Mathematics – Optimization and Control
Scientific paper
2008-07-28
SIAM. J. Matrix Anal. & Appl., Volume 31, Number 3, page 1055--1070 - August 2009
Mathematics
Optimization and Control
the present version is very close to the published one. It contains some corrections with respect to the previous arxiv submss
Scientific paper
This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute.
Bonnabel Silvere
Sepulchre Rodolphe
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