{"ID":2858564,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.06642","arxiv_id":"2510.06642","title":"On the B-subdifferential of proximal operators of affine-constrained $\\ell_1$ regularizer","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2510.06642","url_pdf":"https://arxiv.org/pdf/2510.06642v1","authors":"[\"Xudong Li\",\"Meixia Lin\",\"Kim-Chuan Toh\"]","published":"2025-10-08T04:49:10Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}