高斯多指标模型中的谱相变
Spectral phase transitions in Gaussian multi-index models
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中文总结 AI 辅助
本文针对高斯多指标模型，通过建立随机矩阵理论证明谱方法可达到AMP确定的弱恢复阈值，实现潜在子空间的弱恢复，并证明相关预处理是最优的，验证了通用谱猜想。
中文摘要 AI 辅助
从高维高斯协变量的非线性观测中恢复低维潜在子空间是特征学习领域的基础问题。本文研究高斯多指标模型，其中协变量$\boldsymbol{x}_i$独立同分布服从$\boldsymbol{N}(0,\boldsymbol{I}_d)$，响应变量$\boldsymbol{y}_i$仅通过$\boldsymbol{x}_i$到未知$r$维子空间的投影依赖于$\boldsymbol{x}_i$。早期基于近似消息传递（AMP）的工作确定了弱恢复的尖锐阈值[Troiani等人，2025]，这引发了一个问题：是否可以在无辅助信息的情况下通过谱方法达到该阈值。我们对此给出肯定回答，并为矩阵值谱估计量$\boldsymbol{D}_n=\frac{1}{n}\boldsymbol{\times}_{i=1}^n\boldsymbol{T}(\boldsymbol{y}_i)\boldsymbol{\times}\boldsymbol{x}_i\boldsymbol{x}_i^\top$建立了通用随机矩阵理论，其中$\boldsymbol{T}$是任意固定维度的有界对称矩阵值预处理映射。当$n,d$趋于无穷且$n/d\to\boldsymbol{\times}$时，我们证明$\boldsymbol{D}_n$的经验谱测度几乎必然收敛到由矩阵值自洽方程刻画的确定性紧支撑分布。随后我们针对最大特征值建立了谱相变：阈值以下最大特征值紧贴本体边缘，阈值以上则会出现异常值；我们通过有限维确定性方程刻画异常值位置，并证明相关谱估计量可实现潜在子空间的弱恢复。最后，我们证明[Defilippis等人，2025]提出的基于AMP的预处理在所有固定维度的有界矩阵值预处理映射中是最优的，其相变与AMP弱恢复阈值一致，证明了[Defilippis等人，2025]提出的通用谱猜想。
英文摘要
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 $\boldsymbol{x}_i \stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0,\boldsymbol{I}_d)$ and the responses $\boldsymbol{y}_i$ depend on $\boldsymbol{x}_i$ 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 \[\boldsymbol{D}_n=\frac{1}{n}\sum_{i=1}^n\boldsymbol{T}(\boldsymbol{y}_i)\otimes\boldsymbol{x}_i\boldsymbol{x}_i^\top,\] where $\boldsymbol{T}$ is an arbitrary bounded symmetric matrix-valued preprocessing map of fixed dimension. As $n,d \to \infty$ with $n/d\toα$, we prove that the empirical spectral measure of $\boldsymbol{D}_n$ 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].
