Sharp Spectral Thresholds for Multi-View Spiked Wigner Models

Motivated by multimodal estimation, we study a multi-view spiked Wigner model in which several noisy matrix observations contain correlated latent spikes. We derive a spectral estimator for the latent spikes by linearizing approximate message passing (AMP). Our main result is an explicit sharp transition formula for its spectrum: for $L \geq 2$ views, letting $\lambda$ be the $L$-dimensional vector of spike strengths and $B$ the $L\times L$ limiting Gram matrix of the spikes, the critical parameter is $\mathsf{SNR}(\lambda,B)=\lambda_{\max}[\mathrm{Diag}(\sqrt{\lambda}) (B \odot B) \mathrm{Diag}(\sqrt{\lambda})]$. When $\mathsf{SNR}(\lambda,B)<1$, the linearized AMP matrix has no outlier beyond the right edge of its bulk spectrum. When $\mathsf{SNR}(\lambda,B)>1$, an informative outlier is pinned at the distinguished point $1$, and the associated eigenvector has explicit, nontrivial overlaps with the latent signals. Thus $\mathsf{SNR}(\lambda,B)=1$ gives the exact spectral weak-recovery threshold for the linearized AMP method. To establish our results, we analyze the correlated Gaussian noise matrix through a matrix Dyson equation and combine this deterministic description with finite-rank perturbation arguments adapted to the multi-view spike structure. We also show that, for a broad class of spike priors, the spectral threshold $\mathsf{SNR}(\lambda,B)=1$ coincides with the information-theoretic threshold for weak recovery, ruling out a statistical-computational gap for this class of priors.