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Riemannian optimization on unit sphere with p-norm and its applications

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  • Hiroyuki Sato

    (Kyoto University)

Abstract

This study deals with Riemannian optimization on the unit sphere in terms of p-norm with general $$p> 1$$ p > 1 . As a Riemannian submanifold of the Euclidean space, the geometry of the sphere with p-norm is investigated, and several geometric tools used for Riemannian optimization, such as retractions and vector transports, are proposed and analyzed. Applications to Riemannian optimization on the sphere with nonnegative constraints and $$\textit{L}_{\textit{p}}$$ L p -regularization-related optimization are also discussed. As practical examples, the former includes nonnegative principal component analysis, and the latter is closely related to the Lasso regression and box-constrained problems. Numerical experiments verify that Riemannian optimization on the sphere with p-norm has substantial potential for such applications, and the proposed framework provides a theoretical basis for such optimization.

Suggested Citation

  • Hiroyuki Sato, 2023. "Riemannian optimization on unit sphere with p-norm and its applications," Computational Optimization and Applications, Springer, vol. 85(3), pages 897-935, July.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:3:d:10.1007_s10589-023-00477-0
    DOI: 10.1007/s10589-023-00477-0
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    References listed on IDEAS

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    1. Sakai, Hiroyuki & Sato, Hiroyuki & Iiduka, Hideaki, 2023. "Global convergence of Hager–Zhang type Riemannian conjugate gradient method," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    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. Xiaojing Zhu & Hiroyuki Sato, 2020. "Riemannian conjugate gradient methods with inverse retraction," Computational Optimization and Applications, Springer, vol. 77(3), pages 779-810, December.
    4. 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.
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