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 -