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.

翻译：我们通过随机矩阵理论的视角，讨论非齐次尖峰Wigner模型——一种近期引入用于研究各类学习场景中结构化噪声的理论框架——并特别关注其谱性质。主要目标在于寻找最优谱方法，并将著名的BBP相变判据（在齐次情形中广为人知）推广至我们的非齐次块结构Wigner模型。我们对变换后的矩阵进行了严谨的完整分析，证明：1）谱分布极限谱密度主体外的异常值出现，以及2）相关特征向量与信号之间的正重叠——这两类相变恰好发生在最优阈值处，从而使得所提出的谱方法成为非齐次Wigner问题迭代方法类别中的最优方法。