Authors: Wen Huang PA Absil K A Gallivan
Publish Date: 2016/10/27
Volume: 136, Issue: 2, Pages: 523-543
Abstract
The quasiNewton 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 pframes in mathbb Rn Specifically it requires Onp2 flops which is considerably less expensive than existing alternatives in the frequently encountered case where ngg p We then build on this result to also propose computationally tractable isometric vector transports on other manifolds namely the Grassmann manifold the fixedrank manifold and the positivesemidefinite fixedrank manifold In the process we also propose a convenient way to represent tangent vectors to these manifolds as elements of mathbb Rd where d is the dimension of the manifold We call this an “intrinsic” representation as opposed to “extrinsic” representations as elements of mathbb Rw 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 quasiNewton method applied to minimizing the Brockett cost functionThis 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 T017313
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