Easy, robust approximate message passing for planted spike models

• URL: http://arxiv.org/abs/2606.00500v1
• Date: Sat, 30 May 2026 03:18:52 GMT
• Title: Easy, robust approximate message passing for planted spike models
• Abstract summary: We present a simple and efficient algorithm for robust approximate message passing (AMP) in the spiked matrix setting.<n>Our algorithm consists of a spectral pre-processing step combined with a robust spectral procedure; given these inputs, we prove that (perhaps surprisingly) AMP is robust out-of-the-box.
• Score: 1.0269997499911663
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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