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Fréchet Mean and p-Mean on the Unit Circle: Decidability, Algorithm, and Applications to Clustering on the Flat Torus

Authors: Frédéric Cazals, Bernard Delmas, and Timothee O'Donnell

Published in: LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)


Abstract
The center of mass of a point set lying on a manifold generalizes the celebrated Euclidean centroid, and is ubiquitous in statistical analysis in non Euclidean spaces. In this work, we give a complete characterization of the weighted p-mean of a finite set of angular values on S¹, based on a decomposition of S¹ such that the functional of interest has at most one local minimum per cell. This characterization is used to show that the problem is decidable for rational angular values -a consequence of Lindemann’s theorem on the transcendence of π, and to develop an effective algorithm parameterized by exact predicates. A robust implementation of this algorithm based on multi-precision interval arithmetic is also presented, and is shown to be effective for large values of n and p. We use it as building block to implement the k-means and k-means++ clustering algorithms on the flat torus, with applications to clustering protein molecular conformations. These algorithms are available in the Structural Bioinformatics Library (http://sbl.inria.fr). Our derivations are of interest in two respects. First, efficient p-mean calculations are relevant to develop principal components analysis on the flat torus encoding angular spaces-a particularly important case to describe molecular conformations. Second, our two-stage strategy stresses the interest of combinatorial methods for p-means, also emphasizing the role of numerical issues.

Cite as

Frédéric Cazals, Bernard Delmas, and Timothee O'Donnell. Fréchet Mean and p-Mean on the Unit Circle: Decidability, Algorithm, and Applications to Clustering on the Flat Torus. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{cazals_et_al:LIPIcs.SEA.2021.15,
  author =	{Cazals, Fr\'{e}d\'{e}ric and Delmas, Bernard and O'Donnell, Timothee},
  title =	{{Fr\'{e}chet Mean and p-Mean on the Unit Circle: Decidability, Algorithm, and Applications to Clustering on the Flat Torus}},
  booktitle =	{19th International Symposium on Experimental Algorithms (SEA 2021)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-185-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{190},
  editor =	{Coudert, David and Natale, Emanuele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.15},
  URN =		{urn:nbn:de:0030-drops-137870},
  doi =		{10.4230/LIPIcs.SEA.2021.15},
  annote =	{Keywords: Frech\'{e}t mean, p-mean, circular statistics, decidability, robustness, multi-precision, angular spaces, flat torus, clustering, molecular conformations}
}
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