License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SEA.2021.15
URN: urn:nbn:de:0030-drops-137870
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13787/
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Cazals, Frédéric ; Delmas, Bernard ; O'Donnell, Timothee

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

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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.

BibTeX - Entry

@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/opus/volltexte/2021/13787},
  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}
}

Keywords: Frechét mean, p-mean, circular statistics, decidability, robustness, multi-precision, angular spaces, flat torus, clustering, molecular conformations
Collection: 19th International Symposium on Experimental Algorithms (SEA 2021)
Issue Date: 2021
Date of publication: 31.05.2021
Supplementary Material: Software: https://sbl.inria.fr/
Text (Documentation): https://sbl.inria.fr/doc/Frechet_mean_S1-user-manual.html

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