A centroid (Karcher mean) approach to the joint approximate diagonalisation problem: The real symmetric case

Abstract

A real symmetric matrix is diagonalisable by a suitable orthonormal change of basis. The joint approximate diagonalisation problem is to find an orthonormal change of basis which simultaneously diagonalises, or approximately diagonalises, two or more real symmetric matrices. A number of important signal processing problems require the computation of a joint approximate diagonaliser. However, no algorithm to date is guaranteed to find the optimal diagonaliser. This paper reformulates the diagonalisation problem as a convex optimisation problem on a Riemannian manifold and is thus able to guarantee global convergence to the optimal diagonaliser. © 2005 Elsevier Inc. All rights reserved.