On stability of generalized phase retrieval and generalized affine phase retrieval

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R E S E A R C H

Open Access

On stability of generalized phase retrieval

and generalized afﬁne phase retrieval

Zhitao Zhuang

1*

*_{Correspondence:} [email protected] 1_{School of Mathematics and}

Statistics, North China University of Water Resources and Electric Power, Zhengzhou, P.R. China

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.

MSC: 42C15

Keywords: Bi-Lipschitz; Generalized aﬃne phase retrieval; Cramer–Rao lower bound

1 Introduction

Given a signalx∈Fd(F=CorR), phase retrieval aims to recoverxfrom its intensity mea-surements|x,ϕi|,i= 1, . . . ,N, where{ϕi}N_i₌₁is a frame ofFd. The phase retrieval problem has a long history, which can be traced back as far as 1952 [13]. It has important applica-tions in optics, communication, X-ray crystallography, quantum tomography, signal pro-cessing and more (see e.g. [11,14,16] and the references therein). The task of classical phase retrieval is recovering a signal from its Fourier transform magnitude [5,8]. Using frame theory, Balan et al. constructed a new class of Parseval frames for a Hilbert space in 2006 [2], which allows signal reconstruction from the absolute value of the frame coeffi-cients. Since then lots of theoretical results and practical algorithms emerged in different fields. Generalized phase retrieval was introduced by Yang Wang and Zhiqiang Xu [17], including as special cases the standard phase retrieval as well as the phase retrieval by or-thogonal projection. Explicitly, letHd(F) denote the set ofd×dHermitian matrices overF (ifF=Rthen Hermitian matrices are symmetric matrices). As a standard phase retrieval problem, we consider the equivalence relation∼onFd:x1∼x2if there is a constantb∈F

with|b|= 1 such thatx1=bx2. LetFd:=Fd/∼. For any givenA={Aj}Nj=1⊂Hd(F), deﬁne the mapMA:Fd→RNby

MA(x) =

x∗A1x, . . . ,x∗ANx

,

wherex∗denotes the conjugate transpose ofx. We say thatAis generalized phase retriev-able ifMAis injective onFd. Similarly, ifAjis positive semideﬁnite forj= 1, . . . ,N, we can deﬁne the map√MA:Fd→RN by

MA(x) =

x∗A1x, . . . ,x∗ANx

.

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Affine phase retrieval, introduced by Bing Gao et al., aims to recover signal from the magnitudes of affine measurements. More precisely, instead of recovering x from {|x,ϕj|}N_j₌₁, they consider recoveringx from the absolute values of the affine intensity measurements

x,ϕj+bj, j= 1, . . . ,N,

whereϕj∈Fd_and_b

j∈C. In Sect.3, we consider generalized aﬃne phase retrieval and discuss its basic properties.

Given two vectorsx,y∈Fd, we deﬁne metricsd(x,y) =x–y,d1(x,y) =min{x–y, x+y}and matrix metricd2(x,y) =x+yx–ycorresponding 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. Section2shows that the mappingsMAand

√

MA all have the bi-Lipschitz property with respect to an appropriate metric. However, the generalized aﬃne phase retrieval mappingsMB,b and

√

MB,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

In this section, we discuss the bi-Lipschitz property and Cramer–Rao lower bound of gen-eralized phase retrieval. Given a collection of matrices{Aj}N

j=1⊂Hd(F), deﬁne

a0:= inf

x=y=1

N

j=1

x∗Ajy2 and b0:= sup

x=y=1

N

j=1

x∗Ajy2.

Assuming the collection of vectors{Ajx}Nj=1form a frame ofFdfor anyx= 0, then there

exist constants 0 <αx<βx< +∞such that

αxy2≤

N

j=1

x∗Ajy2≤βxy2, y∈Fd.

