved from synthetic atom probe tomography experiments.

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Acknowledgements

The authors would like to thank the anonymous associate editor and two anonymous reviewers for their insightful comments which substantially improved the manuscript. Moreover, the authors would like to thank Professor David J. Keffer (Department of Materials Science and Engineering at The University of Tennessee) for providing the codes which create the realistic APT datasets and for useful discussions, as well as Professor Kody J.H. Law (School of Mathematics at the University of Manchester) for insightful discussions.

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Correspondence to Vasileios Maroulas.

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This work has been partially supported by the ARO Grant # W911NF-17-1-0313, the NSF DMS-1821241, and UTK 2019 Research Seed Funding-Interdisciplinary.

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Maroulas, V., Micucci, C.P. & Spannaus, A. A stable cardinality distance for topological classification. Adv Data Anal Classif 14, 611–628 (2020). https://doi.org/10.1007/s11634-019-00378-3

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Keywords

  • Stability
  • Classification
  • Persistent homology
  • Persistence diagrams
  • Crystal structure of materials

Mathematics Subject Classification

  • 62H30
  • 62P30
  • 55N99
  • 54H99