TY - JOUR

T1 - Eigenvector Phase Retrieval

T2 - Recovering eigenvectors from the absolute value of their entries

AU - Steinerberger, Stefan

AU - Wu, Hau tieng

N1 - Funding Information:
S.S. was partly supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation (#FG-2021-14114).
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/11/1

Y1 - 2022/11/1

N2 - We consider the eigenvalue problem Ax=λx where A∈Rn×n and the eigenvalue is also real λ∈R. If we are given A, λ and, additionally, the absolute value of the entries of x (the vector (|xi|)i=1n), 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 λ is a simple eigenvalue. The problem should become easier when |λ| is large and we discuss another algorithm for that case as well.

AB - We consider the eigenvalue problem Ax=λx where A∈Rn×n and the eigenvalue is also real λ∈R. If we are given A, λ and, additionally, the absolute value of the entries of x (the vector (|xi|)i=1n), 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 λ is a simple eigenvalue. The problem should become easier when |λ| is large and we discuss another algorithm for that case as well.

KW - Eigenvector phase retrieval

KW - Phase retrieval

KW - Synchronization

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U2 - 10.1016/j.laa.2022.08.002

DO - 10.1016/j.laa.2022.08.002

M3 - Article

AN - SCOPUS:85135702167

SN - 0024-3795

VL - 652

SP - 239

EP - 252

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -