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On stability of generalized phase retrieval and generalized affine phase retrieval
- Zhitao Zhuang^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-019-1968-z
© The Author(s) 2019
- Received: 8 November 2018
- Accepted: 15 January 2019
- Published: 18 January 2019
Abstract
In this paper, we consider the stability of intensity measurement mappings corresponding to generalized phase retrieval and generalized affine phase retrieval in the real case. First, we show the bi-Lipschitz property on measurements of noiseless signals. After that, the stability property as regards a noisy signal is given by the Cramer–Rao lower bound.
Keywords
- Bi-Lipschitz
- Generalized affine phase retrieval
- Cramer–Rao lower bound
MSC
- 42C15
1 Introduction
Given two vectors \(x,y \in F^{d}\), we define metrics \(d(x,y)= \Vert {x-y} \Vert \), \(d_{1}(x,y)=\min\{ \Vert {x-y} \Vert , \Vert {x+y} \Vert \} \) and matrix metric \(d_{2}(x,y)= \Vert {x+y} \Vert \Vert {x-y} \Vert \) corresponding to the nuclear norm. Several robustness bounds to the probabilistic phase retrieval problem in a real case are given in [7]. Stability bounds of a reconstruction for a deterministic frame are studied in [3, 4] with appropriate metrics.
Our study mainly focuses on the stability of generalized phase retrieval and generalized affine phase retrieval in real case in two aspects. The first one addresses the bi-Lipschitz property of generalized phase retrieval. Section 2 shows that the mappings \(M_{A} \) and \(\sqrt{M_{A}} \) all have the bi-Lipschitz property with respect to an appropriate metric. However, the generalized affine phase retrieval mappings \(M_{B,b} \) and \(\sqrt{M_{B,b}} \) only can be controlled by two metrics. The second aspect deals with the Cramer–Rao lower bound of generalized phase retrieval and generalized affine phase retrieval in an additive white Gaussian noise model. The Cramer–Rao lower bound of any unbiased estimator is given by calculating the Fisher information matrix.
2 Stability of generalized phase retrieval
Lemma 2.1
Suppose \(A= \{A_{j}\}_{j=1}^{N} \) is a collection of Hermitian matrices in \(H_{d}(F) \). Then for any \(x \neq0 \), the collection \(\{A_{j}x\}_{j=1}^{N} \) forms a frame for \(F^{d} \) if and only if \(a_{0}>0 \) and \(b_{0}<+\infty\). In this case, (2.1) holds for every \(x,y\in F^{d}\).
For \(A=\{A_{j}\}_{j=1}^{N} \subset H_{d}(\mathbb {R}) \), Yang Wang and Zhiqiang Xu [17] proved that A is phase retrievable if and only if \(\{A_{j}x\}_{j=1}^{N} \) is a frame of \(\mathbb {R}^{d} \) for any nonzero \(x\in \mathbb {R}^{d} \). This incorporating Lemma 2.1 leads to the following theorem.
Theorem 2.1
Let \(A=\{A_{j}\}_{j=1}^{N} \subset H_{d}(\mathbb {R}) \). Then A is phase retrievable if and only if \(a_{0}>0 \) and \(b_{0}<+\infty\).
Since \(| \langle A_{j}x, y \rangle|^{2}=y^{T}A_{j}xx^{T}A_{j}y \) in real case, the above theorem can be rewritten in the quadratic forms as follows.
Corollary 2.1
Theorem 2.2
Suppose \(A_{j}=B_{j}^{T}B_{j}\in H_{d}(\mathbb {R}) \) is a positive semidefinite matrix and \(B_{j}^{T}=(b_{j,1},\ldots,b_{j,r_{j}}) \) for \(j=1,\ldots, N \). If \(\{A_{j}\}_{j=1}^{N} \) is generalized phase retrievable, then the column vectors \(\{b_{j,i}\}_{i=1,j=1}^{r_{j},N} \) satisfy the complementary property and therefore become a phase retrievable frame.
