# Phase Transitions of Spectral Initialization for High-Dimensional Nonconvex Estimation

## TL;DR

This paper analyzes the phase transition behavior of spectral initialization in high-dimensional nonconvex estimation, revealing thresholds for when the method provides meaningful signal estimates.

## Contribution

It offers a precise asymptotic characterization of spectral initialization performance across generalized linear models, extending beyond phase retrieval.

## Key findings

- Performance sharply transitions at specific sample-to-dimension ratios.
- Below threshold, estimates are no better than random.
- Above threshold, estimates align with the true signal.

## Abstract

We study a spectral initialization method that serves a key role in recent work on estimating signals in nonconvex settings. Previous analysis of this method focuses on the phase retrieval problem and provides only performance bounds. In this paper, we consider arbitrary generalized linear sensing models and present a precise asymptotic characterization of the performance of the method in the high-dimensional limit. Our analysis also reveals a phase transition phenomenon that depends on the ratio between the number of samples and the signal dimension. When the ratio is below a minimum threshold, the estimates given by the spectral method are no better than random guesses drawn from a uniform distribution on the hypersphere, thus carrying no information; above a maximum threshold, the estimates become increasingly aligned with the target signal. The computational complexity of the method, as measured by the spectral gap, is also markedly different in the two phases. Worked examples and numerical results are provided to illustrate and verify the analytical predictions. In particular, simulations show that our asymptotic formulas provide accurate predictions for the actual performance of the spectral method even at moderate signal dimensions.