Eigenvector Phase Retrieval: Recovering eigenvectors from the absolute value of their entries

TL;DR

This paper introduces randomized algorithms for recovering eigenvectors from the absolute values of their entries, leveraging phase retrieval techniques, with proven convergence for simple eigenvalues and improved methods for large eigenvalues.

Contribution

It presents novel randomized algorithms for eigenvector recovery from entry magnitudes, providing convergence guarantees and addressing cases with large eigenvalues.

Findings

Algorithm converges in expectation for simple eigenvalues

Recovery is easier when eigenvalue magnitude is large

Provides theoretical analysis of convergence properties

Abstract

We consider the eigenvalue problem $Ax = \lambda x$ where $A \in \mathbb{R}^{n \times n}$ and the eigenvalue is also real $\lambda \in \mathbb{R}$. If we are given $A$, $\lambda$ and, additionally, the absolute value of the entries of $x$ (the vector $(|x_i|)_{i=1}^n$), is there a fast way to recover $x$? In particular, can this be done quicker than computing $x$ from scratch? This may be understood as a special case of the phase retrieval problem. We present a randomized algorithm which provably converges in expectation whenever $\lambda$ is a simple eigenvalue. The problem should become easier when $|\lambda|$ is large and we discuss another algorithm for that case as well.