Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank

Authors:
Silvère Bonnabel;Rodolphe Sepulchre
Affiliations:
bonnabel@montefiore.ulg.ac.be and r.sepulchre@ulg.ac.be;-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2009

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Abstract

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.