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

翻译：块主化最小化是一种简单的非凸优化迭代算法，该方法在固定其他块坐标的同时，依次最小化每个块坐标的目标函数的替代函数。我们考虑一类用于最小化光滑非凸目标的BMM算法，其中每个参数块被约束在黎曼流形的子集内。我们证明该算法渐近收敛于驻点集，并在$\widetilde{O}(\epsilon^{-2})$次迭代内达到$\epsilon$-驻点。特别地，当底层流形为欧几里得或Stiefel流形的乘积时，我们的复杂度结果完全基于欧几里得假设，尽管分析明确利用了黎曼几何。我们的通用分析适用于多种带有黎曼约束的算法：黎曼MM、块投影梯度下降、乐观似然估计、测地线约束子空间跟踪、鲁棒PCA以及黎曼CP字典学习。实验验证表明，我们的算法比应用于黎曼环境的经典欧几里得算法收敛速度更快。