We consider the linear regression problem, where the goal is to recover the vector $\boldsymbol{x}\in \mathbb {R}^n$ from measurements $\boldsymbol{y}=\boldsymbol{A}\boldsymbol{x}+\boldsymbol{w}\in \mathbb {R}^m$ under known matrix $\boldsymbol{A}$ and unknown noise $\boldsymbol{w}$. For large i.i.d. sub-Gaussian $\boldsymbol{A}$, the approximate message passing (AMP) algorithm is precisely analyzable through a state-evolution (SE) formalism, which furthermore shows that AMP is Bayes optimal in certain regimes. The rigorous SE proof, however, is long and complicated. And, although the AMP algorithm can be derived as an approximation of loop belief propagation (LBP), this viewpoint provides little insight into why large i.i.d. $\boldsymbol{A}$ matrices are important for AMP, and why AMP has a state evolution. In this work, we provide a heuristic derivation of AMP and its state evolution, based on the idea of “first-order cancellation,” that provides insights missing from the LBP derivation while being much shorter than the rigorous SE proof.

中文翻译：

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AMP 的简单推导及其通过一阶抵消的状态演化

我们考虑线性回归问题，目标是恢复向量 $\boldsymbol{x}\in \mathbb {R}^n$ 从测量 $\boldsymbol{y}=\boldsymbol{A}\boldsymbol{x}+\boldsymbol{w}\in \mathbb {R}^m$ 在已知矩阵下 $\boldsymbol{A}$ 和未知的噪音 $\boldsymbol{w}$. 对于大 iid 亚高斯$\boldsymbol{A}$，近似消息传递 (AMP) 算法可以通过状态演化 (SE) 形式主义精确分析，这进一步表明 AMP 在某些情况下是贝叶斯最优的。然而，严格的 SE 证明又长又复杂。而且，虽然 AMP 算法可以推导出为循环置信传播 (LBP) 的近似值，但这个观点几乎没有提供为什么大 iid$\boldsymbol{A}$矩阵对 AMP 很重要，以及为什么 AMP 具有状态演化。在这项工作中，我们基于“一阶取消”的思想提供了 AMP 及其状态演化的启发式推导，它提供了 LBP 推导中缺少的见解，同时比严格的 SE 证明要短得多。