Riemannian Optimization via Frank-Wolfe Methods
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10107_2022_Article_1840.pdf
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1.14 MB
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Author(s)
Weber, Melanie
Sra, Suvrit
Date Issued
July 2022
Journal
Mathematical Programming
Publisher
Springer Science and Business Media LLC
Citation
Weber, Melanie and Sra, Suvrit. 2022. "Riemannian Optimization via Frank-Wolfe Methods."
Version
Final published version
Abstract
Abstract
We study projection-free methods for constrained Riemannian optimization. In particular, we propose a Riemannian Frank-Wolfe (RFW) method that handles constraints directly, in contrast to prior methods that rely on (potentially costly) projections. We analyze non-asymptotic convergence rates of RFW to an optimum for geodesically convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize RFW 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 “linear” oracle required by RFW admits a closed form solution; this result may be of independent interest. We complement our theoretical results with an empirical comparison of RFW against state-of-the-art Riemannian optimization methods, and observe that RFW performs competitively on the task of computing Riemannian centroids.
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s10107-022-01840-5