We chooseαxandβxto be the optimal frame bounds corresponding to{Ajx}Nj=1. Obviously,

we have

N

j=1 x∗Ajy

2

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for anyy= 0 andx= 0. Furthermore, the unit sphereS1(Fd_{) =}_{_x_:_x_{= 1,}_x_∈_Fd_}_is com-pact inFd. So isS1(Fd)×S1(Fd) inFd×Fd. Since the mapping

(x,y)−→ N

j=1 x∗Ajy

2

is continuous, it follows that

a0= inf

x=1αx=xinf=y=1

N

j=1

x∗Ajy2> 0

and

b0=sup

x=1

βx= sup

x=y=1

N

j=1 x∗Ajy

2

< +∞.

Conversely, supposea0> 0 andb0< +∞, then, for anyx= 0 andy= 0, we have

a0≤

N

j=1|x∗Ajy|2 x2_y2 =

N

j=1 Aj

x

x, y

y 2

≤b0.

This is equivalent to

a0x2y2≤

N

j=1

Ajx,y

2

≤b0x2y2, (2.1)

which means for any vectorx= 0,Ajxis a frame forFdwith frame boundsa0x2and

b0x2_{. Hence, we have proved the following lemma.}

Lemma 2.1 Suppose A={Aj}N

j=1is a collection of Hermitian matrices in Hd(F).Then for

any x= 0,the collection{Ajx}jN=1forms a frame for Fdif and only if a0> 0and b0< +∞.In this case, (2.1)holds for every x,y∈Fd_.

ForA={Aj}N_j₌₁⊂Hd(R), Yang Wang and Zhiqiang Xu [17] proved thatAis phase re-trievable if and only if{Ajx}Nj=1is a frame ofRdfor any nonzerox∈Rd. This incorporating

Lemma2.1leads to the following theorem.

Theorem 2.1 Let A={Aj}N_j₌₁⊂Hd(R).Then A is phase retrievable if and only if a0> 0and

b0< +∞.

Since|Ajx,y|2=yTAjxxTAjyin real case, the above theorem can be rewritten in the quadratic forms as follows.

Corollary 2.1 Let A={Aj}N

j=1⊂Hd(R).Then A is phase retrieval if and only if there are

two positive real constants a0,b0such that

a0x2I≤Rx≤b0x2I, x∈Rd, (2.2) where the inequality is in the sense of quadratic forms and Rx:=

N

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For any positive semideﬁnite matrixAj∈Hd(R), there is a matrixBj∈Rrj×dsuch that

Aj=BTj Bjwhererj≥1. While the matrixBjcan be taken as the square root ofAj, most of the time, it is not unique. LetBT

j = (bj,1, . . . ,bj,rj), wherebj,iis theith column of the matrix

BT

j , thenAjxcan be expanded as

Ajx=BTj Bjx= rj

i=1

bj,ibTj,ix= rj

i=1

x,bj,ibj,i.

Hence, we have

xTAjx=Ajx,x= rj

i=1

x,bj,i 2

.

Using the above formula, we obtain a relation between generalized phase retrieval and the classical phase retrieval.

Theorem 2.2 Suppose Aj=BTjBj∈Hd(R) is a positive semideﬁnite matrix and BTj = (bj,1, . . . ,bj,rj)for j= 1, . . . ,N.If{Aj}

N

j=1is generalized phase retrievable,then the column vec-tors{bj,i}

rj,N

i=1,j=1satisfy the complementary property and therefore become a phase retrievable frame.

Proof We prove it by contradiction. LetΛ:={(j,i) : 1≤i≤rj, 1≤j≤N}. AssumingS is an arbitrary subset of Λ, it can be divided into two parts:Sj={i|(j,i)∈S}andSCj = {1, 2, . . . ,rj} \Sj. If neither{bj,i}(j,i)∈Snor{bj,i}(j,i)∈SCis a spanning set ofRN, then there exist two nonzero elements x,y∈Rd _{such that}_x_,_b

j,i= 0 for he(j,i)∈Sandy,aj,i= 0 for

(j,i)∈SC_{. Consequently, for}_j_{= 1, . . . ,}_N_{, we have}

Bj(x+y)

2

= rj

i=1

x+y,bj,i 2

=

i∈Sj

x+y,bj,i 2

+

i∈SC_j

x+y,bj,i 2

=

i∈Sj

x–y,bj,i2+

i∈SC_j

x–y,bj,i2

=Bj(x–y)

2

.

