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On stability of generalized (affine) phase retrieval in the complex case
Journal of Inequalities and Applications volume 2019, Article number: 316 (2019)
Abstract
In this paper, we discuss the stability of generalized phase retrieval and generalized affine phase retrieval in the complex case. By the realification method, we obtain the bi-Lipschitz property in the absence of noise case and Cramer–Rao lower bound under noise conditions.
1 Introduction
The problem of phase retrieval aims to recover a signal from its intensity measurements, it naturally arises in various fields of physics, such as crystallography [14], radar [12], electron-microscopy [11]. Most of the successful instances are corresponding to two-dimensional signals. For one-dimensional signals, the uniqueness and algorithms of Fourier phase retrieval problems are discussed in [5]. The frame-based phase retrieval, which addresses the signal reconstruction from the absolute value of frame coefficients, was introduced by Balan et al. [2]. It is crucial that the measurement mapping is of bi-Lipschitz property, since it preserves the density and dispersion point [6]. The Cramer–Rao lower bound (CRLB) is the lower bound on the variance of any unbiased estimation, which allows us to assert that an estimator is minimum variance unbiased estimator. The bi-Lipschitz property and CRLB of frame-based phase retrieval is given in [1, 3, 4]. Generalized phase retrieval (GPR) is introduced by Yang Wang and Zhiqiang Xu [16], which unifies and enhances results from the standard phase retrieval, phase retrieval by projections, and low-rank matrix. Affine phase retrieval aims to recover signals from the magnitudes of affine measurements [8]. Necessary and sufficient conditions as well as minimal number of measurements are given in [10]. The bi-Lipschitz property and CRLB of GPR and GAPR for real signals are discussed in [17]. However, the GPR and GAPR problems with complex signals are also encountered frequently in some fields like optics [15], quantum information [9], interferometry [7], which leads us to addressing the complex case in this paper.
Let \(H_{n}(\mathbb {C}) \) denote the set of \(n \times n \) Hermitian matrices over complex field \(\mathbb {C}\). For any given matrix sequence \(A = \{A_{j}\}^{m}_{j =1}\subset H_{n}(\mathbb {C}) \), define the map \(M_{A} : \mathbb {C}^{n}\rightarrow\mathbb{R}^{m} \) by
where \(x^{*} \) denotes the conjugate transpose of x. We say that A is generalized phase retrievable if \(M_{A} \) is injective up to a global phase, which means
Similarly, if \(A_{j} \) is positive semidefinite for \(j=1,\dots,m \), we can define the map \(\sqrt{M_{A}} : \mathbb {C}^{n}\rightarrow\mathbb{R}^{m} \) by
Let \(B(\mathbb {C}^{n}) \) and \(B(\mathbb {R}^{2n}) \) denote the sets of bounded linear operators on \(\mathbb {C}^{n} \) and \(\mathbb {R}^{2n} \) respectively. For any \(T\in B(\mathbb {C}^{n}) \), the nuclear norm of T denoted by \(\lVert{T} \rVert_{1} \) is given by the 1-norm of its singular values. We still denote the operator norm of T by \(\lVert{T} \rVert\). Given two vectors \(x,y \in \mathbb {C}^{n}\), we define metrics \(d(x,y)= \lVert{x-y} \rVert\), \(d_{1}(x,y)= \min_{|\alpha|=1} \lVert{x-\alpha y} \rVert\) and matrix metric
which is corresponding to the nuclear norm.
Let \(B_{j} \in \mathbb {C}^{r_{j}\times n} \) and \(b_{j} \in \mathbb {C}^{r_{j}} \), where \(r_{j} \) is a positive integer. The GAPR problem aims to recover a signal \(x\in \mathbb {C}^{n} \) from the norms of the affine linear measurements \(\{ \lVert{B_{j}x+b_{j}} \rVert\}_{j=1}^{m} \). Let \(B=\{B_{j}\}_{j=1}^{m} \) and \(b=\{b_{j}\}_{j=1}^{m} \). We define the map \(M_{B,b}:\mathbb {C}^{n} \rightarrow \mathbb {R}^{m} \) by
The pair \((B,b) \) is said to be generalized affine phase retrievable for \(\mathbb {C}^{n} \) if \(M_{B,b} \) is injective on \(\mathbb {C}^{n} \). Note that if \((B,b) \) is generalized affine phase retrievable, one can recover the signal x exactly but not up to a global phase.
