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Riemannian Optimization via Frank-Wolfe Methods
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10107_2022_Article_1840.pdf
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1.14 MB
Format
Adobe PDF
Checksum (MD5)
e18c00a4bb8d85c4e4520d23cfe757eb
Author(s)
Weber, Melanie
Sra, Suvrit
Date Issued
July 14, 2022
Publisher
Springer Berlin Heidelberg
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.
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DOI of Published Version
https://doi.org/10.1007/s10107-022-01840-5
