Sparse Signal Recovery using Log-Sum Regularization and Adaptive Smoothing

2026-05-11

AI Summary

This work addresses the shrinkage bias of ℓ₁ regularization and the algorithmic instability of ℓ₀ approximation in sparse signal recovery from noisy linear measurements. To overcome these limitations, the authors employ log-sum non-convex regularization as a surrogate for the ℓ₀ norm and introduce an adaptive smoothing mechanism to ensure continuity of the proximal operator. They innovatively extend state evolution (SE) theory to this non-convex framework, enabling, for the first time, accurate prediction of phase transition thresholds and mean squared error. Combining AMP and ADMM algorithms, experiments demonstrate that in the noiseless case, ADMM’s success boundary aligns closely with SE predictions, while in noisy settings, AMP tightly tracks the SE trajectory. Moreover, the log-sum penalty significantly outperforms ℓ₁ regularization under low sparsity or high sampling rates.

Abstract

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.

Problem

• Sparse Signal Recovery
• Log-Sum Regularization
• Nonconvex Optimization
• Noise Robustness
• Algorithmic Stability

Innovation

• log-sum regularization
• adaptive smoothing
• approximate message passing
• state evolution
• sparse signal recovery