We study a 1-dimensional discrete signal denoising problem that consists of minimizing a sum of separable convex fidelity terms and convex regularization terms, the latter penalize the differences of adjacent signal values. This problem generalizes the total variation regularization problem. We provide here a unified approach to solve the problem for general convex fidelity and regularization functions that is based on the Karush–Kuhn–Tucker optimality conditions. This approach is shown here to lead to a fast algorithm for the problem with general convex fidelity and regularization functions, and a faster algorithm if, in addition, the fidelity functions are differentiable and the regularization functions are strictly convex. Both algorithms achieve the best theoretical worst case complexity over existing algorithms for the classes of objective functions studied here. Also in practice, our C++ implementation of the method is considerably faster than popular C++ nonlinear optimization solvers for the problem.

我们研究了一维离散信号降噪问题，该问题包括最小化可分离凸保真度项和凸正则化项之和，后者会惩罚相邻信号值的差异。该问题概括了总变化正则化问题。我们在此处提供了一种统一的方法，该方法基于Karush–Kuhn–Tucker最优性条件来解决一般凸保真度和正则化函数的问题。此处显示的这种方法可导致针对具有一般凸保真度和正则化功能的问题的快速算法，以及如果保真度函数是可微的且正则化功能严格为凸的则可提供更快的算法。对于本文研究的目标函数，这两种算法都比现有算法获得了最佳的理论上最坏情况下的复杂度。同样在实践中，对于该问题，我们的方法的C ++实现比流行的C ++非线性优化求解器快得多。