Abstract
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.
Abstract (translated)
稀疏正则化在信号处理和特征提取中至关重要，但通常依赖于非可微分的惩罚项，这与基于梯度的优化器相冲突。我们提出了WEEP（弱凸包络分段惩罚），这是一种新颖的可微正则化方法，源自弱凸包络框架。WEEP提供了一种可调、无偏的稀疏性，并具有简单的闭式形式近端操作符，同时保持完全可微性和L-光滑性，确保了与基于梯度和基于近端算法的兼容性。这解决了统计性能与计算可行性之间的权衡问题。我们在具有挑战性的压缩感知和图像去噪任务中展示了WEEP相较于现有的凸和非凸稀疏正则化方法的优越性能。
URL
https://arxiv.org/abs/2507.20447
