Spectral Initialization for High-Dimensional Phase Retrieval with Biased Spatial Directions
<|reference_start|>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.<|reference_end|>
@article{bousseyroux2024spectral,
    title={Spectral Initialization for High-Dimensional Phase Retrieval with Biased
  Spatial Directions},
    author={Pierre Bousseyroux and Marc Potters},
    journal={arXiv preprint arXiv:2403.15548},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.15548},
    primaryClass={cond-mat.dis-nn cs.IR math.PR}
}