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Practical gradient and conjugate gradient methods on flag manifolds

Author

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  • Xiaojing Zhu

    (Shanghai University of Electric Power)

  • Chungen Shen

    (University of Shanghai for Science and Technology)

Abstract

Flag manifolds, sets of nested sequences of linear subspaces with fixed dimensions, are rising in numerical analysis and statistics. The current optimization algorithms on flag manifolds are based on the exponential map and parallel transport, which are expensive to compute. In this paper we propose practical optimization methods on flag manifolds without the exponential map and parallel transport. Observing that flag manifolds have a similar homogeneous structure with Grassmann and Stiefel manifolds, we generalize some typical retractions and vector transports to flag manifolds, including the Cayley-type retraction and vector transport, the QR-based and polar-based retractions, the projection-type vector transport and the projection of the differentiated polar-based retraction as a vector transport. Theoretical properties and efficient implementations of the proposed retractions and vector transports are discussed. Then we establish Riemannian gradient and Riemannian conjugate gradient algorithms based on these retractions and vector transports. Numerical results on the problem of nonlinear eigenflags demonstrate that our algorithms have a great advantage in efficiency over the existing ones.

Suggested Citation

  • Xiaojing Zhu & Chungen Shen, 2024. "Practical gradient and conjugate gradient methods on flag manifolds," Computational Optimization and Applications, Springer, vol. 88(2), pages 491-524, June.
    • Handle: RePEc:spr:coopap:v:88:y:2024:i:2:d:10.1007_s10589-024-00568-6
    DOI: 10.1007/s10589-024-00568-6
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. P.-A. Absil & Luca Amodei & Gilles Meyer, 2014. "Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries," Computational Statistics, Springer, vol. 29(3), pages 569-590, June.
    4. Du Nguyen, 2022. "Closed-form Geodesics and Optimization for Riemannian Logarithms of Stiefel and Flag Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 142-166, July.
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