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This paper focuses on minimizing a smooth function combined with a nonsmoothregularization term on a compact Riemannian submanifold embedded in theEuclidean space under a decentralized setting. Typically, there are two typesof approaches at present for tackling such composite optimization problems. Thefirst, subgradient-based approaches, rely on subgradient information of theobjective function to update variables, achieving an iteration complexity of$\mathcal{O}(\epsilon^{-4}\log^2(\epsilon^{-2}))$. The second, smoothingapproaches, involve constructing a smooth approximation of the nonsmoothregularization term, resulting in an iteration complexity of$\mathcal{O}(\epsilon^{-4})$. This paper proposes a proximal gradient typealgorithm that fully exploits the composite structure. The global convergenceto a stationary point is established with a significantly improved iterationcomplexity of $\mathcal{O}(\epsilon^{-2})$. To validate the effectiveness andefficiency of our proposed method, we present numerical results in real-worldapplications, showcasing its superior performance.

A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifolds