  We study decentralized smooth optimization problems over compact
submanifolds. Recasting it as a composite optimization problem, we propose a
decentralized Douglas-Rachford splitting algorithm, DDRS. When the proximal
operator of the local loss function does not have a closed-form solution, an
inexact version of DDRS, iDDRS, is also presented. Both algorithms rely on an
ingenious integration of the nonconvex Douglas-Rachford splitting algorithm
with gradient tracking and manifold optimization. We show that our DDRS and
iDDRS achieve the best-known convergence rate of $\mathcal{O}(1/K)$. The main
challenge in the proof is how to handle the nonconvexity of the manifold
constraint. To address this issue, we utilize the concept of proximal
smoothness for compact submanifolds. This ensures that the projection onto the
submanifold exhibits convexity-like properties, which allows us to control the
consensus error across agents. Numerical experiments on the principal component
analysis are conducted to demonstrate the effectiveness of our decentralized
DRS compared with the state-of-the-art ones.
