Weakly Convex Optimization over Stiefel Manifold Using Riemannian Subgradient-Type Methods

We consider a class of nonsmooth optimization problems over the Stiefel manifold, in which objective function is weakly convex ambient Euclidean space. Such are ubiquitous engineering applications but still largely unexplored. present family Riemannian subgradient-type methods---namely subgradient, incremental and stochastic subgradient methods---to solve these show that they all have an iteration complexity $\mathcal{O}(\varepsilon^{-4})$ for driving natural stationarity measure below $\varepsilon$. In addition, we establish local linear convergence methods when problem at hand further satisfies sharpness property algorithms properly initialized use geometrically diminishing stepsizes. To best our knowledge, first guarantees using to optimize nonconvex functions manifold. The fundamental ingredient proof aforementioned results new inequality restrictions on could be independent interest. also can extended handle compact embedded submanifolds Finally, discuss properties various formulations robust subspace recovery orthogonal dictionary learning demonstrate performance both via numerical simulations.