A New Inexact Manifold Proximal Linear Algorithm with Adaptive Stopping Criteria

TL;DR

This paper introduces an inexact manifold proximal linear algorithm with adaptive stopping criteria for nonsmooth, nonconvex optimization on manifolds, achieving optimal complexity and outperforming existing methods in experiments.

Contribution

The paper presents a novel inexact manifold proximal linear algorithm with adaptive stopping, providing convergence guarantees and optimal complexity for nonsmooth nonconvex problems.

Findings

Achieves the best first-order oracle complexity.

Outperforms existing methods in spectral clustering and PCA.

Provides convergence guarantees for the proposed algorithm.

Abstract

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.