基于复近似消息传递（CAMP）的复LASSO的渐近分析
Asymptotic Analysis of Complex LASSO via Complex Approximate Message Passing (CAMP)
摘要：
从一组欠采样的随机线性测量中恢复稀疏信号是压缩感知中的主要问题。在本文中，我们考虑的情况下，信号和测量都是复杂的。本文研究了$\ell_1$中常用的重建方法-正则化最小二乘法（Regularized Least Squares，LASSO）。虽然一些研究表明，LASSO算法在某些条件下提供了理想的解决方案，该算法在复杂的设置中的精确渐近性能尚不清楚。本文将近似消息传递（AMP）算法推广到复信号和复测量，得到复近似消息传递算法（CAMP）。然后，我们推广的状态演化框架最近推出的AMP的分析，复杂的设置。使用状态演化，我们推导出精确的LASSO和CAMP的相变和噪声灵敏度的公式。
英文摘要：
Recovering a sparse signal from an undersampled set of random linear measurements is the main problem of interest in compressed sensing. In this paper, we consider the case where both the signal and the measurements are complex. We study the popular reconstruction method of $\ell_1$-regularized least squares or LASSO. While several studies have shown that the LASSO algorithm offers desirable solutions under certain conditions, the precise asymptotic performance of this algorithm in the complex setting is not yet known. In this paper, we extend the approximate message passing (AMP) algorithm to the complex signals and measurements and obtain the complex approximate message passing algorithm (CAMP). We then generalize the state evolution framework recently introduced for the analysis of AMP, to the complex setting. Using the state evolution, we derive accurate formulas for the phase transition and noise sensitivity of both LASSO and CAMP.
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