RIEMANNIAN METRIC AND GEOMETRIC MEAN FOR POSITIVE SEMIDEFINITE MATRICES OF FIXED RANK

• Published in: SIAM journal on matrix analysis and applications Vol. 31; no. 3; pp. 1055 - 1070
• Main Authors: BONNABEL, Silvère, SEPULCHRE, Rodolphe
• Format: Journal Article Web Resource
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2009; Society for Industrial & Applied Mathematics
• Subjects: Algebra; Algorithms; Approximation; Electrical & electronics engineering; Engineering, computing & technology; Exact sciences and technology; Group theory; Ingénierie électrique & électronique; Ingénierie, informatique & technologie; Invariants; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Matrices; Matrix; Matrix methods; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Preserves; Sciences and techniques of general use; Studies; Topological groups, lie groups; Transformations; positive semidefinite matrices; Riemann metric; Manifold; Geometric mean; Singular value decomposition; Symmetry; 53C20; covariance matrices; Group action; Approximation; Cone; 54B15; Geometrical properties; 53C22; invariant metric; Rank; Riemann geometry; 47A64; Covariance matrix; Matrix decomposition; 58D19; Algorithm; Mean estimation; Orthogonal transformation; Numerical analysis; symmetries; Riemamnian quotient manifold; Linear algebra; Lie group action; Lie group; Riemann space; Quotient
• ISSN: 0895-4798; 1095-7162; 1095-7162
• DOI: 10.1137/080731347

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. [PUBLICATION ABSTRACT]