Beyond $R$-Barycenters: An Effective Averaging Method on Stiefel and Grassmann Manifolds

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.