On the B-subdifferential of proximal operators of affine-constrained $\ell_1$ regularizer

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.