In this work we consider numerical efficiency and convergence rates for solvers of non-convex multi-penalty formulations when reconstructing sparse signals from noisy linear measurements. We extend an existing approach, based on reduction to an augmented single-penalty formulation, to the non-convex setting and discuss its computational intractability in large-scale applications. To circumvent this limitation, we propose an alternative single-penalty reduction based on infimal convolution that shares the benefits of the augmented approach but is computationally less dependent on the problem size. We provide linear convergence rates for both approaches, and their dependence on design parameters. Numerical experiments substantiate our theoretical findings.

在这项工作中，我们在从嘈杂的线性测量重建稀疏信号时考虑了非凸多重惩罚公式的求解器的数值效率和收敛速度。我们将基于减少到增强单惩罚公式的现有方法扩展到非凸设置，并讨论其在大规模应用中的计算难度。为了规避这个限制，我们提出了一种基于 infimal 卷积的替代单惩罚减少，它分享了增强方法的好处，但在计算上较少依赖于问题的大小。我们为这两种方法提供线性收敛率，以及它们对设计参数的依赖。数值实验证实了我们的理论发现。