We consider sparse signal reconstruction via minimization of the smoothly clipped absolute deviation (SCAD) penalty, and develop one-step replica-symmetry-breaking (1RSB) extensions of approximate message passing (AMP), termed 1RSB-AMP. Starting from the 1RSB formulation of belief propagation, we derive explicit update rules of 1RSB-AMP together with the corresponding state evolution (1RSB-SE) equations. A detailed comparison shows that 1RSB-AMP and 1RSB-SE agree remarkably well at the macroscopic level, even in parameter regions where replica-symmetric (RS) AMP, termed RS-AMP, diverges and where the 1RSB description itself is not expected to be thermodynamically exact. Fixed-point analysis of 1RSB-SE reveals a phase diagram consisting of success, failure, and diverging phases, as in the RS case. However, the diverging-region boundary now depends on the Parisi parameter due to the 1RSB ansatz, and we propose a new criterion -- minimizing the size of the diverging region -- rather than the conventional zero-complexity condition, to determine its value. Combining this criterion with the nonconvexity-control (NCC) protocol proposed in a previous RS study improves the algorithmic limit of perfect reconstruction compared with RS-AMP. Numerical solutions of 1RSB-SE and experiments with 1RSB-AMP confirm that this improved limit is achieved in practice, though the gain is modest and remains slightly inferior to the Bayes-optimal threshold. We also report the behavior of thermodynamic quantities -- overlaps, free entropy, complexity, and the non-self-averaging susceptibility -- that characterize the 1RSB phase in this problem.

翻译：本文研究通过平滑截断绝对偏差（SCAD）惩罚最小化实现稀疏信号重构的问题，并发展了近似消息传递（AMP）的一步复本对称破缺（1RSB）扩展方法，称为1RSB-AMP。从信念传播的1RSB表述出发，我们推导了1RSB-AMP的显式更新规则及其对应的状态演化（1RSB-SE）方程。详细对比表明，即使在复本对称（RS）AMP（称为RS-AMP）发散且1RSB描述本身在热力学意义上并不精确的参数区域内，1RSB-AMP与1RSB-SE在宏观层面仍表现出高度一致性。通过对1RSB-SE的定点分析，我们揭示了由成功相、失败相和发散相构成的相图结构，这与RS情形类似。然而，由于采用了1RSB拟设，发散区域的边界现依赖于帕里西参数。为此我们提出一种新准则——最小化发散区域范围（而非传统的零复杂度条件）来确定该参数值。将此准则与先前RS研究中提出的非凸性控制（NCC）协议相结合，相较于RS-AMP，能够提升完美重构的算法极限。1RSB-SE的数值解与1RSB-AMP的实验结果共同证实：该改进极限在实践中得以实现，尽管增益较为有限且仍略逊于贝叶斯最优阈值。我们还报告了描述该问题中1RSB相的热力学量行为——包括重叠度、自由熵、复杂度及非自平均敏感性。