Riemannian Optimization via Frank-Wolfe Methods

• DOI: 10.1007/s10107-022-01840-5
• Publication Date: 2022-07-14T16:03:02Z
• Authors: Melanie Weber; Suvrit Sra
• Keywords: Centroid

Abstract We study projection-free methods for constrained Riemannian optimization. In particular, we propose a Frank-Wolfe ( RFW ) method that handles constraints directly, in contrast to prior rely on (potentially costly) projections. analyze non-asymptotic convergence rates of an optimum geodesically convex problems, and critical point nonconvex objectives. also present practical setting under which can attain linear rate. As concrete example, specialize the manifold positive definite matrices apply it two tasks: (i) computing matrix geometric mean (Riemannian centroid); (ii) Bures-Wasserstein barycenter. Both tasks involve interval constraints, show “linear” oracle required by admits closed form solution; this result may be independent interest. complement our theoretical results with empirical comparison against state-of-the-art optimization methods, observe performs competitively task centroids.