用于去偏信号恢复的基于上层图松弛的稀疏增强多层非凸正则化
Sparsity-Enhanced Multilayered Non-Convex Regularization with Epigraphical Relaxation for Debiased Signal Recovery
AI总结：
本文提出一种基于上层图松弛（ER）的多层非凸正则化方法，通过分解正则化函数以计算近端算子并集成到凸优化模型中，解决了信号恢复中多层非凸正则化闭式算子不可用和易陷入局部最优的问题，有效减少了估计偏差。
摘要：
本文提出了一种具有多层非凸正则化的精确信号恢复方法，增强了包括图像和视频在内的高维信号的稀疏性/低秩性。在基于优化的信号恢复中，基于L1范数和核范数的多层凸正则化函数不仅保证了全局最优解，而且由于其对高维信号中结构化稀疏性和低秩性的忠实建模，比单层正则化提供了更准确的估计。然而，已知这些函数会产生有偏的解（估计的幅度值小于真实值）。为了解决这个问题，人们考虑了多层非凸正则化函数，尽管它们面临着自身的挑战：1）它们的闭式近端算子不可用，以及2）收敛可能导致局部最优解。在本文中，我们通过一种基于上层图松弛（ER）的方法解决了这两个问题。首先，ER将多层非凸正则化函数分解为最外层函数和内部函数的上层图约束，从而便于计算近端算子。其次，由ER松弛的正则化函数被集成到一个非凸正则化的凸优化模型中，以估计具有较小偏差的全局最优解。数值实验证明了该方法在图像恢复和主成分分析中实现的偏差减少。
英文摘要：
This paper proposes a precise signal recovery method with multilayered non-convex regularization, enhancing sparsity/low-rankness for high-dimensional signals including images and videos. In optimization-based signal recovery, multilayered convex regularization functions based on the L1 and nuclear-norms not only guarantee a global optimal solution but also offer more accurate estimation than single-layered ones, thanks to their faithful modeling of structured sparsity and low-rankness in high-dimensional signals. However, these functions are known to yield biased solutions (estimated with smaller amplitude values than the true ones). To address this issue, multilayered non-convex regularization functions have been considered, although they face their own challenges: 1) their closed-form proximity operators are unavailable, and 2) convergence may result in a local optimal solution. In this paper, we resolve the two issues with an approach based on epigraphical relaxation (ER). First, ER decomposes a multilayered non-convex regularization function into the outermost function and epigraph constraints for the inner functions, facilitating the computation of proximity operators. Second, the relaxed regularization functions by ER are integrated into a non-convexly regularized convex optimization model to estimate a global optimal solution with less bias. Numerical experiments demonstrate the bias reduction achieved by the proposed method in image recovery and principal component analysis.
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