Classical sparsity promoting methods rely on the l0 norm, which treats all nonzero components as equally significant. In practical inverse problems, however, solutions often exhibit many small amplitude components that have little effect on reconstruction but lead to an overestimation of signal complexity. We address this limitation by shifting the paradigm from discrete cardinality to effective sparsity. Our approach introduces the effective number of nonzeros (ENZ), a unified class of normalized entropy-based regularizers, including Shannon and Renyi forms, that quantifies the concentration of significant coefficients. We show that, unlike the classical l0 norm, the ENZ provides a stable and continuous measure of effective sparsity that is insensitive to negligible perturbations. For noisy linear inverse problems, we establish theoretical guarantees under the Restricted Isometry Property (RIP), proving that ENZ based recovery is unique and stable. We also derive a decomposition showing that the ENZ equals the support cardinality times a distributional efficiency term, thereby linking entropy with l0 regularization. Numerical experiments show that this effective sparsity framework outperforms traditional cardinality based methods in robustness and accuracy.

翻译：经典稀疏促进方法依赖于l0范数，其将所有非零分量视为同等重要。然而在实际逆问题中，解往往包含大量对重构影响甚微的小幅值分量，这会导致对信号复杂性的高估。我们通过将范式从离散基数转向有效稀疏性来解决这一局限。本文提出的方法引入了非零有效数（ENZ）——一类基于归一化熵的统一正则化器（包括香农熵与雷尼熵形式），用于量化显著系数的集中程度。我们证明，与经典l0范数不同，ENZ能提供稳定且连续的有效稀疏性度量，且对微小扰动不敏感。针对含噪线性逆问题，我们在限制等距性（RIP）条件下建立了理论保证，证明了基于ENZ的恢复具有唯一性与稳定性。我们还推导出一个分解式，表明ENZ等于支撑基数乘以分布效率项，从而建立了熵与l0正则化之间的理论联系。数值实验表明，该有效稀疏性框架在鲁棒性与精度方面均优于传统的基于基数的方法。