A Riemannian ADMM
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866929604147544064 |
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| author | Li, Jiaxiang Ma, Shiqian Srivastava, Tejes |
| author_facet | Li, Jiaxiang Ma, Shiqian Srivastava, Tejes |
| contents | 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 the 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 $ε$-stationary point is analyzed under mild assumptions. Existing ADMM for solving nonconvex problems either does not allow nonconvex constraint set, or does not allow 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_02163 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Riemannian ADMM Li, Jiaxiang Ma, Shiqian Srivastava, Tejes Optimization and Control Machine Learning 90C30 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 the 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 $ε$-stationary point is analyzed under mild assumptions. Existing ADMM for solving nonconvex problems either does not allow nonconvex constraint set, or does not allow 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. |
| title | A Riemannian ADMM |
| topic | Optimization and Control Machine Learning 90C30 |
| url | https://arxiv.org/abs/2211.02163 |