An inexact Riemannian proximal gradient method

Wen Huang () and Ke Wei ()
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Wen Huang: Xiamen University
Ke Wei: Fudan University

Computational Optimization and Applications, 2023, vol. 85, issue 1, No 1, 32 pages

Abstract: 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.

Keywords: Riemannian optimization; Riemannian proximal gradient; Sparse PCA (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-023-00451-w

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