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A Riemannian conjugate gradient method for optimization on the Stiefel manifold

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  • Xiaojing Zhu

    (Shanghai University of Electric Power)

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.

Suggested Citation

  • Xiaojing Zhu, 2017. "A Riemannian conjugate gradient method for optimization on the Stiefel manifold," Computational Optimization and Applications, Springer, vol. 67(1), pages 73-110, May.
    • Handle: RePEc:spr:coopap:v:67:y:2017:i:1:d:10.1007_s10589-016-9883-4
    DOI: 10.1007/s10589-016-9883-4
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    References listed on IDEAS

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    1. Hiroyuki Sato, 2016. "A Dai–Yuan-type Riemannian conjugate gradient method with the weak Wolfe conditions," Computational Optimization and Applications, Springer, vol. 64(1), pages 101-118, May.
    2. Igor Grubisic & Raoul Pietersz, 2005. "Efficient Rank Reduction of Correlation Matrices," Finance 0502007, University Library of Munich, Germany.
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    Cited by:

    1. Harry Oviedo, 2023. "Proximal Point Algorithm with Euclidean Distance on the Stiefel Manifold," Mathematics, MDPI, vol. 11(11), pages 1-17, May.
    2. Lei Wang & Xin Liu & Yin Zhang, 2023. "A communication-efficient and privacy-aware distributed algorithm for sparse PCA," Computational Optimization and Applications, Springer, vol. 85(3), pages 1033-1072, July.
    3. Hiroyuki Sakai & Hideaki Iiduka, 2021. "Sufficient Descent Riemannian Conjugate Gradient Methods," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 130-150, July.
    4. Ke Wang & Zhuo Chen & Shihui Ying & Xinjian Xu, 2023. "Low-Rank Matrix Completion via QR-Based Retraction on Manifolds," Mathematics, MDPI, vol. 11(5), pages 1-17, February.

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