Optimization on product manifolds under a preconditioned metric
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917926234226688 |
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| author | Gao, Bin Peng, Renfeng Yuan, Ya-xiang |
| author_facet | Gao, Bin Peng, Renfeng Yuan, Ya-xiang |
| contents | Since optimization on Riemannian manifolds relies on the chosen metric, it is appealing to know that how the performance of a Riemannian optimization method varies with different metrics and how to exquisitely construct a metric such that a method can be accelerated. To this end, we propose a general framework for optimization problems on product manifolds endowed with a preconditioned metric, and we develop Riemannian methods under this metric. Generally, the metric is constructed by an operator that aims to approximate the diagonal blocks of the Riemannian Hessian of the cost function. We propose three specific approaches to design the operator: exact block diagonal preconditioning, left and right preconditioning, and Gauss--Newton type preconditioning. Specifically, we tailor new preconditioned metrics and adapt the proposed Riemannian methods to the canonical correlation analysis and the truncated singular value decomposition problems, which provably accelerate the Riemannian methods. Additionally, we adopt the Gauss--Newton type preconditioning to solve the tensor ring completion problem. Numerical results among these applications verify that a delicate metric does accelerate the Riemannian optimization methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08873 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimization on product manifolds under a preconditioned metric Gao, Bin Peng, Renfeng Yuan, Ya-xiang Optimization and Control Numerical Analysis 15A69, 65K05, 65F30, 90C30 Since optimization on Riemannian manifolds relies on the chosen metric, it is appealing to know that how the performance of a Riemannian optimization method varies with different metrics and how to exquisitely construct a metric such that a method can be accelerated. To this end, we propose a general framework for optimization problems on product manifolds endowed with a preconditioned metric, and we develop Riemannian methods under this metric. Generally, the metric is constructed by an operator that aims to approximate the diagonal blocks of the Riemannian Hessian of the cost function. We propose three specific approaches to design the operator: exact block diagonal preconditioning, left and right preconditioning, and Gauss--Newton type preconditioning. Specifically, we tailor new preconditioned metrics and adapt the proposed Riemannian methods to the canonical correlation analysis and the truncated singular value decomposition problems, which provably accelerate the Riemannian methods. Additionally, we adopt the Gauss--Newton type preconditioning to solve the tensor ring completion problem. Numerical results among these applications verify that a delicate metric does accelerate the Riemannian optimization methods. |
| title | Optimization on product manifolds under a preconditioned metric |
| topic | Optimization and Control Numerical Analysis 15A69, 65K05, 65F30, 90C30 |
| url | https://arxiv.org/abs/2306.08873 |