Incorporating xT_A

jx=Bjx2 for any x∈ Rd, the above equation indicates that (x+

y)T_A

j(x+y) = (x–y)TAj(x–y) for allj. Since{Aj}Nj=1is phase retrievable, we havex+y=

±(x–y). This contradicts the nonzeroness ofxandy.

If{Aj}N_j₌₁is generalized phase retrievable, then{Aj,i}

N,rj

j=1,i=1={bj,ibTj,i} N,rj

j=1,i=1is generalized

phase retrievable by Theorem2.2. Again, by Theorem2.1, there exist positive constants

a1,b1such that

a1x2y2≤

N

j=1

rj

i=1

Aj,ix,y

2

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On the other hand,

N

j=1

xTAjy2= N j=1 rj i=1

Aj,ix,y

2 ≤r N j=1 rj i=1

Aj,ix,y2≤rb1x2y2,

withr=maxj{rj}. Therefore, we have the upper bound relationb0≤rb1.

2.1 Bi-Lipschitz property

In this subsection, we consider the bi-Lipschitz property of tmappingsMAand √

MA. At ﬁrst, we show the stability of the mappingMAwith respect to the metricd2.

Theorem 2.3 Let {Aj}N_j₌₁ ⊂ Hd(R) be generalized phase retrievable. Then MA is

bi-Lipschitz with respect to matrix metric d2(x,y) =x+yx–y.

Proof For anyx,y∈Rd_{, we have}

_M_A(x) –MA(y)

2

= N

j=1

_A_j(x+y),x–y2.

By Lemma2.1, we have

a0x+y2x–y2≤

N

j=1

Aj(x+y),x–y

2

≤b0x+y2x–y2.

This is equivalent to

a0d22(x,y)≤MA(x) –MA(y)

2

≤b0d22(x,y). (2.3)

Now, we consider the stability of the mapping√MAwith respect to the metricd1.

Lemma 2.2 Let{Aj}N

j=1⊂Hd(R)be a collection of positive semideﬁnite matrices and

gener-alized phase retrievable.Then√MAis upper bounded with respect to the metric d1(x,y) =

min{x+y,x–y}.

Proof First, by the deﬁnition of√MA, we have

MA(x) –

MA(y)2= N

j=1

xT_A jx–

yT_A

jy 2 = N j=1

Bjx–Bjy

2

,

whereBjis the square root ofAj. Then, by the reverse triangle inequality,

N

j=1

Bjx–Bjy

2 ≤ N j=1

minBj(x–y),Bj(x+y)

2

≤min

_N

j=1

Bj(x–y)

2

, N

j=1

Bj(x+y)

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=min

_N

j=1

(x–y)TAj(x–y), N

j=1

(x+y)TAj(x+y)

=min

(x–y)T

_N

j=1 Aj

(x–y), (x+y)T

_N

j=1 Aj

(x+y)

.

Since Aj is positive semideﬁnite, so is

N

j=1Aj. Let λ1 be the maximum eigenvalue of

N

j=1Aj, Then we have

min

(x–y)T

_N

j=1 Aj

(x–y), (x+y)T

_N

j=1 Aj

(x+y)

≤λ1d₁2(x,y).

Combining all the above inequalities, we have

MA(x) –

MA(y)2≤λ1d21(x,y).

This demonstrates the mapping√MAis upper bounded byλ1. Furthermore, picking an

eigenvectorxofN_j₌₁Ajcorresponding toλ1, then we have

MA(x) –

MA(0)

2

= N

j=1

Bjx2=xT

_N

j=1 Aj

x=λ1x2,

which meansλ1is the optimal upper bound.

For the lower bound, we consider the parallelogram law

x+y2+x–y2= 2x2+y2

in two cases. At ﬁrst, ifx+y ≤ x–y, we have

x+y2_x_–_y2

x2₊_y2 ≥ x+y 2_.

