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Block majorization-minimization (BMM) is a simple iterative algorithm fornonconvex optimization that sequentially minimizes a majorizing surrogate ofthe objective function in each block coordinate while the other blockcoordinates are held fixed. We consider a family of BMM algorithms forminimizing smooth nonconvex objectives, where each parameter block isconstrained within a subset of a Riemannian manifold. We establish that thisalgorithm converges asymptotically to the set of stationary points, and attainsan $\epsilon$-stationary point within $\widetilde{O}(\epsilon^{-2})$iterations. In particular, the assumptions for our complexity results arecompletely Euclidean when the underlying manifold is a product of Euclidean orStiefel manifolds, although our analysis makes explicit use of the Riemanniangeometry. Our general analysis applies to a wide range of algorithms withRiemannian constraints: Riemannian MM, block projected gradient descent,optimistic likelihood estimation, geodesically constrained subspace tracking,robust PCA, and Riemannian CP-dictionary-learning. We experimentally validatethat our algorithm converges faster than standard Euclidean algorithms appliedto the Riemannian setting.

Convergence and complexity of block majorization-minimization for constrained block-Riemannian optimization