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arXiv · 2109.11143

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

Stefan Steinerberger·Hau-Tieng Wu
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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.

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Stefan Steinerberger, Hau-Tieng Wu. 2021-09-23. Eigenvector Phase Retrieval: Recovering eigenvectors from the absolute value of their entries. https://arxiv.org/abs/2109.11143

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