On the B-subdifferential of proximal operators of affine-constrained $\ell_1$ regularizer
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908580869832704 |
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| author | Li, Xudong Lin, Meixia Toh, Kim-Chuan |
| author_facet | Li, Xudong Lin, Meixia Toh, Kim-Chuan |
| contents | In this work, we study the affine-constrained $\ell_1$ regularizers, which frequently arise in statistical and machine learning problems across a variety of applications, including microbiome compositional data analysis and sparse subspace clustering. With the aim of developing scalable second-order methods for solving optimization problems involving such regularizers, we analyze the associated proximal mapping and characterize its generalized differentiability, with a focus on its B-subdifferential. The revealed structured sparsity in the B-subdifferential enables us to design efficient algorithms within the proximal point framework. Extensive numerical experiments on real applications, including comparisons with state-of-the-art solvers, further demonstrate the superior performance of our approach. Our findings provide new insights into the sensitivity and stability properties of affine-constrained nonsmooth regularizers, and contribute to the development of fast second-order methods for a class of structured, constrained sparse learning problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_06642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the B-subdifferential of proximal operators of affine-constrained $\ell_1$ regularizer Li, Xudong Lin, Meixia Toh, Kim-Chuan Optimization and Control In this work, we study the affine-constrained $\ell_1$ regularizers, which frequently arise in statistical and machine learning problems across a variety of applications, including microbiome compositional data analysis and sparse subspace clustering. With the aim of developing scalable second-order methods for solving optimization problems involving such regularizers, we analyze the associated proximal mapping and characterize its generalized differentiability, with a focus on its B-subdifferential. The revealed structured sparsity in the B-subdifferential enables us to design efficient algorithms within the proximal point framework. Extensive numerical experiments on real applications, including comparisons with state-of-the-art solvers, further demonstrate the superior performance of our approach. Our findings provide new insights into the sensitivity and stability properties of affine-constrained nonsmooth regularizers, and contribute to the development of fast second-order methods for a class of structured, constrained sparse learning problems. |
| title | On the B-subdifferential of proximal operators of affine-constrained $\ell_1$ regularizer |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2510.06642 |