Fréchet mean and p-mean on the unit circle: characterization, decidability, and algorithm

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 note, 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. This implementation is effective for large values of n and p. Experiments on random sets of angles and protein dihedral angles consistently show that the Fréchet mean (p = 2) yields a variance reduction of ∼ 20% with respect to the classically used circular mean. 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. The implementation is available in the Structural Bioinformatics Library (http://sbl.inria.fr).