Easy, robust approximate message passing for planted spike models

2026-05-29

AI Summary

This work addresses the problem of recovering high-accuracy approximate message passing (AMP) solutions from a spiked matrix model corrupted by adversarial sparse perturbations. The authors propose an efficient algorithm that integrates spectral preprocessing with robust spectral initialization, enabling accurate reconstruction using only the contaminated observation matrix. For the first time, they theoretically establish that AMP and several of its variants—including sparse PCA, non-negative PCA, and ℤ₂ synchronization—are inherently robust when equipped with appropriate initialization. Specifically, under adversarial perturbations affecting an εn×εn submatrix, the recovered vector achieves an error of Õ(√ε) relative to the ideal AMP output, providing a rigorous theoretical guarantee of robustness.

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.

Problem

• approximate message passing
• spiked matrix models
• robustness
• adversarial corruption
• spectral initialization

Innovation

• approximate message passing
• robustness
• spiked matrix models
• spectral initialization
• adversarial corruption