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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, Timothee O'Donnell

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Author Details

Frédéric Cazals
  • Université Côte d'Azur, France
  • Inria, Sophia Antipolis, France
Bernard Delmas
  • INRAe, Jouy-en-Josas, France
Timothee O'Donnell
  • Université Côte d'Azur, France
  • Inria, Sophia Antipolis, France


Chee Yap and Sylvain Pion are acknowledged for discussions on irrational number theory and number types, respectively.

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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)


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 ( 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Frechét mean
  • p-mean
  • circular statistics
  • decidability
  • robustness
  • multi-precision
  • angular spaces
  • flat torus
  • clustering
  • molecular conformations


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