Our study mainly focuses on the stability of GPR and GAPR in the complex case. In Sect. 2, we establish the bi-Lipschitz inequalities of GPR and GAPR with appropriate metrics. In Sect. 3, we present the Cramer–Rao lower bound of noised GPR and GAPR.
2 Bi-Lipschitz stability
In this section, we discuss the bi-Lipschitz property of GPR and GAPR in the complex case. Realification of the complex vectors and operators is our main method to deal with the phase retrieval problems in the complex case. We consider the \(\mathbb{R} \)-linear map \(\mathbf{j} : \mathbb{C}^{n} \rightarrow\mathbb{R}^{2 n} \) defined by
where \(\Re(z) \) and \(\Im(z) \) are the real part and the imaginary part of z respectively. Then, for any \(x, y \in\mathbb{R}^{n} \), the inverse operator of j is given by
We transfer operators in \(B(\mathbb {C}^{n}) \) to operators in \(B(\mathbb {R}^{2n}) \) by
As in reference [1], for any two vectors \(u, v \in \mathbb {C}^{n} \), we define their symmetric outer product by
Let \(I_{n} \) be the \(n \times n \) identity matrix and
Then the transpose of J is \(J^{T}=-J \), and by direct computation we have
where \(\xi=\mathbf {j}(u) \), \(\eta=\mathbf {j}(v) \), and \([\![ \xi, \eta]\!]=\frac{1}{2}(\xi\eta^{T}+\eta\xi^{T}) \).
Denote the real part and the imaginary part of a complex matrix \(A_{j}\in H_{n}(\mathbb {C}) \) by \(D_{j} \) and \(C_{j} \) respectively. Then we have \(A_{j}=D_{j}+iC_{j} \), \(D_{j}^{T}=D_{j} \), \(C_{j}^{T}=-C_{j} \), and
Let \(\operatorname {Tr}(A_{j}) \) be the trace of \(A_{j} \). Then it is only related to the real part of \(A_{j} \) since \(\operatorname {Tr}(A_{j})=\operatorname {Tr}(D_{j}+iC_{j})=\operatorname {Tr}(D_{j}) \). Furthermore, the trace of the realification of \(A_{j} \) can be computed by \(\operatorname {Tr}(\tau(A_{j}))=2\operatorname {Tr}(D_{j})=2\operatorname {Tr}(A_{j}) \). Let \(\langle T, G \rangle_{\mathrm{HS}}=\operatorname {Tr}(TG^{*}) \) be the Hilbert–Schmidt inner product of T and G. Since the trace of the product of a symmetric matrix and an antisymmetric matrix equals zero, one can easily prove that
Since \(J^{T}\tau(A_{j})J=\tau(A_{j}) \), we have
Therefore, we obtain the relationship before and after the realification:
Since \(\tau(A_{j}) \) is symmetric, the Hilbert–Schmidt inner product can be simplified as follows:
It follows that
Let
The summation can be written as
The following theorem gives an equivalent condition for a set of matrices to be phase retrievable.
Theorem 2.1
([16])
Let \(A= \{A_{j} \}_{j=1}^{m} \subset\mathbf {H}_{n}(\mathbb{C}) \). The following are equivalent:
- (1)
A has the phase retrieval property.
- (2)
There exist no \(v, u \neq0 \)in \(\mathbb{C}^{n} \)with \(u \neq i c v \)for any \(c \in\mathbb{R} \)such that \(\Re (v^{*} A_{j} u )=0\)for all \(1 \leq j \leq m \).
2.1 Stability of GPR
In this subsection, we discuss the bi-Lipschitz property of GPR in the complex case. For a set \(A= \{A_{j} \}_{j=1}^{m} \subset\mathbf{H}_{n}(\mathbb{C})\), we define the map \(\mathcal{A }\) on the set \(B(\mathbb {C}^{n}) \) by
We denote by \(S^{1,1} \) the set of Hermitian matrices that have at most one positive eigenvalue and at most one negative eigenvalue. Then we have the following equivalent conditions for the set A to be phase retrievable.
Theorem 2.2
Let \(A= \{A_{j} \}_{j=1}^{m} \subset\mathbf{H}_{n}(\mathbb{C}) \), then the following are equivalent:
- (1)
Ahas the phase retrieval property.
