The Intrinsic Riemannian Proximal Gradient Method for Nonconvex Optimization
Provides a new optimization tool for nonconvex problems on Riemannian manifolds, particularly those that are not embedded, which are currently out of reach for other methods.
The authors propose an intrinsic Riemannian proximal gradient method for nonconvex optimization on manifolds, avoiding reliance on embeddings. They prove convergence and demonstrate effectiveness on nonconvex/nonembedded problems where existing methods fail.
We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses proximal maps defined on the manifold and therefore does not require or work in the embedding. We investigate its convergence properties and illustrate its numerical performance, particularly for nonconvex or nonembedded problems that are hence out of reach for other methods.