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