Secondly, ifx+y ≥ x–y, we have

x+y2_x_–_y2

x2₊_y2 ≥ x–y 2_.

Combining the above two cases,

d₁2(x,y) =minx+y2,x–y2≤x+y

2_x_–_y2

x2₊_y2 = d2

2(x,y)

x2₊_y2, (2.4)

which indicates the relation between two metrics. This allows us to estimate the lower bound of√MA.

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Proof By the formula for the diﬀerence of a square, we have

MA(x) –

MA(y)

2

= N

j=1

Bjx–Bjy

2

= N

j=1

Bjx2–Bjy2 Bjx+Bjy

2

,

whereBjis the square root ofAj. LetCis the uniform upper operator bound for{Aj}Nj=1,

that is,Ajx ≤Cxforx∈Rdandj= 1, . . . ,N. ThereforeBjx ≤ √

Cxand we have

N

j=1

Bjx2–Bjy2 Bjx+Bjy

2

≥

N

j=1(Bjx2–Bjy2)2

C(x+y)2

≥ a0d22(x,y)

2C(x2₊_y2₎

≥ a0 2Cd

2 1(x,y),

where the last inequality is due to (2.4). Thus we have proved that a0

2C is a lower bound of

the mapping√MA.

Our discussion of the bi-Lipschitz property of √MA is summarized in the following theorem by combining Lemma2.2and Lemma2.3.

Theorem 2.4 Let{Aj}N

j=1⊂HdN(R)be generalized phase retrievable and positive

semideﬁ-nite.Then√MAis bi-Lipschitz with respect to the metric d1(x,y) =min{x+y,x–y}as follows:

a0

2Cd 2

1(x,y)≤MA(x) –

MA(y)

2

≤λ1d2₁(x,y).

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. SincexTPjx=Pjx2whenPjis a projection to an appropriate subspace ofRd, phase retrieval by projections is a special case of generalized phase retrieval withAj=Pj. Therefore, Theorem2.3and Theorem2.4also hold for phase retrieval by projections. In this special case,λ1can be upper bounded byNandCequals one.

2.2 Cramer–Rao lower bound

Given signalx∈Rd_{, we take measurements of the form}_Y₌_ϕ₍_x_{) +}_Z_{, where the entries} ofZare independent Gaussian random variables with mean value 0 and varianceσ2. The generalized phase retrieval problem with noise is to estimatexfrom measurementsY. In this case, we apply the theory of Fisher information to evaluate the stability ofϕ(x). The Fisher information matrix is deﬁned entry-wise by

I(x)_m_,= –E

∂2logL(x) ∂xm∂x

,

whereL(x) is the likelihood function. By assumption,Yis a random vector with

L(x) = 1 (2π σ2₎N/2e

– 1 2σ2y–ϕ(x)

2

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By some simple computations, the Fisher information matrix entry (I(x))m,equals

1 σ2

N

j=1

E

∂(ϕ(x))j ∂xm

∂(ϕ(x))j ∂x

– 1 σ2

N

j=1

Eyj–

ϕ(x)_j∂

2₍_ϕ₍_x₎₎

j ∂xm∂x

,

where (ϕ(x))jis thejth component ofϕ(x). SinceE[yj] = (ϕ(x))j, the second expectation equals zero and

I(x)_m_,= 1 σ2

N

j=1

∂ ∂xm

ϕ(x)_j ∂ ∂x

ϕ(x)_j. (2.5)

In terms of generalized phase retrieval problem in real case, we haveϕ(x) = (xT_A

jx)Nj=1and

in order to obtain a unique solution, we make an assumption of the signalxintroduced in [1]: the signalxis in a half super plane with respect to a vectore∈Rd_{, i.e.}_x_,_e_{> 0. See} Ref. [4] for another assumption to guarantee uniqueness. Substitutingϕ(x) = (xT_A

jx)Nj=1

into (2.5), we have

I(x)_m_,= 4 σ2

N

j=1

(Ajx)m(Ajx),

where (Ajx)mis themth element of vectorAjx. This indicates the Fisher information matrix can be expressed as

I(x) = 4 σ2

N

j=1

(Ajx)(Ajx)T= 4 σ2

N

j=1

AjxxTAj= 4

σ2Rx. (2.6)

Since Corollary2.1implies the matrixRxis positive deﬁnite, we obtain the Cramer–Rao lower bound by Theorem 3.2 in [12], as incorporated in the following theorem.

