We explore a spectral initialization method that plays a central role in contemporary research on signal estimation in nonconvex scenarios. In a noiseless phase retrieval framework, we precisely analyze the method's performance in the high-dimensional limit when sensing vectors follow a multivariate Gaussian distribution for two rotationally invariant models of the covariance matrix C. In the first model C is a projector on a lower dimensional space while in the second it is a Wishart matrix. Our analytical results extend the well-established case when C is the identity matrix. Our examination shows that the introduction of biased spatial directions leads to a substantial improvement in the spectral method's effectiveness, particularly when the number of measurements is less than the signal's dimension. This extension also consistently reveals a phase transition phenomenon dependent on the ratio between sample size and signal dimension. Surprisingly, both of these models share the same threshold value.

翻译：我们探讨了一种谱初始化方法，该方法在非凸场景下信号估计的现代研究中发挥着核心作用。在无噪声相位恢复框架中，我们精确分析了该方法在高维极限下的性能，其中感知向量遵循多元高斯分布，且协方差矩阵C满足两种旋转不变模型。在第一种模型中，C是低维空间上的投影算子；在第二种模型中，C是Wishart矩阵。我们的分析结果将C为单位矩阵的经典案例进行了推广。研究表明，引入有偏空间方向可显著提升谱方法的有效性，特别是在测量次数少于信号维度时。该推广还一致揭示了依赖于样本量与信号维度比值的相变现象。令人惊讶的是，这两种模型具有相同的阈值。