Sparse regularization is fundamental in signal processing and feature extraction but often relies on non-differentiable penalties, conflicting with gradient-based optimizers. We propose WEEP (Weakly-convex Envelope of Piecewise Penalty), a novel differentiable regularizer derived from the weakly-convex envelope framework. WEEP provides tunable, unbiased sparsity and a simple closed-form proximal operator, while maintaining full differentiability and L-smoothness, ensuring compatibility with both gradient-based and proximal algorithms. This resolves the tradeoff between statistical performance and computational tractability. We demonstrate superior performance compared to established convex and non-convex sparse regularizers on challenging compressive sensing and image denoising tasks.

翻译：稀疏正则化在信号处理和特征提取中至关重要，但通常依赖于不可微的惩罚项，这与基于梯度的优化器存在冲突。本文提出WEEP（分段惩罚的弱凸包络），一种基于弱凸包络框架的新型可微正则化器。WEEP提供可调、无偏的稀疏性以及简单的闭式近端算子，同时保持完全可微性和L-光滑性，确保与基于梯度的算法和近端算法均兼容。这解决了统计性能与计算可处理性之间的权衡问题。在具有挑战性的压缩感知和图像去噪任务中，我们证明了WEEP相较于现有凸和非凸稀疏正则化器具有更优的性能。