提出高效计算分组稀疏惩罚项近似算子的方法，提升变量自动聚类的信号恢复效果。
Proximal Operators of Sorted Nonconvex Penalties
- 利用排序非凸正则化促进变量聚类，结合近似算子求解优化问题。
- 对弱凸情况用PAV算法精确求解，对非凸情况改进算法实现高效计算。
- 适用于需要自动分组的高维稀疏建模任务，如基因数据分析。
本文研究基于排序非光滑惩罚项的稀疏信号恢复与变量自动分组问题。针对广义线性模型，提出一类推广排序L1范数（SLOPE）的排序非凸惩罚函数，其排序特性有助于变量聚类，而非凸性可减少系数收缩。目标是提供高效的近似算子计算方法，以支持常用近似算法求解复合优化问题。区分两类情形：弱凸情况（如排序MCP、SCAD）下近似算子仍为凸问题，可通过池相邻违背者（PAV）算法精确求解；非凸情况（如排序Lq，q∈(0,1)）下则为复杂非凸组合问题，本文通过改进PAV算法实现高效求解。同时给出非凸近似问题极小值的新理论分析。实验验证了该方法在多组数据上的实际有效性。
原文摘要 · Abstract (English)
This work studies the problem of sparse signal recovery with automatic grouping of variables. To this end, we investigate sorted nonsmooth penalties as a regularization approach for generalized linear models. We focus on a family of sorted nonconvex penalties which generalizes the Sorted L1 Norm (SLOPE). These penalties are designed to promote clustering of variables due to their sorted nature, while the nonconvexity reduces the shrinkage of coefficients. Our goal is to provide efficient ways to compute their proximal operator, enabling the use of popular proximal algorithms to solve composite optimization problems with this choice of sorted penalties. We distinguish between two classes of problems: the weakly convex case where computing the proximal operator remains a convex problem, and the nonconvex case where computing the proximal operator becomes a challenging nonconvex combinatorial problem. For the weakly convex case (e.g. sorted MCP and SCAD), we explain how the Pool Adjacent Violators (PAV) algorithm can exactly compute the proximal operator. For the nonconvex case (e.g. sorted Lq with q in ]0,1[), we show that a slight modification of this algorithm turns out to be remarkably efficient to tackle the computation of the proximal operator. We also present new theoretical insights on the minimizers of the nonconvex proximal problem. We demonstrate the practical interest of using such penalties on several experiments.
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