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DOI: 10.4230/LIPIcs.SoCG.2024.42

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

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

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

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.

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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)

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@InProceedings{conradi_et_al:LIPIcs.SoCG.2024.42,
  author =	{Conradi, Jacobus and Kolbe, Benedikt and Psarros, Ioannis and Rohde, Dennis},
  title =	{{Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{42:1--42:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.42},
  URN =		{urn:nbn:de:0030-drops-199875},
  doi =		{10.4230/LIPIcs.SoCG.2024.42},
  annote =	{Keywords: Dynamic time warping, coreset, median clustering, approximation algorithm}
}

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DOI: 10.4230/LIPIcs.SoCG.2023.53

Random Projections for Curves in High Dimensions

Authors: Ioannis Psarros and Dennis Rohde

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract

Modern time series analysis requires the ability to handle datasets that are inherently high-dimensional; examples include applications in climatology, where measurements from numerous sensors must be taken into account, or inventory tracking of large shops, where the dimension is defined by the number of tracked items. The standard way to mitigate computational issues arising from the high dimensionality of the data is by applying some dimension reduction technique that preserves the structural properties of the ambient space. The dissimilarity between two time series is often measured by "discrete" notions of distance, e.g. the dynamic time warping or the discrete Fréchet distance. Since all these distance functions are computed directly on the points of a time series, they are sensitive to different sampling rates or gaps. The continuous Fréchet distance offers a popular alternative which aims to alleviate this by taking into account all points on the polygonal curve obtained by linearly interpolating between any two consecutive points in a sequence. We study the ability of random projections à la Johnson and Lindenstrauss to preserve the continuous Fréchet distance of polygonal curves by effectively reducing the dimension. In particular, we show that one can reduce the dimension to O(ε^{-2} log N), where N is the total number of input points while preserving the continuous Fréchet distance between any two determined polygonal curves within a factor of 1± ε. We conclude with applications on clustering.

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Ioannis Psarros and Dennis Rohde. Random Projections for Curves in High Dimensions. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{psarros_et_al:LIPIcs.SoCG.2023.53,
  author =	{Psarros, Ioannis and Rohde, Dennis},
  title =	{{Random Projections for Curves in High Dimensions}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{53:1--53:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.53},
  URN =		{urn:nbn:de:0030-drops-179030},
  doi =		{10.4230/LIPIcs.SoCG.2023.53},
  annote =	{Keywords: polygonal curves, time series, dimension reduction, Johnson-Lindenstrauss lemma, Fr\'{e}chet distance}
}

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Documents authored by Rohde, Dennis

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

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Random Projections for Curves in High Dimensions

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