论文标题
使用$ \ ell_1 $ norm的稀疏正规化参数选择
Parameter Choices for Sparse Regularization with the $\ell_1$ Norm
论文作者
论文摘要
我们考虑一个正规化问题，其目标函数由凸保真项组成，以及由$ \ ell_1 $ norms确定的正则化项，该术语由线性变换组成。经验结果表明，使用$ \ ell_1 $ norm的正则化可以促进正规解决方案的稀疏性。本文的目的是从理论上理解正则化参数对正则化解决方案的稀疏性的影响。我们在溶液的变换矩阵下建立了稀疏性的表征。当保真度项具有特殊的结构，并且转换矩阵与身份矩阵重合时，可以将结果表征视为正则化参数选择策略，正则化问题具有具有一定级别的稀疏性的解决方案。我们研究正则化参数的选择，以便调整术语减轻不良性并促进所得正则化解决方案的稀疏性。数值实验表明，正规化参数的选择可以平衡正则化问题解决方案的稀疏性及其与保真度函数最小化器的近似值。
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.
