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 o...

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Bibliographic Details
Published in:Numerische Mathematik 2017-06, Vol.136 (2), p.523-543
Main Authors: Huang, Wen, Absil, P.-A., Gallivan, K. A.
Format: Article
Language:English
Subjects:
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Matrix algebra
Matrix methods
Numerical Analysis
Numerical and Computational Physics
Quasi Newton methods
Representations
Riemann manifold
Simulation
Theoretical
Transport
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Summary: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 n . Specifically, it requires O ( n p 2 ) 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 positive-semidefinite fixed-rank manifold. In the process, we also propose a convenient way to represent tangent vectors to these manifolds as elements of R d , where d is the dimension of the manifold. We call this an “intrinsic” representation, as opposed to “extrinsic” representations as elements of R w , 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.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-016-0848-4