We study sparse signal recovery from noisy linear observations using nonconvex log-sum regularization. The log-sum penalty reduces the shrinkage bias of $\ell_1$ regularization and more closely approximates the $\ell_0$ regularization, but its nonconvexity can make reconstruction algorithms unstable. To mitigate this instability, we use an adaptive smoothing strategy that determines the smoothing parameter so that the scalar proximal operator remains continuous. Using this proximal operator, we formulate the approximate message passing (AMP) algorithm and derive the corresponding state evolution (SE) recursion. The fixed point of the SE recursion predicts the final mean squared error (MSE) and, in the noiseless limit, the exact-recovery phase transition. To further investigate finite-dimensional reconstruction behavior, we implement an alternating direction method of multipliers (ADMM) algorithm. In the noiseless setting, we find that the empirical success boundary of ADMM closely agrees with the SE-predicted phase transition. In the noisy setting, we observe that AMP closely follows the SE prediction, whereas ADMM qualitatively reproduces the SE-predicted dependence of the final MSE on the regularization parameter. A comparison with $\ell_1$ regularization shows that log-sum regularization is beneficial in low-density or high-measurement-rate regimes, whereas $\ell_1$ regularization remains preferable at higher densities and lower measurement rates.

翻译：我们研究基于非凸对数求和正则化的含噪线性观测稀疏信号恢复问题。对数求和惩罚项可减少$\ell_1$正则化的收缩偏差，更接近$\ell_0$正则化的效果，但其非凸性可能导致重构算法不稳定。为解决此不稳定性，我们采用自适应平滑策略来确定平滑参数，确保标量近端算子保持连续性。利用该近端算子，我们构建近似消息传递算法并推导其状态演化递归过程。状态演化递归的不动点可预测最终均方误差，并在无噪极限下预测精确恢复的相变边界。为进一步分析有限维重构行为，我们实现了交替方向乘子算法。在无噪情形下，交替方向乘子法的经验成功边界与状态演化预测的相变高度吻合。在有噪情形下，观察发现近似消息传递密切跟随状态演化预测，而交替方向乘子法定性再现了状态演化预测的最终均方误差对正则化参数的依赖关系。与$\ell_1$正则化的对比表明：在低密度或高测量率场景中，对数求和正则化更具优势；而在高密度或低测量率场景中，$\ell_1$正则化仍为更优选择。