- (2)
\(\ker(\mathcal{A}) \cap S^{1,1}=\{0\} \).
- (3)
There is a constant \(a_{0}>0 \)such that, for every \(u,v \in \mathbb {C}^{n} \),
$$ \sum_{j=1}^{m} \bigl( \Re \bigl(v^{*} A_{j} u \bigr) \bigr)^{2}\geq a_{0} \bigl[ \Vert u \Vert ^{2} \Vert v \Vert ^{2} -\bigl( \Im\langle u, v\rangle\bigr)^{2} \bigr] =a_{0} \bigl\lVert { [\![ u, v ]\!]} \bigr\rVert _{1}^{2}. $$(2.3) - (4)
There is a constant \(a_{0}>0 \)such that, for all nonzero \(\xi\in \mathbb {R}^{2n} \),
$$ R(\xi) \geq a_{0} \Vert \xi \Vert ^{2} P_{J \xi}^{\perp}, $$(2.4)where the inequality is in the sense of quadratic forms and \(P_{J \xi}^{\perp}=I-\frac{1}{\|\xi\|^{2}} J \xi\xi^{T} J^{T} \).
- (5)
For any nonzero \(\xi\in \mathbb {R}^{2n} \), \(\operatorname {rank}(R(\xi))=2n-1 \).
Proof
(1) ⇔ (2). If there exists a nonzero operator \(T\in \operatorname{ker}(\mathcal{A}) \cap S^{1,1} \), by [1, Lemma 3.7], there exist nonzero vectors \(u,v \) such that \(T=[\![ u, v ]\!] \). Since \(T\ne0 \), we have \(u\ne icv \) for any \(c\in \mathbb {R}\). Furthermore, \(T\in\ker(\mathcal{A}) \) implies
which contradicts (2) of Theorem 2.1. The necessity can be proved similarly.
(3) ⇒ (2). If \(T\in\operatorname{ker}(\mathcal{A}) \cap S^{1,1} \), then there exist vectors u, v such that \(T=[\![ u, v ]\!]\) and
which means \(T=0 \).
(2) ⇒ (3). Since \(\operatorname{ker}(\mathcal{A}) \cap S^{1,1}=\{0\} \), we have \(\lVert{\mathcal{A}(T)} \rVert^{2}= \sum_{j=1}^{m}| \langle A_{j}, [\![ u, v ]\!] \rangle_{\mathrm{HS}}|^{2} >0 \) for all nonzero \(T=[\![ u, v ]\!]\in S^{1,1} \). Let
Since \(S^{1,1} \) is a cone in \(B(\mathbb {C}^{n}) \), the homogeneity and continuousness of mapping \(T\rightarrow\sum_{j=1}^{m} | \langle A_{j}, T \rangle_{\mathrm{HS}}|^{2} \) implies
By [1, Lemma 3.8], we have
Substituting \(\langle A_{j}, [\![ u, v ]\!] \rangle_{\mathrm{HS}}= \Re(v^{*}A_{j}u) \) and (2.6) to (2.5), we get the desired inequality.
(3) ⇔ (4). By formula (2.1), we have
Hence (2.3) is equivalent to
Since \(\|u\|=\|\xi\|,\|v\|=\|\eta\| \), and \((\Im \langle u, v \rangle)^{2}=\eta^{T} J\xi\xi^{T}J^{T}\eta\), we have
Substituting (2.2) and (2.8) to (2.7), we obtain the desired inequality.
