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Documents authored by Kurlin, Vitaliy

Document
DOI: 10.4230/LIPIcs.SoCG.2021.32
The Density Fingerprint of a Periodic Point Set

Authors: Herbert Edelsbrunner, Teresa Heiss, Vitaliy Kurlin, Philip Smith, and Mathijs Wintraecken

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.
Cite as

Herbert Edelsbrunner, Teresa Heiss, Vitaliy Kurlin, Philip Smith, and Mathijs Wintraecken. The Density Fingerprint of a Periodic Point Set. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2021.32,
  author =	{Edelsbrunner, Herbert and Heiss, Teresa and Kurlin, Vitaliy and Smith, Philip and Wintraecken, Mathijs},
  title =	{{The Density Fingerprint of a Periodic Point Set}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{32:1--32:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.32},
  URN =		{urn:nbn:de:0030-drops-138310},
  doi =		{10.4230/LIPIcs.SoCG.2021.32},
  annote =	{Keywords: Lattices, periodic sets, isometries, Dirichlet-Voronoi domains, Brillouin zones, bottleneck distance, stability, continuity, crystal database}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.32
The Mergegram of a Dendrogram and Its Stability

Authors: Yury Elkin and Vitaliy Kurlin

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
This paper extends the key concept of persistence within Topological Data Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case the distance-based persistence is determined by a single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stable-under-noise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay.
Cite as

Yury Elkin and Vitaliy Kurlin. The Mergegram of a Dendrogram and Its Stability. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{elkin_et_al:LIPIcs.MFCS.2020.32,
  author =	{Elkin, Yury and Kurlin, Vitaliy},
  title =	{{The Mergegram of a Dendrogram and Its Stability}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{32:1--32:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.32},
  URN =		{urn:nbn:de:0030-drops-127281},
  doi =		{10.4230/LIPIcs.MFCS.2020.32},
  annote =	{Keywords: clustering dendrogram, topological data analysis, persistence, stability}
}
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