The Approximate Message Passing (AMP) algorithm has garnered significant attention in recent years for solving linear inverse problems, particularly in the field of Bayesian inference for high-dimensional models. In this paper, we consider sampling from the posterior in the linear inverse problem, with an i.i.d. random design matrix. We develop a sampling algorithm by integrating the AMP algorithm and stochastic localization. We give a proof for the convergence in smoothed KL divergence between the distribution of the samples generated by our algorithm and the target distribution, whenever the noise variance $\Delta$ is below $\Delta_{\rm AMP}$, which is the computation threshold for mean estimation introduced in (Barbier et al., 2020).

翻译：近似消息传递（AMP）算法近年来在求解线性逆问题方面受到广泛关注，特别是在高维模型的贝叶斯推断领域。本文考虑在独立同分布随机设计矩阵的线性逆问题中从后验分布采样。我们通过整合AMP算法与随机局部化技术，提出了一种采样算法。当噪声方差$\Delta$低于$\Delta_{\rm AMP}$（Barbier等人于2020年提出的均值估计计算阈值）时，我们证明了算法生成样本的分布与目标分布之间的平滑KL散度收敛性。