Coordinate Descent Without Coordinates: Tangent Subspace Descent on Riemannian Manifolds

We extend coordinate descent to manifold domains, and provide convergence analyses for geodesically convex and non-convex smooth objective functions. Our key insight is to draw an analogy between coordinate blocks in Euclidean space and tangent subspaces of a manifold. Hence, our method is called tangent subspace descent (TSD). The core principle behind ensuring convergence of TSD is the appropriate choice of subspace at each iteration. To this end, we propose two novel conditions, the gap ensuring and $C$-randomized norm conditions on deterministic and randomized modes of subspace selection respectively, that promise convergence for smooth functions and that are satisfied in practical contexts. We propose two subspace selection rules of particular practical interest that satisfy these conditions: a deterministic one for the manifold of square orthogonal matrices, and a randomized one for the Stiefel manifold. Our proof-of-concept numerical experiments on the orthogonal Procrustes problem demonstrate TSD's efficacy.