CUED Publications database

Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank

Bonnabel, S and Sepulchre, R (2009) Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank. SIAM Journal on Matrix Analysis and Applications, 31. pp. 1055-1070. ISSN 0895-4798

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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. © 2009 Society for Industrial and Applied Mathematics.

Item Type: Article
Uncontrolled Keywords: Covariance matrices Geometric mean Invariant metric Lie group action Matrix decomposition Positive semidefinite matrices Riemannian quotient manifold Singular value decomposition Symmetries
Subjects: UNSPECIFIED
Divisions: Div F > Control
Depositing User: Cron Job
Date Deposited: 07 Mar 2014 11:25
Last Modified: 08 Dec 2014 02:18
DOI: 10.1137/080731347