Intrinsic representation of tangent vectors and vector transports on matrix manifolds

The quasi-Newton methods on Riemannian manifolds proposed thus far do not appear to lend themselves to satisfactory convergence analyses unless they resort to an isometric vector transport. This prompts us to propose a computationally tractable isometric vector transport on the Stiefel manifold of orthonormal p-frames in R. Specifically, it requires O(np) flops, which is considerably less expensive than existing alternatives in the frequently encountered case where n ≫ p. We then build on this result to also propose computationally tractable isometric vector transports on other manifolds, namely the Grassmann manifold, the fixed-rank manifold, and the positivesemidefinite fixed-rank manifold. In the process, we also propose a convenient way to represent tangent vectors to these manifolds as elements of R, where d is the dimension of the manifold. We call this an “intrinsic” representation, as opposed to “extrinsic” representations as elements of R, where w is the dimension of the embedding space. Finally, we demonstrate the performance of the proposed isometric vector transport in the context of a Riemannian quasi-Newton method applied to minimizing the Brockett cost function. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office. This work was supported by grant FNRS PDR T.0173.13. Wen Huang Department of Mathematical Engineering, ICTEAM Institute, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium. Tel.: +32-10-478005 Fax: +32-10-472180 E-mail: [email protected] P.-A. Absil Department of Mathematical Engineering, ICTEAM Institute, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium. K. A. Gallivan Department of Mathematics, 208 Love Building, 1017 Academic Way, Florida State University, Tallahassee FL 32306-4510, USA.