A Riemannian Alternating Direction Method of Multipliers

• 成果类型: Article; Early Access
• 署名作者: Li, Jiaxiang; Ma, Shiqian; Srivastava, Tejes
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities; Rice University; University of Chicago
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2023.0068
• 发表日期: 2024
• 关键词: locally lipschitz functions; subgradient algorithm; optimization

摘要

We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function considered in the ambient space. This class of problems finds important applications in machine learning and statistics, such as sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an epsilon-stationary point is analyzed under mild assumptions. Existing ADMMs for solving nonconvex problems either do not allow a nonconvex constraint set or do not allow a nonsmooth objective function. Our algorithm is the first ADMM-type algorithm that minimizes a nonsmooth objective over manifold-a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method.