First-order methods on bounded-rank tensors converging to stationary points
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918498705342464 |
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| author | Gao, Bin Peng, Renfeng Yuan, Ya-xiang |
| author_facet | Gao, Bin Peng, Renfeng Yuan, Ya-xiang |
| contents | Provably finding stationary points on bounded-rank tensors turns out to be an open problem [E. Levin, J. Kileel, and N. Boumal, Math. Program., 199 (2023), pp. 831--864] due to the inherent non-smoothness of the set of bounded-rank tensors. In contrast with bounded-rank matrices, tensors where some but not all modes are of full rank render essential difficulties in developing provable first-order methods. We resolve this problem by proposing two first-order methods with guaranteed convergence to stationary points. Specifically, we revisit the variational geometry of bounded-rank tensors and explicitly characterize its normal cones. Moreover, we propose gradient-related approximate projection methods that are provable to find stationary points, where the decisive ingredients are gradient-related vectors from tangent cones, line search along approximate projections, and rank-decreasing mechanisms near rank-deficient points. Numerical experiments on tensor completion validate that the proposed methods converge to stationary points across various rank parameters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_04523 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First-order methods on bounded-rank tensors converging to stationary points Gao, Bin Peng, Renfeng Yuan, Ya-xiang Optimization and Control Numerical Analysis Algebraic Geometry 15A69, 40A05, 65K05, 90C30 Provably finding stationary points on bounded-rank tensors turns out to be an open problem [E. Levin, J. Kileel, and N. Boumal, Math. Program., 199 (2023), pp. 831--864] due to the inherent non-smoothness of the set of bounded-rank tensors. In contrast with bounded-rank matrices, tensors where some but not all modes are of full rank render essential difficulties in developing provable first-order methods. We resolve this problem by proposing two first-order methods with guaranteed convergence to stationary points. Specifically, we revisit the variational geometry of bounded-rank tensors and explicitly characterize its normal cones. Moreover, we propose gradient-related approximate projection methods that are provable to find stationary points, where the decisive ingredients are gradient-related vectors from tangent cones, line search along approximate projections, and rank-decreasing mechanisms near rank-deficient points. Numerical experiments on tensor completion validate that the proposed methods converge to stationary points across various rank parameters. |
| title | First-order methods on bounded-rank tensors converging to stationary points |
| topic | Optimization and Control Numerical Analysis Algebraic Geometry 15A69, 40A05, 65K05, 90C30 |
| url | https://arxiv.org/abs/2503.04523 |