arXiv subjects

Qianru Liu

Publications and source records attributed to Qianru Liu.

2 recordsLinked to original sources

arXiv2026

Sparsity-Guided Multi-Parameter Selection in $\ell_1$-Regularized Models via a Fixed-Point Proximity Approach

We study a regularization framework that combines a convex fidelity term with multiple $\ell_1$-based regularizers, each linked to a distinct linear transform. This multi-penalty model enhances flexibility in promoting structured sparsity. We analyze how the choice of regularization parameters governs the sparsity of solutions under the given transforms and derive a precise relationship between the parameters and resulting sparsity patterns. This insight enables the development of an iterative strategy for selecting parameters to achieve prescribed sparsity levels. A key computational challenge arises in practice: effective parameter tuning requires simultaneous access to the regularized solution and two auxiliary vectors derived from the sparsity analysis. To address this, we propose a fixed-point proximity algorithm that jointly computes all three vectors. Together with our theoretical characterization, this algorithm forms the basis of a practical multi-parameter selection scheme. Numerical experiments demonstrate that the proposed method reliably produces solutions with desired sparsity patterns and strong approximation accuracy.

math.NA↗

arXiv2022

Parameter Choices for Sparse Regularization with the $\ell_1$ Norm

We consider a regularization problem whose objective function consists of a convex fidelity term and a regularization term determined by the $\ell_1$ norm composed with a linear transform. Empirical results show that the regularization with the $\ell_1$ norm can promote sparsity of a regularized solution. It is the goal of this paper to understand theoretically the effect of the regularization parameter on the sparsity of the regularized solutions. We establish a characterization of the sparsity under the transform matrix of the solution. When the fidelity term has a special structure and the transform matrix coincides with a identity matrix, the resulting characterization can be taken as a regularization parameter choice strategy with which the regularization problem has a solution having a sparsity of a certain level. We study choices of the regularization parameter so that the regularization term alleviates the ill-posedness and promote sparsity of the resulting regularized solution. Numerical experiments demonstrate that choices of the regularization parameters can balance the sparsity of the solutions of the regularization problem and its approximation to the minimizer of the fidelity function.

math.NA↗