Sparse Signal Recovery using Log-Sum Regularization and Adaptive Smoothing

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.