Title: Simple and Resilient Approximate Message Passing for Planted Spike Models

Abstract:

This paper introduces a straightforward and highly efficient algorithm designed to ensure robustness in approximate message passing (AMP) within the context of spiked matrix models. Specifically, consider a scenario where $\varepsilon$ represents a sufficiently small constant. Let $X \in \mathbb R^{n \times n}$ denote a Gaussian matrix containing a planted rank-1 spike, while $E \in \mathbb R^{n \times n}$ is an adversarially selected matrix that is non-zero only on an $\varepsilon n \times \varepsilon n$ principal minor. If $v_{\mathrm{AMP}}(X)$ signifies the result of an AMP iteration performed on the pristine matrix $X$, we propose a method that, using solely the corrupted matrix $Y = X + E$, generates a vector $v_{\mathrm{ALG}}(Y)$. This output vector is shown to be $\tilde{O}(\sqrt{\varepsilon})$-close to $v_{\mathrm{AMP}}(X)$. This guarantee holds for a broad class of AMP iterations, encompassing sparse Principal Component Analysis (PCA), non-negative PCA, and $\mathbb Z_2$ synchronization. The proposed algorithm leverages a spectral pre-processing stage alongside a robust spectral initialization technique. Given these inputs, our analysis demonstrates that AMP exhibits inherent robustness, a result that may seem counterintuitive.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC