The minimal measurement number for generalized conjugate phase retrieval
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https://doi.org/10.1007/s11425-020-1757-6
The minimal measurement number for generalized conjugate phase retrieval Wei Dan∗ College of Big Data and Internet, Shenzhen Technology University, Shenzhen 518118, China Email: [email protected] Received March 10, 2020; accepted August 6, 2020
Abstract
The generalized conjugate phase retrieval problem aims to reconstruct a complex signal x ∈ Cn
from quadratic measurements x∗ A1 x, . . . , x∗ Am x, where A1 , . . . , Am ∈ Rn×n are real symmetric matrices. The equivalent formulation for generalized conjugate phase retrieval along with the minimal measurement number required for accurate retrieval (up to a global phase factor as well as conjugacy) are derived in this paper. We present a set of nine vectors in R4 and prove that it is conjugate phase retrievable on C4 . This result implies the measurement number bound 4n − 6 is not optimal for some n, which confirms a conjecture in the article by Evans and Lai (2019). Keywords MSC(2010)
phase retrieval, measurement number, conjugate 15A63, 94A12, 42C15
Citation: Dan W. The minimal measurement number for generalized conjugate phase retrieval. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-020-1757-6
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Introduction
The phase retrieval problem [9, 10] has attracted enormous attention in numerous fields, such as X-ray imaging [5], crystallography [11, 16], astronomy [7], coded diffraction patterns [1], optics [18] and so on. Mathematically, we want to reconstruct a complex signal x ∈ Cn from the sampling measurements |⟨x, w1 ⟩|2 , |⟨x, w2 ⟩|2 , . . . , |⟨x, wm ⟩|2 with w1 , . . . , wm the measuring vectors in the finite dimensional Hilbert space Fn (F equals C or R) and where ⟨·, ·⟩ stands for the inner product. When F = C, it is the conventional complex phase retrieval problem. We refer interested readers to [2–4,14,17,19] and the references therein. When F = R, the phase retrieval problem on Cn by real vectors was considered in the recent publication [8]. At the moment, it is important to emphasize that |⟨eiθ x, wk ⟩| = |⟨x, wk ⟩| = |⟨eiθ x, wk ⟩| holds for any x ∈ Cn , θ ∈ [0, 2π) and 1 6 k 6 m. Here, x represents the entrywise conjugate. Therefore, we can only hope to recover the signal x ∈ Cn up to a global phase factor as well as conjugacy. This is the reason why it was called conjugate phase retrieval in [8, 13, 15]. c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Dan W
Sci China Math
The present article investigates the problem to recover x ∈ Rn from quadratic measurements LA (x) = (x∗ A1 x, . . . , x∗ Am x), where x∗ is the conjugate transpose of x and A = (Ak )m k=1 is a set of real symmetric matrices. We call this the generalized conjugate phase retrieval problem, because the standard conjugate phase retrieval problem can be viewed as a special case by placing restrictions Ak = wk wkT for all 1 6 k 6 m. Here, wkT denotes the transpose vector of wk . This article focuses on the following two questions abou
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