The Intrinsic Riemannian Proximal Gradient Method for Nonconvex Optimization
arXiv:2506.09775
The paper proposes an intrinsic Riemannian proximal gradient algorithm that operates directly on manifolds without requiring an embedding, and analyzes its convergence for possibly non‑geodesically convex functions, demonstrating its performance on nonconvex problems.
Abstract
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.