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arXiv · 2507.20447

WEEP: A Differentiable Nonconvex Sparse Regularizer via Weakly-Convex Envelope

Takanobu Furuhashi·Hidekata Hontani·Qibin Zhao·Tatsuya Yokota
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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.

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Takanobu Furuhashi, Hidekata Hontani, Qibin Zhao, Tatsuya Yokota. 2025-07-28. WEEP: A Differentiable Nonconvex Sparse Regularizer via Weakly-Convex Envelope. https://arxiv.org/abs/2507.20447

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