  We study the convergence properties of Riemannian gradient method for solving
the consensus problem (for an undirected connected graph) over the Stiefel
manifold. The Stiefel manifold is a non-convex set and the standard notion of
averaging in the Euclidean space does not work for this problem. We propose
Distributed Riemannian Consensus on Stiefel Manifold (DRCS) and prove that it
enjoys a local linear convergence rate to global consensus. More importantly,
this local rate asymptotically scales with the second largest singular value of
the communication matrix, which is on par with the well-known rate in the
Euclidean space. To the best of our knowledge, this is the first work showing
the equality of the two rates. The main technical challenges include (i)
developing a Riemannian restricted secant inequality for convergence analysis,
and (ii) to identify the conditions (e.g., suitable step-size and
initialization) under which the algorithm always stays in the local region.
