Quantitative Universality of Approximate Message Passing for Rank-One Quadratic Sensing

2026-10-04

AI Summary

This study addresses the long-standing open conjecture posed by Maillard et al. regarding the lack of a rigorous proof for the quantitative universality of the Approximate Message Passing (AMP) algorithm under rank-one quadratic sensing matrices. To resolve this, we propose a column-wise Lindeberg replacement and leave-one-out analysis framework, integrating smooth spectral expansions with higher-order moment concentration inequalities to systematically compare general ensembles against Gaussian ensembles and quantify their error discrepancies. This work provides the first rigorous proof of AMP’s quantitative universality in this setting, thereby resolving the aforementioned conjecture. We demonstrate that differences in statistics such as the normalized signal overlap are of order O(d^{-1/2}), establishing universal predictions for the mean squared error and spectral distribution. These findings offer solid theoretical guarantees for AMP algorithms in quadratic regression.

Abstract

Approximate Message Passing (AMP) algorithms are attractive as they are computationally efficient and simultaneously admit a precise characterization in terms of the low-dimensional"state-evolution"recursion. In this work, we establish quantitative universality for AMP with centered rank-one sensing matrices $Z_i=(x_ix_i^\top-I_d)/\sqrt d$, where $x_i$ are independent standard Gaussian vectors. These matrices arise in quadratic regression and learning quadratic neural networks. Their normalized vectorizations have isotropic covariance in dimension $p=d(d+1)/2$, but strongly dependent coordinates. For linear observations with Gaussian noise and prescribed smooth, bounded spectral denoisers, we compare rank-one sensing with a covariance-matched isotropic Gaussian ensemble. At a fixed iteration horizon, with $n\leq Cp$ and uniform bounds on the normalized signal energy, initialization, coefficients, and first four denoiser derivatives, normalized signal overlaps, cross-time overlaps, and smooth linear spectral statistics differ by $O(d^{-1/2})$ in every fixed $L^m$. This yields universality of the normalized mean-squared error, fixed spectral moments, and empirical spectral distributions. Whenever the Gaussian observables admit a joint state-evolution limit, the same predictions hold under rank-one sensing. The proof uses columnwise Lindeberg replacement, leave-one-out analysis, smooth spectral expansions, and high-moment concentration. For prescribed smooth spectral denoisers, the result establishes the AMP universality conjectured by Maillard et al. (arXiv:2408.03733).

Problem

• Approximate Message Passing
• Quantitative Universality
• Rank-One Quadratic Sensing
• State Evolution

Innovation

• Approximate Message Passing
• Quantitative Universality
• Rank-One Quadratic Sensing
• Lindeberg Replacement
• State Evolution