  This paper focuses on minimizing a smooth function combined with a nonsmooth
regularization term on a compact Riemannian submanifold embedded in the
Euclidean space under a decentralized setting. Typically, there are two types
of approaches at present for tackling such composite optimization problems. The
first, subgradient-based approaches, rely on subgradient information of the
objective function to update variables, achieving an iteration complexity of
$\mathcal{O}(\epsilon^{-4}\log^2(\epsilon^{-2}))$. The second, smoothing
approaches, involve constructing a smooth approximation of the nonsmooth
regularization term, resulting in an iteration complexity of
$\mathcal{O}(\epsilon^{-4})$. This paper proposes a proximal gradient type
algorithm that fully exploits the composite structure. The global convergence
to a stationary point is established with a significantly improved iteration
complexity of $\mathcal{O}(\epsilon^{-2})$. To validate the effectiveness and
efficiency of our proposed method, we present numerical results in real-world
applications, showcasing its superior performance.
