The paper introduces a new iterative thresholding algorithm for compressed sensing, which aims to achieve the same reconstruction performance as convex optimization methods but with significantly faster computation. The algorithm, inspired by belief propagation in graphical models, is designed to improve the sparsity-undersampling tradeoff, a critical aspect of compressed sensing. The authors demonstrate that their algorithm performs as well as convex optimization methods in terms of reconstruction quality, while being much faster. They provide theoretical and empirical evidence to support their findings, including a state evolution formalism that accurately predicts the algorithm's behavior. The paper also discusses the phase transitions in reconstruction success and failure, showing that the new algorithm achieves the same phase transitions as convex optimization methods. Additionally, the authors explore the optimality of the algorithm's parameters and provide explicit expressions for the optimal tuning parameters. The results highlight the potential of the new algorithm for large-scale applications in fields such as spectroscopy and medical imaging.The paper introduces a new iterative thresholding algorithm for compressed sensing, which aims to achieve the same reconstruction performance as convex optimization methods but with significantly faster computation. The algorithm, inspired by belief propagation in graphical models, is designed to improve the sparsity-undersampling tradeoff, a critical aspect of compressed sensing. The authors demonstrate that their algorithm performs as well as convex optimization methods in terms of reconstruction quality, while being much faster. They provide theoretical and empirical evidence to support their findings, including a state evolution formalism that accurately predicts the algorithm's behavior. The paper also discusses the phase transitions in reconstruction success and failure, showing that the new algorithm achieves the same phase transitions as convex optimization methods. Additionally, the authors explore the optimality of the algorithm's parameters and provide explicit expressions for the optimal tuning parameters. The results highlight the potential of the new algorithm for large-scale applications in fields such as spectroscopy and medical imaging.