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Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping

Authors Jacobus Conradi , Benedikt Kolbe , Ioannis Psarros , Dennis Rohde

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Author Details

Jacobus Conradi
  • University of Bonn, Germany
Benedikt Kolbe
  • Hausdorff Center for Mathematics, University of Bonn, Germany
Ioannis Psarros
  • Archimedes, Athena Research Center, Greece
Dennis Rohde
  • University of Bonn, Germany

Funding

  • Conradi, Jacobus: Partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 313421352 (FOR 2535 Anticipating Human Behavior) and the iBehave Network: Sponsored by the Ministry of Culture and Science of the State of North Rhine-Westphalia. Affiliated with Lamarr Institute for Machine Learning and Artificial Intelligence.
  • Kolbe, Benedikt: Partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 459420781.
  • Psarros, Ioannis: This work has been partially supported by project MIS 5154714 of the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program.
  • Rohde, Dennis: Partially supported by the Hausdorff Center for Mathematics. Partially funded by the iBehave Network: Sponsored by the Ministry of Culture and Science of the State of North Rhine-Westphalia.

Acknowledgements

We thank Anne Driemel of the University of Bonn for detailed discussion and guidance.

Cite AsGet BibTex

Jacobus Conradi, Benedikt Kolbe, Ioannis Psarros, and Dennis Rohde. Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.42

Abstract

We present algorithms for the computation of ε-coresets for k-median clustering of point sequences in ℝ^d under the p-dynamic time warping (DTW) distance. Coresets under DTW have not been investigated before, and the analysis is not directly accessible to existing methods as DTW is not a metric. The three main ingredients that allow our construction of coresets are the adaptation of the ε-coreset framework of sensitivity sampling, bounds on the VC dimension of approximations to the range spaces of balls under DTW, and new approximation algorithms for the k-median problem under DTW. We achieve our results by investigating approximations of DTW that provide a trade-off between the provided accuracy and amenability to known techniques. In particular, we observe that given n curves under DTW, one can directly construct a metric that approximates DTW on this set, permitting the use of the wealth of results on metric spaces for clustering purposes. The resulting approximations are the first with polynomial running time and achieve a very similar approximation factor as state-of-the-art techniques. We apply our results to produce a practical algorithm approximating (k,𝓁)-median clustering under DTW.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Dynamic time warping
  • coreset
  • median clustering
  • approximation algorithm

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