Numerous renowned algorithms for tackling the compressed sensing problem employ an alternating strategy, which typically involves data matching in one module and denoising in another. Based on an in-depth analysis of the connection between the message passing and operator splitting, we present a novel approach, the Alternating Subspace Method (ASM), which intuitively combines the principles of the greedy methods (e.g., the orthogonal matching pursuit type methods) and the splitting methods (e.g., the approximate message passing type methods). Essentially, ASM modifies the splitting method by achieving fidelity in a subspace-restricted fashion. We reveal that such confining strategy still yields a consistent fixed point iteration and establish its local geometric convergence on the lasso problem. Numerical experiments on both the lasso and channel estimation problems demonstrate its high convergence rate and its capacity to incorporate different prior distributions. Further theoretical analysis also demonstrates the advantage of the motivated message-passing splitting by incorporating quasi-variance degree of freedom even for the classical lasso optimization problem. Overall, the proposed method is promising in efficiency, accuracy and flexibility, which has the potential to be competitive in different sparse recovery applications.

翻译：解决压缩感知问题的众多著名算法采用交替策略，通常包含一个模块进行数据匹配，另一个模块进行去噪。基于对消息传递与算子分裂之间关联的深入分析，我们提出了一种新方法——交替子空间方法（ASM），该方法直观地结合了贪婪方法（如正交匹配追踪类方法）与分裂方法（如近似消息传递类方法）的原理。本质上，ASM通过以子空间受限的方式实现保真度来改进分裂方法。我们揭示了这种限制策略仍能产生一致的定点迭代，并在LASSO问题上建立了其局部几何收敛性。在LASSO和信道估计问题上的数值实验证明了其高收敛率以及整合不同先验分布的能力。进一步的理论分析也表明，即使对于经典的LASSO优化问题，通过引入准方差自由度，所启发的消息传递分裂策略仍具优势。总体而言，所提方法在效率、精度和灵活性方面前景广阔，有望在不同稀疏恢复应用中具备竞争力。