Riemannian Frank-Wolfe with application to the geometric mean of positive definite matrices

We study projection free methods for constrained geodesically convex optimization. In particular, we propose a Riemannian version of the Frank-Wolfe (RFW) method. We analyze RFWu0027s convergence and provide a global, non-asymptotic sublinear convergence rate. We also present a setting under which RFW can attain a rate. Later, we specialize RFW to the manifold of positive definite matrices, where we are motivated by the specific task of computing the geometric mean (also known as Karcher mean or Riemannian centroid). For this task, RFW requires access to a linear oracle that turns out to be a nonconvex semidefinite program. Remarkably, this nonconvex program is shown to admit a closed form solution, which may be of independent interest too. We complement this result by also studying a nonconvex Euclidean Frank-Wolfe approach, along with its global convergence analysis. Finally, we empirically compare Rfw against recently published methods for the Riemannian centroid and observe strong performance gains.