Phase transition in compressed sensing using log-sum penalty and adaptive smoothing

2026-04-15

AI Summary

This work addresses the challenge of sparse signal recovery in underdetermined linear systems, where conventional ℓ₁-norm minimization suffers from estimation bias and falls short of the Bayesian optimal limit. The authors propose an approximate message passing algorithm that integrates a log-sum penalty with an adaptive smoothing strategy to enhance the stability of non-convex optimization. Leveraging the replica method and state evolution theory, they rigorously characterize the exact recovery phase transition threshold under Gaussian measurement matrices. Compared to ℓ₁-based approaches, the proposed algorithm achieves exact reconstruction over a significantly broader parameter regime. Although it remains constrained by metastable states and does not yet attain the information-theoretic limit, it substantially expands the region of recoverable signals.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• compressed sensing
• sparse signal recovery
• log-sum penalty
• phase transition
• algorithmic stability

Innovation

• log-sum penalty
• adaptive smoothing
• approximate message passing
• phase transition
• replica method