Spectral initialization for nonconvex estimation: High-dimensional limit and phase transitions
We study a simple spectral method that serves as a key ingredient in a growing line of work using efficient iterative algorithms for estimating signals in nonconvex settings. Unlike previous work, which focuses on the phase retrieval setting and provides only bounds on the performance, we consider arbitrary generalized linear sensing models and provide an exact characterization of the performance of the spectral method in the high-dimensional regime. Our analysis reveals a phase transition phenomenon that depends on the sampling ratio. When the ratio is below a critical threshold, the estimates given by the spectral method are no better than random guesses drawn uniformly from the hypersphere; above the threshold, however, the estimates become increasingly aligned with the underlying signal. Worked examples and numerical simulations are provided to illustrate and verify the analytical predictions.