Theorem 2.5 The Fisher information matrix for the noisy generalized phase retrieval model in real case is given by(2.6).Consequently,for any unbiased estimatorΦ(y)for x,

the covariance matrix is bounded below by the Cramer–Rao lower bound as follows:

CovΦ(y)≥I(x)–1=σ

2

4 (Rx)

–1_.

Therefore,the mean square error of any unbiased estimatorΦ(y)is given by

EΦ(y) –x2|x≥σ 2

4 Tr

R–1_x .

Taking inverse operator to matrices in (2.2) leads to

I b0x2 ≤R

–1

x ≤

I a0x2.

Then taking trace of every matrix yields

d b0x2

≤TrR–1_x ≤ d a0x2

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Therefore, using Theorem (2.5), we get the mean square error bounds of unbiased esti-mator as in the following corollary.

Corollary 2.2 If A={Aj}N_j₌₁is generalized phase retrievable,then,for any unbiased esti-matorΦ(y)for x,we have

EΦ(y) –x2|x≥ σ 2_d

4b0x2

.

Furthermore, any unbiased estimate that achieves the Cramer–Rao lower bound has a mean square error that is bounded above by

EΦ(y) –x2|x≤ σ 2_d

4a0x2.

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.

LetBj∈Frj×d, whererjis a positive integer. We consider recovering signalxfrom the norm of the aﬃne linear measurementsBjx+bj,j= 1, . . . ,N, wherebj∈Frj andx∈Fd. LetB={Bj}N

j=1andb={bj}Nj=1, we deﬁne the mappingMB,b:Fd→RN+ by MB,b(x) =

B1x+b12,B2x+b22, . . . ,BNx+bN2

.

The pair (B,b) is said to be generalized aﬃne phase retrieval forFd_if_M

B,bis injective on

Fd_{. This deﬁnition is a little bit diﬀerent from the one in [10] where all}_r

j equals same integerr≥1. Similar to generalized phase retrieval, we deﬁne the mapping√MB,bby

MB,b(x) =

B1x+b1,B2x+b2, . . . ,BNx+bN

.

Theorem 3.1 Let Bj∈Rrj×dand bj∈Rrj.Then the following are equivalent:

(A) The pair(B,b)is generalized aﬃne phase retrievable forRd_.

(B) There exist no nonzerou∈Rd_{such that}_B

ju,Bjv+bj= 0for all1≤j≤Nand

v∈Rd_.

(C) Ifvis the solution of equationsBjv+bj= 0forj∈S⊂ {1, 2, . . . ,N},then {BT

jBjv+BTjbj}j∈SC is a spanning set ofRd.

(D) The Jacobian ofMB,bhas rankdeverywhere onRd.

Proof (A)⇔(B). Assume thatMB,b(x) =MB,b(y) for somex=yinRd. For anyj, we have Bjx+bj2–Bjy+bj2=

Bj(x–y),Bj(x+y) + 2bj

.

Set 2u=x–yand 2v=x+y. Thenu= 0 and for allj,

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Conversely, assume that (3.1) hold for allj. Letx,y∈Rd_{be given by}_x_–_y_{= 2}_u_and_x₊_y₌ 2v. Thenx=y. However, we haveMB,b(x) =MB,b(y). Hence (B,b) cannot be aﬃne phase retrievable.

(B)⇔(C). Assume{BT

j Bjv+BTjbj}j∈SCis not a spanning set ofRd, then there is a nonzero vectoru∈Rd_{such that}_B

ju,Bjv+bj=u,BTjBjv+BTjbj= 0 forj∈SC. Forj∈S, sincev is the solution of equationsBjv+bj= 0, the inner productBju,Bjv+bjalso equals zero, which contradicts (B). The converse can be proven similarly.

