A New Inexact Manifold Proximal Linear Algorithm with Adaptive Stopping Criteria

This paper proposes a new inexact manifold proximal linear (IManPL) algorithm for solving nonsmooth, nonconvex composite optimization problems over an embedded submanifold. At each iteration, IManPL solves a convex subproblem inexactly, guided by two adaptive stopping criteria. We establish convergence guarantees and show that IManPL achieves the best first-order oracle complexity for solving this class of problems. Numerical experiments on sparse spectral clustering and sparse principal component analysis demonstrate that our methods outperform existing approaches.