A stable cardinality distance for topological classification
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A stable cardinality distance for topological classification Vasileios Maroulas1
· Cassie Putman Micucci1 · Adam Spannaus1
Received: 12 December 2018 / Revised: 20 September 2019 / Accepted: 18 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
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. Keywords Stability · Classification · Persistent homology · Persistence diagrams · Crystal structure of materials Mathematics Subject Classification 62H30 · 62P30 · 55N99 · 54H99
1 Introduction A crucial first step in understanding properties of a crystalline material is determining its crystal structure. For highly disordered metallic alloys, such as high entropy
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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Vasileios Maroulas [email protected] Cassie Putman Micucci [email protected] Adam Spannaus [email protected]
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Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
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Fig. 1 Example of a body-centered cubic, and b face-centered cubic unit cells without additive noise or sparsity. Notice there is an essential topological difference between the two structures: The body-centered cubic structure has one atom at its center, whereas the face-centered cubic is hollow in its center, but has one atom in the middle of each of its faces
alloys (HEAs), atom probe tomography (APT) gives a snapshot of the local atomic environment. APT has two main drawbacks: experimental noise and missing data. Approximately 65% of the atoms in a sample are not registered in a typical experiment, and those atoms that are captured have their spatial coordinates corrupted by experimental noise. As noted by Kelly et al. (2013) and Miller et al. (2012), APT has a spatial resolution approximately the length of the unit cell we consider, as seen in Fig. 1. Hence the process is unable to see the finer details of a material, making the determination of a lattice structure a challenging problem. Existing algorithms for detecting the crystal structure (Chisholm and Motherwell 2004; Hicks et al. 2018; Honeycutt and Andersen 1987; Larsen et al. 2016; Moody et al. 2011
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