(C)⇔(D). The Jacobian ofMB,batxis exactly

JB,b(x) = 2

BT₁B1x+B1Tb1,BT2B2x+BT2b2, . . . ,BTNBNx+BTNbN

,

which means thejth column ofJB,bis preciselyBjTBjx+BTjbj. This indicates the equivalence

of (C) and (D).

Minimality problems have attracted much attention from different areas recently. For generalized affine phase retrieval, the answer is related to different constraints onBj,bj and prior knowledge of signalx. The following theorem is given in [10].

Theorem 3.2([10]) Let N≥2d and r> 1.Then a generic{(Bj,bj)}Nj=1⊂Rr×(d+1)has the generalized aﬃne phase retrieval property inRd_.

Letr=maxjrj. Therj×(d+1) matrix (Bj,bj) in Theorem3.1can be extended tor×(d+1) matrix by ﬁlling with zero rows. The extended matrix can be viewed as an aﬃne phase retrieval matrix where allrj=rand hence leads to the following corollary by Theorem3.2.

Corollary 3.1 LetA˜j= (BTj ,bTj )T(Bj,bj),where bj∈Rrjand Bj∈Rrj×dis a nonzero matrix.

If N≥2d andA˜ = (A˜j)Nj=1is a generic set in HdN(R),Then the pair(B,b)is generalized aﬃne

phase retrievable.

Example3.1 LetB1=B2be the 2×2 identity matrix,B3= (1, 0),b1= (0, 0)T_,_b2_{= (0, 1)}T_,

b3= 1. Then the pair (B,b) is generalized aﬃne phase retrievable inR2. In fact, assuming u= (x,y)T_∈_R2_{, then}

B1u+b12=x2+y2, B2u+b22=x2+ (y+ 1)2_,

B3u+b32= (x+ 1)2.

By simple computation, one can easily solve the equations with respect tox,y. The number of measurements equals 3 withr1=r2= 2,r3= 1.

Now, we consider the stability of generalized aﬃne phase retrieval. LetB˜j= (Bj,bj),A˜j= ˜

BT

jB˜j, andx˜= (xT, 1)T. We have the following theorem.

Theorem 3.3 SupposeA˜ ={ ˜Aj}N

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on(B,b)such that,for any x,y∈Rd_,

c0d2₂(x,y) +d2(x,y)≤MB,b(x) –MB,b(y)2≤c1

d2₂(x,y) +d2(x,y), (3.2)

C0d2₁(x,y)≤MB,b(x) –

MB,b(y)2≤C1d2(x,y). (3.3)

Proof The generalized aﬃne phase retrievable property of pair (B,b) is due to Corol-lary 3.1. Noticed that Bjx+bj2 = ˜Bj˜x2 =x˜TA˜x˜ implies MB,b(x) =MA˜(x˜), we have MB,b(x) –MB,b(y)2=MA˜(x˜) –MA˜(y˜)2. By Theorem2.3, we have

_M_B_,_b₍_x_{) –}_M_B_,_b₍_y₎2

d2₂(x˜,y˜),

where the symbol “” denotes the bi-Lipschitz relation. Sinced2₂(x˜,y˜) =˜x+y˜2_˜_x_–_y_˜2₌

(x+y2_{+ 4)}_x_–_y2_{, there exist constants}_c0_,_c1_{such that (3.2) holds. Similarly, by}

The-orem2.4, we have

MB,b(x) –

Mb,b(y)2d21(x˜,y˜).

Sinced2

1(x˜,y˜) =min{˜x+˜y2,˜x–y˜2}=min{x+y2+ 4,x–y2}, there exist constants

C0,C1such that (3.3) holds.

In contrast to Theorem 4.1 in [9], Theorem3.4leads to a slack constraint of the signal from a compact set toRd_{. Although aﬃne phase retrieval is not bi-Lipschitz with respect} to one metric, the mappingsMB,band√MB,bis bounded by two metrics.

