We discuss the inhomogeneous spiked Wigner model, a theoretical framework
recently introduced to study structured noise in various learning scenarios,
through the prism of random matrix theory, with a specific focus on its
spectral properties. Our primary objective is to find an optimal spectral
method and to extend the celebrated \cite{BBP} (BBP) phase transition criterion
-- well-known in the homogeneous case -- to our inhomogeneous,
block-structured, Wigner model. We provide a thorough rigorous analysis of a
transformed matrix and show that the transition for the appearance of 1) an
outlier outside the bulk of the limiting spectral distribution and 2) a
positive overlap between the associated eigenvector and the signal, occurs
precisely at the optimal threshold, making the proposed spectral method optimal
within the class of iterative methods for the inhomogeneous Wigner problem.

我们通过随机矩阵理论，重点研究了非齐次分组伪威格纳模型中的结构噪声，目标是找到一个最优的频谱方法，并将均匀情况下著名的 BBP 相变准则扩展到非齐次块状威格纳模型，我们对一个变换矩阵进行了彻底而严格的分析，并展示了在最佳阈值处 1）超出极限谱分布的异常值的出现，和 2）相应特征向量与信号之间的正重叠，在最佳阈值处同时出现，使得所提议的频谱方法在非齐次威格纳问题的迭代方法类中是最优的。