Nonsmooth Optimization over the Stiefel Manifold and Beyond: Proximal Gradient Method and Recent Variants
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AbstractWe consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this class of problems converge slowly in practice, involve subproblems that can be as difficult as the original problem, or lack rigorous convergence guarantees. In this paper, we propose a manifold proximal gradient method (ManPG) for solving this class of problems. We prove that the proposed method converges globally to a stationary point and establish its iteration complexity for obtaining an ��
-stationary point. Furthermore, we present numerical results on the sparse PCA and compressed modes problems to demonstrate the advantages of the proposed method. We also discuss some recent advances related to ManPG for Riemannian optimization with nonsmooth objective functions.
-stationary point. Furthermore, we present numerical results on the sparse PCA and compressed modes problems to demonstrate the advantages of the proposed method. We also discuss some recent advances related to ManPG for Riemannian optimization with nonsmooth objective functions.
All Author(s) ListShixiang Chen, Shiqian Ma, Anthony Man-Cho So, Tong Zhang
Journal nameSIAM Review
Year2024
Month5
Volume Number66
Issue Number2
PublisherSociety for Industrial and Applied Mathematics
Pages319 - 352
ISSN0036-1445
eISSN1095-7200
LanguagesEnglish-United States
