Fréchet mean and $p$-mean on the unit circle: decidability, algorithm, and applications to clustering on the flat torus
Résumé
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 $S^1$ , based on a decomposition of $S^1$ 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.
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