(4) ⇔ (5). The symmetry of \(D_{j} \) and the antisymmetry of \(C_{j} \) imply \(\xi^{T}\tau(A_{j})J\xi=0 \). As a result, we have
Considering that \(R(\xi) \) is a \(2n\times2n \) real matrix, the rank of \(R(\xi) \) can not be greater than \(2n-1 \). If it is less than \(2n-1 \), then there exists a nonzero vector η such that \(\langle\eta, J\xi \rangle=0 \) and \(R(\xi)\eta=0 \). Consequently, we have \(\eta^{T}R(\xi)\eta=0 \) and
which contradicts (2.4). This proves (5). Conversely, assume \(\operatorname {rank}(R(\xi))=2n-1 \) for all \(\xi\neq0 \). Let \(a(\xi) \) be the smallest nonzero eigenvalue of \(R(\xi) \). Then we have \(R(\xi)\geq a(\xi)P_{J\xi}^{\perp}\). Define \(a_{0}=\min_{ \lVert{\xi} \rVert=1}a(\xi) \). Then the constant \(a_{0}>0 \). By the homogeneity of \(R(\xi) \), we have \(a(\xi) =a_{0} \lVert{\xi } \rVert^{2}\). This proves (4). □
Theorem 2.3
Let \(\{A_{j}\}_{j=1}^{m} \subset H_{n} (\mathbb {C}) \)be generalized phase retrievable. Then \(M_{A} \)is bi-Lipschitz with respect to metric \(\lVert{xx^{*}-yy^{*}} \rVert_{1}^{2} \)with the upper Lipschitz bound
and the lower Lipschitz bound
where \(a_{2n-1}(R(\xi)) \)is the smallest nonzero eigenvalue of \(R(\xi) \).
Proof
For any \(x,y\in \mathbb {C}^{n} \), the definition of \(M_{A} \) gives
Substituting \(u=x+y \) and \(v=x-y \) into the above equation and applying (2.1), we have
where \(\xi=\mathbf {j}(u) \), \(\eta=\mathbf {j}(v) \). As shown in (2.8), \(\|[\![ u, v ]\!]\|_{1}^{2}= \|\xi\|^{2}\langle P_{J \xi}^{\perp} \eta, P_{J \xi}^{\perp} \eta\rangle\) holds true. This implies that, for \(\eta\in\{ \operatorname {span} \{J\xi\} \}^{\perp}\), we have
Since \(\langle R(\xi)\eta, \eta \rangle=0 \) and \(\|[\![ u, v ]\!]\|_{1}^{2}=0 \) for \(\eta\in \operatorname {span} \{J\xi\} \), we conclude that
Similarly, we have
□
The following lemma states the relationship between the metrics \(d_{1}\) and \(d_{2} \) defined in Sect. 1.
Lemma 2.1
For any \(x,y \in \mathbb {C}^{n} \)with \(\lVert{x} \rVert+ \lVert{y} \rVert\neq0 \), the metrics \(d_{1}\)and \(d_{2} \)have the relationship
Proof
Take \(\alpha_{0}= \alpha_{0}(x,y)= \frac{\langle x, y\rangle}{ |\langle x, y \rangle|}\). Then the modulus of \(\alpha_{0} \) equals one and
The parallelogram law gives \(\Vert x-\alpha_{0} y \Vert ^{2}+ \Vert x+\alpha_{0} y \Vert ^{2}=2 (\|x\|^{2}+\| y\|^{2} )\). It follows that
Since \(\langle x, \alpha_{0} y \rangle =|\langle x, y \rangle|\), direct computation gives
Combining the above inequalities, we get the relationship of two metrics:
□
With Lemma 2.1 in hand, one can prove the bi-Lipschitz property of map \(\sqrt{M_{A}} \) by the similar process of Theorem 2.4 in [17].
Theorem 2.4
Let \(\{A_{j}\}_{j=1}^{m} \subset H_{n}(\mathbb {C}) \)be generalized phase retrievable and all \(A_{j} \)be positive semidefinite. Then \(\sqrt{ M_{A}} \)is bi-Lipschitz with respect to metric \(d_{1}(x,y)=\min_{\alpha=1} \{ \lVert{x-\alpha y} \rVert \} \)as follows:
whereCis the uniform upper operator bound for \(\{A_{j}\}_{j=1}^{m} \)and \(\lambda_{1} \)is the maximum eigenvalue of matrix \(\sum_{j=1}^{m}A_{j} \).
2.2 Stability of GAPR
In this subsection, we discuss the bi-Lipschitz property of GAPR in the complex case. We show that the bi-Lipschitz bound is related to two metrics.