We now consider the additive white Gaussian noise model

Y=ϕ(x) +Z,

withϕ(x) = (Bjx+bj2)Nj=1. Then, by Eq. (2.5), the Fisher information of this model isσ42Rax whereRa_x=N_j₌₁BT_j(Bjx+bj)(Bjx+bj)TBj.

Lemma 3.1 If the pair(B,b)is generalized aﬃne phase retrievable,then the Fisher infor-mation Ra

xis positive deﬁnite for any x∈Rd.

Proof It is easy to see thatRa

xis positive semideﬁnite. AssumeyTRaxy= 0 for somey∈Rd, that is,

yTRa_xy= N

j=1

(Bjx+bj)TBjy2= 0.

Therefore, for allj= 1, . . . ,N,

(Bjx+bj)TBjy=

BT_j(Bjx+bj),y

= 0.

Since (B,b) is generalized aﬃne phase retrievable, by Theorem3.1, the collection{BT j (Bjx+

bj)}Nj=1is a spanning set ofRdand hencey= 0.

(12)

Theorem 3.4 The Fisher information matrix for the noisy generalized aﬃne phase re-trieval model in real case is _σ42Rax.Consequently,for any unbiased estimatorΦ(y)for x,

The covariance matrix is bounded below by the Cramer–Rao lower bound as follows:

CovΦ(y)≥I(x)–1=σ

2

4

Ra_x–1.

Therefore,the mean square error of any unbiased estimatorΦ(y)is given by

EΦ(y) –x2|x≥σ 2

4 Tr

Ra_x–1.

As a generalization of frame, g-frame is introduced by Wenchang Sun in [15]. The op-erator sequence{Λj}N_j₌₁is a g-frame if there are two positive constantscandCsuch that

cx2≤

N

j=1

Λjx2≤Cx2, ∀x∈Rd.

Lemma 3.2 If the pair(B,b)is generalized aﬃne phase retrievable, then the collection

{BT

jBj}Nj=1is a g-frame forRd. Proof SinceBT

jBjx2=xT(BTjBj)2x, the summation

N

j=1BTjBjx2is upper bounded by :=λmax(Nj=1(BTjBj)2). For the lower bound, we prove by contradiction. If the summation is not lower bounded, we can ﬁnd a vectory∈Rd_{such that}_y_{= 1 and}N

j=1BTjBjy2= 0, which meansBT_jBjy= 0 for allj= 1, . . . ,N. Therefore, we have

yTBT_jBjy=Bjy2= 0,

which impliesBjy= 0 for allj= 1, . . . ,N. Consequently, we havey= 0 and

Bjy+bj2=Bj0 +bj2,

which contradicts the assumption that (B,b) is generalized aﬃne phase retrievable.

Corollary 3.2 If the pair(B,b)is generalized aﬃne phase retrievable,then,for any unbi-ased estimatorΦ(y)for nonzero x,we have

EΦ(y) –x2|x≥ σ 2_d2

8(x2₊_C₎.

Proof Since the matrix Ra

x is positive deﬁnite by Lemma 3.1, the inequality Tr(Rax)· Tr((Ra

x)–1)≥d2holds and we can estimate the lower bound of mean square error by The-orem3.4as follows:

EΦ(Y) –x2|x≥ σ 2_d2

4Tr(Ra x)

. (3.4)

By Lemma3.2, the trace can be estimated as

TrRa_x= N

j=1

BT_jBjx+BTj bj

2

≤2 N

j=1

BT_j Bjx

2

+ 2 N

j=1 BT_jbj

2

(13)

whereC=N_j₌₁BT

jbj2. Substituting it into (3.4), we have

EΦ(Y) –x2|x≥ σ 2_d2

8(x2₊_C₎.

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.

Acknowledgements

The authors would like to thank the referees for their useful comments and remarks.

Funding

This study was partially supported by National Natural Science Foundation of China (Grant No. 11601152).

Availability of data and materials Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 8 November 2018 Accepted: 15 January 2019

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