Proof
2.1 Bi-Lipschitz property
In this subsection, we consider the bi-Lipschitz property of tmappings \(M_{A} \) and \(\sqrt{M_{A}} \). At first, we show the stability of the mapping \(M_{A} \) with respect to the metric \(d_{2} \).
Theorem 2.3
Let \(\{A_{j}\}_{j=1}^{N} \subset H_{d} (\mathbb {R}) \) be generalized phase retrievable. Then \(M_{A} \) is bi-Lipschitz with respect to matrix metric \(d_{2}(x,y)= \Vert {x+y} \Vert \Vert {x-y} \Vert \).
Proof
Now, we consider the stability of the mapping \(\sqrt{M_{A}} \) with respect to the metric \(d_{1} \).
Lemma 2.2
Let \(\{A_{j}\}_{j=1}^{N} \subset H_{d}(\mathbb {R}) \) be a collection of positive semidefinite matrices and generalized phase retrievable. Then \(\sqrt{ M_{A}} \) is upper bounded with respect to the metric \(d_{1}(x,y)=\min\{ \Vert {x+y} \Vert , \Vert {x-y} \Vert \} \).
Proof
Lemma 2.3
Let \(\{A_{j}\}_{j=1}^{N} \subset H_{d}^{N}(\mathbb {R}) \) be generalized phase retrievable and positive semidefinite. Then \(\sqrt{ M_{A}} \) is lower bounded with respect to the metric \(d_{1}(x,y)=\min\{ \Vert {x+y} \Vert , \Vert {x-y} \Vert \} \).
Proof
Our discussion of the bi-Lipschitz property of \(\sqrt{M_{A}} \) is summarized in the following theorem by combining Lemma 2.2 and Lemma 2.3.
Theorem 2.4
Phase retrieval by projections introduced by Cahill et al. in [6] aims at recovering a signal from measurements consisting of norms of its orthogonal projections onto a family of subspaces. Since \(x^{T}P_{j}x= \Vert {P_{j}x} \Vert ^{2} \) when \(P_{j} \) is a projection to an appropriate subspace of \(\mathbb {R}^{d} \), phase retrieval by projections is a special case of generalized phase retrieval with \(A_{j}=P_{j} \). Therefore, Theorem 2.3 and Theorem 2.4 also hold for phase retrieval by projections. In this special case, \(\lambda_{1} \) can be upper bounded by N and C equals one.
2.2 Cramer–Rao lower bound
Theorem 2.5
Corollary 2.2
3 Stability of generalized affine phase retrieval
The standard affine phase retrieval introduced by Bing Gao et al. in [9] can be used for recovering signals with prior knowledge. In this section, we consider generalized affine phase retrieval theoretically and give some basic mathematical properties at first, then we focus on its stability property.
Theorem 3.1
- (A)
The pair \((B,b) \) is generalized affine phase retrievable for \(\mathbb {R}^{d} \).
- (B)
There exist no nonzero \(u \in \mathbb {R}^{d} \) such that \(\langle B_{j}u, B_{j}v+b_{j} \rangle=0 \) for all \(1\leq j \leq N \) and \(v\in \mathbb {R}^{d} \).
- (C)
If v is the solution of equations \(B_{j}v+b_{j}=0 \) for \(j\in S\subset\{1,2,\ldots,N\} \), then \(\{B_{j}^{T}B_{j}v+B_{j}^{T}b_{j}\}_{j\in S^{C}} \) is a spanning set of \(\mathbb {R}^{d} \).
- (D)
The Jacobian of \(M_{B,b} \) has rank d everywhere on \(\mathbb {R}^{d}\).
