线性逆问题中带稀疏性约束的迭代阈值算法
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
AI总结：
本文提出一种迭代阈值算法，用于求解带稀疏性约束的线性逆问题，证明其收敛性并展示应用前景。
摘要：
我们考虑线性逆问题，其中解被假定在任意预先指定的标准正交基上具有稀疏展开。我们证明，用系数上的加权l^p惩罚（其中1≤p≤2）替换通常的二次正则化惩罚，仍然能使问题正则化。如果p<2，此类l^p惩罚问题的正则化解在所考虑的基上将具有更稀疏的展开。为了计算相应的正则化解，我们提出了一种迭代算法，该算法相当于在每一步迭代中应用阈值（或非线性收缩）的Landweber迭代。我们证明了该算法在范数意义下收敛。我们还回顾了该方法的一些潜在应用。
英文摘要：
We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.
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