The spiked Wigner ensemble is a prototypical model for high-dimensional inference. We study the spectral properties of an inhomogeneous rank-one spiked Wigner model in which the variance of each entry of the noise matrix is itself a random variable. In the high-dimensional limit, we derive exact equations for the spectral edges, the outlier eigenvalue, and the distribution of the components of the outlier eigenvector. These equations determine the BBP transition line that separates the gapped phase, where the signal is detectable, from the gapless phase. In the gapped regime, the distribution of the outlier eigenvector provides a natural estimator of the spike. We solve the equations for a noise matrix whose variances are generated from a truncated power-law distribution. In this case, the BBP transition line is non-monotonic, showing that an inhomogeneous noise can enhance signal detectability.

翻译：尖峰Wigner系综是高维推断的原型模型。研究一类非均匀秩一尖峰Wigner模型的谱性质，其中噪声矩阵每个分量的方差本身是随机变量。在高维极限下，推导出谱边缘、异常特征值以及异常特征向量分量分布的精确方程。这些方程确定了BBP相变线，它将信号可检测的有隙相与无隙相区分开来。在有隙相中，异常特征向量的分布提供了尖峰的自然估计。我们求解了方差由截断幂律分布生成的噪声矩阵的方程。在此情形下，BBP相变线呈现非单调性，表明非均匀噪声可以增强信号的可检测性。