Proof
(B) ⇔ (C). Assume \(\{B_{j}^{T}B_{j}v+B_{j}^{T}b_{j}\}_{j\in S^{C}} \) is not a spanning set of \(\mathbb {R}^{d} \), then there is a nonzero vector \(u\in \mathbb {R}^{d} \) such that \(\langle B_{j}u, B_{j}v+b_{j} \rangle= \langle u, B_{j}^{T}B_{j}v+B_{j}^{T}b_{j} \rangle=0 \) for \(j\in S^{C} \). For \(j\in S \), since v is the solution of equations \(B_{j}v+b_{j}=0 \), the inner product \(\langle B_{j}u, B_{j}v+b_{j} \rangle \) also equals zero, which contradicts (B). The converse can be proven similarly.
Minimality problems have attracted much attention from different areas recently. For generalized affine phase retrieval, the answer is related to different constraints on \(B_{j} \), \(b_{j} \) and prior knowledge of signal x. The following theorem is given in [10].
Theorem 3.2
([10])
Let \(N\geq2d \) and \(r>1 \). Then a generic \(\{(B_{j},b_{j})\}_{j=1}^{N} \subset \mathbb {R}^{r\times(d+1)} \) has the generalized affine phase retrieval property in \(\mathbb {R}^{d} \).
Let \(r=\max_{j} r_{j} \). The \(r_{j}\times(d+1) \) matrix \((B_{j},b_{j}) \) in Theorem 3.1 can be extended to \(r\times(d+1) \) matrix by filling with zero rows. The extended matrix can be viewed as an affine phase retrieval matrix where all \(r_{j}=r \) and hence leads to the following corollary by Theorem 3.2.
Corollary 3.1
Let \(\tilde{A}_{j}=(B_{j}^{T},b_{j}^{T})^{T}(B_{j},b_{j})\), where \(b_{j}\in \mathbb {R}^{r_{j}} \) and \(B_{j}\in \mathbb {R}^{r_{j}\times d} \) is a nonzero matrix. If \(N\geq2d \) and \(\tilde{A}=(\tilde{A}_{j})_{j=1}^{N} \) is a generic set in \(H_{d}^{N}(\mathbb {R}) \), Then the pair \((B,b) \) is generalized affine phase retrievable.
Example 3.1
Now, we consider the stability of generalized affine phase retrieval. Let \(\tilde{B}_{j}=(B_{j},b_{j}) \), \(\tilde{A}_{j}=\tilde{B}_{j}^{T}\tilde{B}_{j} \), and \(\tilde{x}=(x^{T},1)^{T} \). We have the following theorem.
Theorem 3.3
Proof
In contrast to Theorem 4.1 in [9], Theorem 3.4 leads to a slack constraint of the signal from a compact set to \(\mathbb{R}^{d} \). Although affine phase retrieval is not bi-Lipschitz with respect to one metric, the mappings \(M_{B,b} \) and \(\sqrt{M_{B,b}} \) is bounded by two metrics.
Lemma 3.1
If the pair \((B,b) \) is generalized affine phase retrievable, then the Fisher information \(R_{x}^{a} \) is positive definite for any \(x\in \mathbb {R}^{d} \).
Proof
Similar to generalized phase retrieval, we have the following theorem.
Theorem 3.4
Lemma 3.2
If the pair \((B,b) \) is generalized affine phase retrievable, then the collection \(\{B_{j}^{T}B_{j}\}_{j=1}^{N} \) is a g-frame for \(\mathbb {R}^{d} \).
Proof
Corollary 3.2
Proof
We discussed the stability of generalized phase retrieval and affine generalized phase retrieval in this paper. The first one can be viewed as a generalization of stability of phase retrieval in [3, 4], or as a continuation of the work in [17]. The second one is an extension of the work in [9, 10]. As all the results in this paper are obtained in real Hilbert space, the stability property in complex Hilbert space still needs to be addressed.
Declarations
Acknowledgements
The authors would like to thank the referees for their useful comments and remarks.
Availability of data and materials
Not applicable.
Funding
This study was partially supported by National Natural Science Foundation of China (Grant No. 11601152).
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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