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A Riemannian conjugate gradient method for optimization on the Stiefel manifold

Xiaojing Zhu (042563zxj***at***163.com)

Abstract: In this paper we propose a new Riemannian conjugate gradient method for optimization on the Stiefel manifold. We introduce two novel vector transports associated with the retraction constructed by the Cayley transform. Both of them satisfy the Ring-Wirth nonexpansive condition, which is fundamental for convergence analysis of Riemannian conjugate gradient methods, and one of them is also isometric. It is known that the Ring- Wirth nonexpansive condition does not hold for traditional vector transports as the differentiated retractions of QR and polar decompositions. Practical formulae of the new vector transports for low-rank matrices are obtained. Dai's nonmonotone conjugate gradient method is generalized to the Riemannian case and global convergence of the new algorithm is established under standard assumptions. Numerical results on a variety of low-rank test problems demonstrate the effectiveness of the new method.

Keywords: Riemannian optimization, Stiefel manifold, conjugate gradient method, retraction, vector transport, Cayley transform

Category 1: Nonlinear Optimization

Category 2: Applications -- Science and Engineering

Category 3: Linear, Cone and Semidefinite Programming

Citation: Tech.report SHIEP-SLXY-2016.09.04, Shanghai University of Electric Power, Sep 2016

Download: [Compressed Postscript][PDF]

Entry Submitted: 09/04/2016
Entry Accepted: 09/04/2016
Entry Last Modified: 11/20/2016

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