Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds
This provides a more efficient averaging method for applications involving manifold data, though it is incremental as it builds on prior R-barycenters.
The paper tackles the problem of averaging data on Stiefel and Grassmann manifolds by proposing RL-barycenters, which simplify to arithmetic means projected onto the manifold, making them computationally cheaper and competitive with existing methods.
In this paper, the issue of averaging data on a manifold is addressed. While the Fréchet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this, R-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However, R-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call RL-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.