Support matrix machine: exploring sample sparsity, low rank, and adaptive sieving in high-performance computing
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| Format: | Preprint |
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2024
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| author | Wu, Can Li, Dong-Hui Sun, Defeng |
| author_facet | Wu, Can Li, Dong-Hui Sun, Defeng |
| contents | Support matrix machine (SMM) is a successful supervised classification model for matrix-type samples. Unlike support vector machines, it employs low-rank regularization on the regression matrix to effectively capture the intrinsic structure embedded in each input matrix. When solving a large-scale SMM, a major challenge arises from the potential increase in sample size, leading to substantial computational and storage burdens. To address these issues, we design a semismooth Newton-CG (SNCG) based augmented Lagrangian method (ALM) for solving the SMM. The ALM exhibits an asymptotic R-superlinear convergence if a strict complementarity condition is satisfied. The SNCG method is employed to solve the ALM subproblems, achieving at least a superlinear convergence rate under the nonemptiness of an index set. Furthermore, the sparsity of samples and the low-rank nature of solutions enable us to reduce the computational cost and storage demands for the Newton linear systems. Additionally, we develop an adaptive sieving strategy that generates a solution path for the SMM by exploiting sample sparsity. The finite convergence of this strategy is also demonstrated. Numerical experiments on both large-scale real and synthetic datasets validate the effectiveness of the proposed methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_08023 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Support matrix machine: exploring sample sparsity, low rank, and adaptive sieving in high-performance computing Wu, Can Li, Dong-Hui Sun, Defeng Optimization and Control Support matrix machine (SMM) is a successful supervised classification model for matrix-type samples. Unlike support vector machines, it employs low-rank regularization on the regression matrix to effectively capture the intrinsic structure embedded in each input matrix. When solving a large-scale SMM, a major challenge arises from the potential increase in sample size, leading to substantial computational and storage burdens. To address these issues, we design a semismooth Newton-CG (SNCG) based augmented Lagrangian method (ALM) for solving the SMM. The ALM exhibits an asymptotic R-superlinear convergence if a strict complementarity condition is satisfied. The SNCG method is employed to solve the ALM subproblems, achieving at least a superlinear convergence rate under the nonemptiness of an index set. Furthermore, the sparsity of samples and the low-rank nature of solutions enable us to reduce the computational cost and storage demands for the Newton linear systems. Additionally, we develop an adaptive sieving strategy that generates a solution path for the SMM by exploiting sample sparsity. The finite convergence of this strategy is also demonstrated. Numerical experiments on both large-scale real and synthetic datasets validate the effectiveness of the proposed methods. |
| title | Support matrix machine: exploring sample sparsity, low rank, and adaptive sieving in high-performance computing |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2412.08023 |