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An inexact Riemannian proximal gradient method

Author

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  • Wen Huang

    (Xiamen University)

  • Ke Wei

    (Fudan University)

Abstract

This paper considers the problem of minimizing the summation of a differentiable function and a nonsmooth function on a Riemannian manifold. In recent years, proximal gradient method and its variants have been generalized to the Riemannian setting for solving such problems. Different approaches to generalize the proximal mapping to the Riemannian setting lead different versions of Riemannian proximal gradient methods. However, their convergence analyses all rely on solving their Riemannian proximal mapping exactly, which is either too expensive or impracticable. In this paper, we study the convergence of an inexact Riemannian proximal gradient method. It is proven that if the proximal mapping is solved sufficiently accurately, then the global convergence and local convergence rate based on the Riemannian Kurdyka–Łojasiewicz property can be guaranteed. Moreover, practical conditions on the accuracy for solving the Riemannian proximal mapping are provided. As a byproduct, the proximal gradient method on the Stiefel manifold proposed in Chen et al. [SIAM J Optim 30(1):210–239, 2020] can be viewed as the inexact Riemannian proximal gradient method provided the proximal mapping is solved to certain accuracy. Finally, numerical experiments on sparse principal component analysis are conducted to test the proposed practical conditions.

Suggested Citation

  • Wen Huang & Ke Wei, 2023. "An inexact Riemannian proximal gradient method," Computational Optimization and Applications, Springer, vol. 85(1), pages 1-32, May.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:1:d:10.1007_s10589-023-00451-w
    DOI: 10.1007/s10589-023-00451-w
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
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