Intrinsic representation of tangent vectors and vector transport on matrix manifolds
(2016) Numerische Mathematik — Vol. 136, n° 2, p. 523-543 (2016)
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- Abstract
- 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 Rn. Specifically, it requires O(np2)flops, which is considerably less expensive than existing alternatives in the frequently encountered case where np. 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 Rd, where d is the dimension of the manifold. We call this an “intrinsic” representation, as opposed to “extrinsic” representations as elements of Rw, where wis 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.
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Huang, W., Absil, P.-A., & Gallivan, K. A. (2016). Intrinsic representation of tangent vectors and vector transport on matrix manifolds. Numerische Mathematik, 136(2), 523-543. https://doi.org/10.1007/s00211-016-0848-4 (Original work published 2016)