In many real-world problems, recovering sparse signals from underdetermined linear systems remains a fundamental challenge. Although $\ell_1$ norm minimization is widely used, it suffers from estimation bias that prevents it from reaching the Bayes-optimal reconstruction limit. Nonconvex alternatives, such as the log-sum penalty, have been proposed to promote stronger sparsity. However, maintaining their algorithmic stability is challenging. To address this challenge, we introduce an adaptive smoothing strategy within an approximate message passing framework to mitigate algorithmic instability. Furthermore, we evaluate the typical exact-recovery threshold for Gaussian measurement matrices using the replica method and state evolution. The results indicate that the adaptive method achieves exact recovery over a broader region than $\ell_1$ norm minimization, although metastable states hinder reaching the information-theoretic limit.

翻译：在许多实际问题中，从欠定线性系统中恢复稀疏信号仍然是一项基本挑战。尽管$\ell_1$范数最小化被广泛使用，但它存在估计偏差，阻碍其达到贝叶斯最优重构极限。非凸替代方案，例如对数求和惩罚，已被提出以促进更强的稀疏性。然而，保持其算法稳定性具有挑战性。为了应对这一挑战，我们在近似消息传递框架内引入了一种自适应平滑策略，以缓解算法不稳定性。此外，我们使用复制方法和状态演化评估了高斯测量矩阵的典型精确恢复阈值。结果表明，自适应方法在比$\ell_1$范数最小化更广泛的区域实现了精确恢复，尽管亚稳态阻碍其达到信息论极限。