We present a simple and efficient algorithm for robust approximate message passing (AMP) in the spiked matrix setting. In particular, let $\varepsilon$ be a sufficiently small constant, and suppose that $X \in \mathbb R^{n \times n}$ is a Gaussian matrix with a planted rank-$1$ spike, and $E \in \mathbb R^{n \times n}$ is an adversarially chosen matrix supported on an $\varepsilon n \times \varepsilon n$ principal minor. Let $v_{\mathrm{AMP}}(X)$ be the output of an AMP iteration on the uncorrupted matrix $X$. We give a procedure that, given access only to the corrupted matrix $Y = X + E$, computes a vector $v_{\mathrm{ALG}}(Y)$ which is $\tilde{O}(\sqrt{\varepsilon})$-close to $v_{\mathrm{AMP}}(X)$, for any of a class of AMP iterations which includes sparse Principal Component Analysis (PCA), non-negative PCA, and $\mathbb Z_2$ synchronization. Our algorithm consists of a spectral pre-processing step combined with a robust spectral initialization procedure; given these inputs, we prove that (perhaps surprisingly) AMP is robust out-of-the-box.

翻译：本文提出一种用于尖峰矩阵设定下鲁棒近似消息传递(AMP)的简洁高效算法。具体而言，设$\varepsilon$为充分小的常数，假设$X \in \mathbb R^{n \times n}$为具有植入秩-$1$尖峰的高斯矩阵，$E \in \mathbb R^{n \times n}$为支撑在$\varepsilon n \times \varepsilon n$主主子阵上的对抗性选择矩阵。记$v_{\mathrm{AMP}}(X)$为对未损坏矩阵$X$进行AMP迭代的输出。我们给出一个算法流程：仅通过访问损坏矩阵$Y = X + E$，即可计算向量$v_{\mathrm{ALG}}(Y)$，该向量与$v_{\mathrm{AMP}}(X)$的误差为$\tilde{O}(\sqrt{\varepsilon})$，适用于包含稀疏主成分分析(PCA)、非负PCA和$\mathbb Z_2$同步在内的AMP迭代类。算法由谱预处理步骤与鲁棒谱初始化过程组成；基于这些输入，我们证明（或许令人惊讶的是）AMP在开箱即用条件下具有鲁棒性。