数学
质心
数学优化
正多边形
收敛速度
趋同(经济学)
应用数学
算法
计算机科学
几何学
钥匙(锁)
计算机安全
经济增长
经济
作者
Melanie Weber,Suvrit Sra
标识
DOI:10.1007/s10107-022-01840-5
摘要
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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