凸稀疏信号重构正则化常数的上界
Upper-Bounding the Regularization Constant for Convex Sparse Signal Reconstruction
摘要：
考虑通过最小化凸可微负对数似然（NLL）（数据保真度）项和凸正则化项的加权和来重建信号$x$，该凸正则化项对$x$施加凸集约束并使用$\ell_1$范数分析正则化来强制其稀疏性。我们计算正则化调整常数的上限，超过该上限，正则化项压倒性地主导NLL项，使目标函数的最小值点的集合不会改变。建立了稀疏信号正则化无关性的充分必要条件和有限上界存在的条件。我们制定了一个优化问题，找到这些界限时，正则项可以全局最小化的可行$x$，也开发了交替方向的乘法器（ADMM）型方法的计算。仿真例子表明，推导的和经验的界限相匹配。
英文摘要：
Consider reconstructing a signal $x$ by minimizing a weighted sum of a convex differentiable negative log-likelihood (NLL) (data-fidelity) term and a convex regularization term that imposes a convex-set constraint on $x$ and enforces its sparsity using $\ell_1$-norm analysis regularization. We compute upper bounds on the regularization tuning constant beyond which the regularization term overwhelmingly dominates the NLL term so that the set of minimum points of the objective function does not change. Necessary and sufficient conditions for irrelevance of sparse signal regularization and a condition for the existence of finite upper bounds are established. We formulate an optimization problem for finding these bounds when the regularization term can be globally minimized by a feasible $x$ and also develop an alternating direction method of multipliers (ADMM) type method for their computation. Simulation examples show that the derived and empirical bounds match.
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