We consider a general ill-posed inverse problem in a Hilbert space setting by minimizing a misfit functional coupling with a multi-penalty regularization for stabilization. For solving this minimization problem, we investigate two proximal algorithms with stepsize controls: a proximal fixed point algorithm and an alternating proximal algorithm. We prove the decrease of the objective functional and the convergence of both update schemes to a stationary point under mild conditions on the stepsizes. These algorithms are then applied to the sparse and non-negative matrix factorization problems. Based on a priori information of non-negativity and sparsity of the exact solution, the problem is regularized by corresponding terms. In both cases, the implementation of our proposed algorithms is straight-forward since the evaluation of the proximal operators in these problems can be done explicitly. Finally, we test the proposed algorithms for the non-negative sparse matrix factorization problem with both simulated and real-world data and discuss reconstruction performance, convergence, as well as achieved sparsity.

我们通过将不匹配的函数耦合与用于稳定的多罚正则化最小化来考虑希尔伯特空间设置中的一般不适定逆问题。为了解决此最小化问题，我们研究了两种带有逐步控制的近端算法：近端固定点算法和交替近端算法。我们证明了在逐步调整的条件下，目标函数的减少以及两种更新方案在收敛条件下的收敛性都达到了固定点。然后将这些算法应用于稀疏和非负矩阵分解问题。根据精确解的非负性和稀疏性的先验信息，可以通过相应的条件对问题进行正则化。在这两种情况下 由于可以明确完成对这些问题中近端算子的评估，因此我们提出的算法的实现非常简单。最后，我们使用模拟数据和实际数据测试针对非负稀疏矩阵分解问题的拟议算法，并讨论重建性能，收敛性和实现的稀疏性。