Frank-Wolfe methods for geodesically convex optimization with application to the matrix geometric mean

We consider optimization of geodesically convex objectives over geodesically convex subsets of the manifold of positive definite matrices. In particular, for this task we develop Euclidean and Riemannian Frank-Wolfe (FW) algorithms. For both settings we analyze non-asymptotic convergence rates to global optimality. To our knowledge, these are the first results on Riemannian FW and its convergence. We specialize our algorithms for the task of computing the matrix geometric mean, i.e., the Riemannian centroid of a set of positive definite matrices. For this problem, we provide concrete, closed-form realizations of the crucial “linear oracle” required by FW that may be of independent interest. Moreover, under an additional hypothesis we prove how Riemannian FW can even attain a linear rate of convergence. Experiments against recently published methods for the matrix geometric mean substantiate the competitiveness of the proposed FW algorithms.