# Perfect reconstruction of sparse signals using nonconvexity control and one-step RSB message passing

arXiv: https://arxiv.org/abs/2512.17426 · HTML: https://mlsanity.com/paper/2512.17426/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 19 Dec 2025.

## Builds on (most cited first, 7 of 7)

- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Optimization of the Sherrington-Kirkpatrick Hamiltonian](https://mlsanity.com/paper/1812.10897/) (2018)
- [The backtracking survey propagation algorithm for solving random K-SAT problems](https://mlsanity.com/paper/1508.05117/) (2015)
- [Approximate Survey Propagation for Statistical Inference](https://mlsanity.com/paper/1807.01296/) (2018)
- [Generalized Approximate Survey Propagation for High-Dimensional Estimation](https://mlsanity.com/paper/1905.05313/) (2019)
- [L1-Penalized Censored Gaussian Graphical Model](https://mlsanity.com/paper/1801.07981/) (2018)
- [Perfect reconstruction of sparse signals with piecewise continuous nonconvex penalties and nonconvexity control](https://mlsanity.com/paper/1902.07436/) (2019)

## Abstract (arXiv)

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.