A randomized submanifold descent on the Stiefel manifold reduces retraction cost to O(r^3) and achieves O(n^2 r^{-2} / k) expected convergence for smooth nonconvex functions.
Optimization algorithms on matrix manifolds
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Efficient Optimization with Orthogonality Constraint: a Randomized Riemannian Submanifold Method
A randomized submanifold descent on the Stiefel manifold reduces retraction cost to O(r^3) and achieves O(n^2 r^{-2} / k) expected convergence for smooth nonconvex functions.