On the Local Linear Rate of Consensus on the Stiefel Manifold
This work addresses distributed optimization on manifolds for applications like robotics and machine learning, providing a novel theoretical guarantee that bridges Euclidean and Riemannian settings, though it is incremental in extending known results to a specific manifold.
The paper tackles the consensus problem on the non-convex Stiefel manifold by proposing the Distributed Riemannian Consensus on Stiefel Manifold (DRCS) algorithm, proving it achieves local linear convergence with a rate that asymptotically matches the Euclidean case, scaling with the second largest singular value of the communication matrix.
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.