Linearly involved generalized Moreau enhanced models and their proximal splitting algorithm under overall convexity condition

The convex envelopes of the direct discrete measures, for the sparsity of vectors or for the low-rankness of matrices, have been utilized extensively as practical penalties in order to compute a globally optimal solution of the corresponding regularized least-squares models. Motivated mainly by the ideas in Zhang (2010 Ann. Stat. 38 894–942; Selesnick 2017 IEEE Trans. Signal Process. 65 4481–94; Yin et al 2019 IEEE Trans. Signal Process. 67 2595–607) to exploit nonconvex penalties in the regularized least-squares models without losing their overall convexities, this paper presents the linearly involved generalized Moreau enhanced (LiGME) model as a unified extension of such utilizations of nonconvex penalties. The proposed model can admit multiple nonconvex penalties without losing its overall convexity and thus is applicable to much broader scenarios in the sparsity-rank-aware signal processing. Under the general overall-convexity condition of the LiGME model, we also present a novel proximal splitting type algorithm of guaranteed convergence to a globally optimal solution. Numerical experiments in typical examples of the sparsity-rank-aware signal processing demonstrate the effectiveness of the LiGME models and the proposed proximal splitting algorithm.

对于向量的稀疏性或矩阵的低秩性，直接离散测度的凸包已被广泛用作实际的惩罚项，以便计算相应正则化最小二乘模型的全局最优解。主要受Zhang（2010 Ann. Stat. 38 894–942；Selesnick 2017 IEEE Trans. Signal Process. 65 4481–94；Yin等人2019 IEEE Trans. Signal Process. 67 2595–607）中在正则化最小二乘模型中利用非凸惩罚项而不损失其整体凸性的思想启发，本文提出了线性相关广义莫罗增强（LiGME）模型，作为此类非凸惩罚项应用的统一扩展。所提出的模型可以容纳多个非凸惩罚项而不损失其整体凸性，因此适用于稀疏 - 秩感知信号处理中更广泛的场景。在LiGME模型的一般整体凸性条件下，我们还提出了一种新的近端分裂型算法，该算法保证收敛到全局最优解。在稀疏 - 秩感知信号处理的典型示例中的数值实验证明了LiGME模型和所提出的近端分裂算法的有效性。