Spectral phase transitions in Gaussian multi-index models

Recovering a low-dimensional latent subspace from nonlinear observations of Gaussian covariates in high dimensions is a fundamental problem in feature learning. Here, we consider Gaussian multi-index models in which the covariates ${\mathit{x}}_i\mathcal{N}(0,{\mathit{I}}_d)$ and the responses ${\mathit{y}}_i$ depend on ${\mathit{x}}_i$ only through its projection onto an unknown $r$-dimensional subspace. Earlier work based on approximate message passing (AMP) identified a sharp threshold for weak recovery [Troiani et al., 2025], raising the question of whether it can be attained, without side information, by a spectral method. We answer this affirmatively and develop a general random matrix theory for matrix-valued spectral estimators of the form $${\mathit{D}}_n=\frac{1}{n}\sum _{i=1}^n\mathit{T}({\mathit{y}}_i)\otimes {\mathit{x}}_i{\mathit{x}}_i^{\mathrm{\top }},$$ where $\mathit{T}$ is an arbitrary bounded symmetric matrix-valued preprocessing map of fixed dimension. As $n,d\to \mathrm{\infty }$ with $n\mathrm{/}d\to \alpha$, we prove that the empirical spectral measure of ${\mathit{D}}_n$ converges almost surely to a deterministic compactly supported distribution characterized by a matrix-valued self-consistent equation. We then establish a spectral phase transition for the largest eigenvalue: below threshold it sticks to the bulk edge, while above threshold an outlier emerges. We characterize the outlier location through a finite-dimensional deterministic equation and show that the associated spectral estimator achieves weak recovery of the latent subspace. Finally, we prove that the AMP-derived preprocessing of [Defilippis et al., 2025] is optimal among all bounded matrix-valued preprocessing maps of any fixed dimension. Its transition coincides with the AMP weak-recovery threshold, proving the general spectral conjecture of [Defilippis et al., 2025].