Theorem 2.5
Let \(\tilde{A}_{j}=(B_{j}^{*},b_{j}^{*})^{*}(B_{j},b_{j})\). Suppose that \(\tilde{A}=\{\tilde{A}_{j}\}_{j=1}^{m} \)is a generic set with \(m\geq4n \), then \((B, b) \)is generalized affine phase retrievable. Furthermore, there exist positive constants \(c_{0}\), \(c_{1}\), \(C_{0}\), \(C_{1} \)depending on \((B, b)\)such that, for any \(x,y \in \mathbb {C}^{n} \),
Proof
Since à is generic, so is \((B,b) \). By Theorem 4.3 in [16], the pair \((B,b) \) is generalized affine phase retrievable. Notice that the equation \(\lVert{B_{j}x+b_{j}} \rVert^{2}= {\tilde{x}}^{*}\tilde{A}\tilde{x}\) implies that the equation \(M_{B,b}(x)=M_{\tilde{A}}(\tilde{x}) \) holds true for \(\tilde{x}=(x^{*},1)^{*} \) with \(x\in \mathbb {C}^{n} \). Combining with Theorem 2.3, we have
where the symbol “≃” denotes the bi-Lipschitz relationship. Direct computation of the metric yields
Therefore, there exist constants \(c_{0}\), \(c_{1} \) such that (2.9) holds. Similarly, by Theorem 2.4, we have
Since \(d_{1}^{2}(\tilde{x},\tilde{y})= \min_{|\alpha|=1} \{ \lVert{x-\alpha y} \rVert^{2}+|1-\alpha|^{2}\} \), we have the inequities
Therefore, there exist constants \(C_{0}\), \(C_{1} \) such that (2.10) holds. □
3 Cramer–Rao stability
In this section, we discuss the stability of GPR and GAPR in the noised measurement case. Given signal \(x\in \mathbb {R}^{n} \) and \(\varphi(x) \) is a real differentiable vector-valued function of x. Assume that the measurement has the form \(Y=\varphi(x)+Z \), where the entries of Z are independent Gaussian random variables with mean value 0 and variance \(\sigma^{2} \). The noised generalized phase retrieval problem is to estimate x from measurement Y. In this scenario, we apply the Fisher information theory to evaluate the Cramer–Rao lower bound of any unbiased estimator of signal x. The Fisher information matrix is defined entrywise by
where \(p(y;x) \) is the probability density function of random vector Y with vector parameter x and the expectation \(\mathbb {E}\) is taken with respect to \(p(y;x) \), resulting in a function of x only. As assumption, Y is a random vector with probability density function
By some regular computations, the Fisher information matrix entry \((\mathbb {I}(x))_{m,\ell} \) equals
where \(\varphi_{j}(x) \) is the jth element of \(\varphi(x) \).
We are now ready to state the Cramer–Rao lower bound theorem.
Theorem 3.1
It is assumed that the probability density function \(p(y;x) \)satisfies conditions in an open set \(\varTheta\subset \mathbb {R}^{n} \)as follows:
- (1)
For any \(y\in\mathfrak{X} \)and \(x\in\varTheta\), the probability density function \(p(y;x)>0 \);
- (2)
The derivative \(\partial p(y;x)/\partial x_{j} \)exists and
$$\int_{\mathfrak{X}}\frac{\partial p(y;x)}{\partial x_{j}}\,dy=0,\quad 1\le j \le n; $$ - (3)
The expectation \(\mathbb {E}\vert \frac{\partial\log p(y;x)}{\partial x_{i}} \frac{\partial \log p(y;x)}{\partial x_{j}} \vert \)is finite and the Fisher information matrix is positive semidefinite.
For any unbiased estimator \(\hat{\delta}(y) \)of a differentiable function \(g(x) \), if integration and differentiation byxcan be interchanged in \(\int_{\mathfrak{X}}\hat{\delta}_{j}(y)p(y;x)\,dy \), then we have
where \(\mathbb {I}^{\dagger}(x) \)is the Moore–Penrose inverse of the Fisher information matrix \(\mathbb {I}(x) \)and \(D(x) \)is the matrix with elements \(\partial g_{i}(x)/\partial x_{j} \).
