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

Sparse regularization is fundamental in signal processing for efficient signal recovery and feature extraction. However, it faces a fundamental dilemma: the most powerful sparsity-inducing penalties are often non-differentiable, conflicting with gradient-based optimizers that dominate the field. We introduce WEEP (Weakly-convex Envelope of Piecewise Penalty), a novel, fully differentiable sparse regularizer derived from the weakly-convex envelope framework. WEEP provides strong, unbiased sparsity while maintaining full differentiability and L-smoothness, making it natively compatible with any gradient-based optimizer. This resolves the conflict between statistical performance and computational tractability. We demonstrate superior performance compared to the L1-norm and other established non-convex sparse regularizers on challenging signal and image denoising tasks.

Adaptive Block Sparse Regularization under Arbitrary Linear Transform

We propose a convex signal reconstruction method for block sparsity under arbitrary linear transform with unknown block structure. The proposed method is a generalization of the existing method LOP-$\ell_2$/$\ell_1$ and can reconstruct signals with block sparsity under non-invertible transforms, unlike LOP-$\ell_2$/$\ell_1$. Our work broadens the scope of block sparse regularization, enabling more versatile and powerful applications across various signal processing domains. We derive an iterative algorithm for solving proposed method and provide conditions for its convergence to the optimal solution. Numerical experiments demonstrate the effectiveness of the proposed method.