Fast, robust approximate message passing

TL;DR

This paper introduces a spectral pre-processing method for approximate message passing algorithms that enhances robustness against localized perturbations in the input data, ensuring solutions remain close to the uncorrupted case.

Contribution

The authors develop a spectral procedure that makes AMP algorithms robust to small, localized input perturbations, a novel approach in the context of quadratic optimization.

Findings

The method guarantees solutions close to the unperturbed AMP output.

Performance degrades gracefully as the size of the perturbation increases.

The approach applies to any separable AMP algorithm with subgaussian entries.

Abstract

We give a fast, spectral procedure for implementing approximate-message passing (AMP) algorithms robustly. For any quadratic optimization problem over symmetric matrices $X$ with independent subgaussian entries, and any separable AMP algorithm $\mathcal A$, our algorithm performs a spectral pre-processing step and then mildly modifies the iterates of $\mathcal A$. If given the perturbed input $X + E \in \mathbb R^{n \times n}$ for any $E$ supported on a $\varepsilon n \times \varepsilon n$ principal minor, our algorithm outputs a solution $\hat v$ which is guaranteed to be close to the output of $\mathcal A$ on the uncorrupted $X$, with $\|\mathcal A(X) - \hat v\|_2 \le f(\varepsilon) \|\mathcal A(X)\|_2$ where $f(\varepsilon) \to 0$ as $\varepsilon \to 0$ depending only on $\varepsilon$.