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Program."],"published-print":{"date-parts":[[2023,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study projection-free methods for constrained Riemannian optimization. In particular, we propose a Riemannian Frank-Wolfe (<jats:sc>RFW<\/jats:sc>) method that handles constraints directly, in contrast to prior methods that rely on (potentially costly) projections. We analyze non-asymptotic convergence rates of <jats:sc>RFW<\/jats:sc> to an optimum for geodesically convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which <jats:sc>RFW<\/jats:sc> can attain a linear convergence rate. As a concrete example, we specialize <jats:sc>RFW<\/jats:sc> to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian \u201clinear\u201d oracle required by <jats:sc>RFW<\/jats:sc> admits a closed form solution; this result may be of independent interest. We complement our theoretical results with an empirical comparison of <jats:sc>RFW<\/jats:sc> against state-of-the-art Riemannian optimization methods, and observe that <jats:sc>RFW<\/jats:sc> performs competitively on the task of computing Riemannian centroids.<\/jats:p>","DOI":"10.1007\/s10107-022-01840-5","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T16:03:02Z","timestamp":1657814582000},"page":"525-556","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Riemannian Optimization via Frank-Wolfe Methods"],"prefix":"10.1007","volume":"199","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1104-7181","authenticated-orcid":false,"given":"Melanie","family":"Weber","sequence":"first","affiliation":[]},{"given":"Suvrit","family":"Sra","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,7,14]]},"reference":[{"key":"1840_CR1","volume-title":"Optimization algorithms on matrix manifolds","author":"PA Absil","year":"2009","unstructured":"Absil, P.A., Mahony, R., Sepulchre, R.: Optimization algorithms on matrix manifolds. 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