Spectral Initialization for High-Dimensional Phase Retrieval with Biased Spatial Directions

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.