Proof
Define the vector-valued function \(S=(S_{1},S_{2},\ldots,S_{n})^{T} \), where \(S_{j}= \partial p(y;x)/\partial x_{j} \). Condition (2) implies that \(\mathbb {E}[S]=0 \). By condition (3), the covariance matrix \(\operatorname {Cov}(S) \) exists and is exactly the Fisher information matrix \(\mathbb {I}(x) \). The condition assumed on the estimator δ̂ gives
Therefore, we have
Since \(\mathbb {I}(x) \) is positive semidefinite and the set Θ is open, we have the desired inequality (3.2). In fact, if this is not the case, there exists a nonzero vector ξ such that
By taking \(\eta=(\xi^{T}, (-\mathbb {I}^{\dagger}(x)D(x)^{T}\xi)^{T})^{T} \), matrix multiplication shows that
This contradicts . □
3.1 Cramer–Rao lower bound of GPR
Since we consider phase retrieval of complex signals, the above theorem can not be used directly. We still deal with this by realification of complex vectors and matrices. Let \(\xi=\mathbf {j}(x) \) be the realification of x. Then we have \(\varphi(x)=x^{*}A_{j}x=\xi^{T}\tau(A_{j})\xi\). We use ξ to take the place of x in formula (3.1). Consequently, we get the Fisher information matrix
In order to obtain a unique solution, we make an assumption of the signal x introduced in [1]: the signal x satisfies \(\langle x, z_{0} \rangle>0 \) for a fixed normalized vector \(z_{0}\in \mathbb {C}^{n} \). Define \(H_{z_{0}}=\{\xi=\mathbf {j}(x): \langle x, z_{0} \rangle >0,x\in \mathbb {C}^{n}\} \). We have the Cramer–Rao lower bound of the noised GPR problem as follows.
Theorem 3.2
Assume that the nonlinear map \(M_{A} \)is injective and fix a vector \(z_{0} \in \mathbb {C}^{n}\). For any vector \(x \in \mathbb {C}^{n} \)with \(\langle x, z_{0} \rangle> 0 \), the covariance of any unbiased estimatorδ̂of x is bounded below by the Cramer–Rao lower bound given by
where † denotes the Moore–Penrose pseudoinverse operator. In particular, the mean-square error ofδ̂is bounded below by
Proof
By taking \(g(\xi)=\xi\), inequality (3.3) is the direct result of applying Theorem 3.1 to \(\mathbf {j}(\hat{\delta}) \), the realification of δ̂. Taking trace of inequality (3.3) and considering \(\|x-\hat{\delta}\|= \|\xi-\mathbf {j}(\hat{\delta})\| \), we get inequality (3.4). □
3.2 Cramer–Rao lower bound of GAPR
In this subsection, we need not make an extra assumption to guarantee the uniqueness since the signal can be retrieved exactly with the generalized affine phase retrievable set. Furthermore, one can prove that if the pair \((B,b) \) is generalized affine phase retrievable, then the collection \(\{B^{T}_{j} B_{j}\}^{m}_{ j=1} \) is a g-frame for \(\mathbb {C}^{n} \). We denote the upper frame bound by Δ, which means
In the noised GAPR problem, we have
Define \(R_{x}^{a}=\sum_{j=1}^{m} B_{j}^{T} (B_{j} x+b_{j} ) (B_{j} x+b_{j} )^{T} B_{j} \). By direct computation, the Fisher information matrix of the noised GAPR problem is given by
Theorem 3.3
The Fisher information matrix for the noised generalized affine phase retrieval model is \(\frac{4}{\sigma^{2}}\tau(R_{x}^{a}) \). Consequently, for any unbiased estimatorδ̂ofx, the covariance matrix is bounded below by the Cramer–Rao lower bound as follows:
Therefore, the mean square error of any unbiased estimatorδ̂is given by
Proof
By applying Theorem 3.2 in [13] to the realification model \(Y=\varphi(x)+Z \), we get inequality (3.5) directly. Taking trace of the two sides of inequality (3.5) yields
Since \(\operatorname {Tr}(Q) \cdot \operatorname {Tr}(Q^{-1}) \geq n^{2}\) holds for any invertible \(n\times n \) matrix Q, we have
Furthermore, the g-frame property of the collection \(\{B^{T}_{j} B_{j}\}^{m}_{ j=1} \) gives
where \(C=\sum_{j=1}^{m} \lVert{B_{j}^{T}b_{j}} \rVert^{2} \). Substituting the above inequality into (3.6), we have
□
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The authors would like to thank the referees for their useful comments and remarks.
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This study was partially supported by the National Natural Science Foundation of China (Grant No. 11601152).
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Zhuang, Z. On stability of generalized (affine) phase retrieval in the complex case. J Inequal Appl 2019, 316 (2019). https://doi.org/10.1186/s13660-019-2270-9
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DOI: https://doi.org/10.1186/s13660-019-2270-9