  In this paper, we focus on the decentralized optimization problem over the
Stiefel manifold, which is defined on a connected network of $d$ agents. The
objective is an average of $d$ local functions, and each function is privately
held by an agent and encodes its data. The agents can only communicate with
their neighbors in a collaborative effort to solve this problem. In existing
methods, multiple rounds of communications are required to guarantee the
convergence, giving rise to high communication costs. In contrast, this paper
proposes a decentralized algorithm, called DESTINY, which only invokes a single
round of communications per iteration. DESTINY combines gradient tracking
techniques with a novel approximate augmented Lagrangian function. The global
convergence to stationary points is rigorously established. Comprehensive
numerical experiments demonstrate that DESTINY has a strong potential to
deliver a cutting-edge performance in solving a variety of testing problems.
