# Analysis of Spectral Methods for Phase Retrieval with Random Orthogonal Matrices

## TL;DR

This paper analyzes the effectiveness of spectral initialization methods for phase retrieval when using random orthogonal matrices, providing precise asymptotic characterizations for practical measurement models.

## Contribution

It extends the theoretical understanding of spectral methods in phase retrieval to isotropically random orthogonal matrices, a more realistic model for practical systems.

## Key findings

- Derived a simple expression for the overlap between spectral estimator and true signal.
- Provided asymptotic analysis for large measurement and signal dimensions.
- Enhanced understanding of spectral initialization performance in practical measurement models.

## Abstract

Phase retrieval refers to algorithmic methods for recovering a signal from its phaseless measurements. Local search algorithms that work directly on the non-convex formulation of the problem have been very popular recently. Due to the nonconvexity of the problem, the success of these local search algorithms depends heavily on their starting points. The most widely used initialization scheme is the spectral method, in which the leading eigenvector of a data-dependent matrix is used as a starting point. Recently, the performance of the spectral initialization was characterized accurately for measurement matrices with independent and identically distributed entries. This paper aims to obtain the same level of knowledge for isotropically random column-orthogonal matrices, which are substantially better models for practical phase retrieval systems. Towards this goal, we consider the asymptotic setting in which the number of measurements $m$, and the dimension of the signal, $n$, diverge to infinity with $m/n = \delta\in(1,\infty)$, and obtain a simple expression for the overlap between the spectral estimator and the true signal vector.