Almost Phase Retrieval with Few Intensity Measurements

In signal processing, the phases of the measurements are often unknown. To recover a signal of length N, Balan, Casazza, and Edidin showed that it suffices to know at least 2N − 1 intensity measurements. We give another characterization of frames which give phase retrieval for almost all signals with only N + 1 intensity measurements. We provide a method to test if a frame has this property or not. With our method, we can construct tight frames or modify any frames such that they give almost phase retrieval. Numerical results show that frames with this property can recover signals with high probability.


Introduction
Given a set of vectors {  } 1≤≤ ⊂ K  (K = R or C), the phase retrieval problem is to recover a signal from {|⟨,   ⟩|} 1≤≤ which is called intensity measurements.Phase retrieval is a key step in speech processing [1], X-ray crystallography [2], and so forth.Since frames are redundant systems of vectors in a Hilbert space, it is possible to reconstruct signals with suitable frames.
First, we introduce the definition of phase retrieval.
In fact, this definition is equivalent to the following.
Let  be an × matrix of rank .If the set of columns of  gives phase retrieval, we also say that the matrix  gives phase retrieval.Balan et al. [4] studied the properties of such frames with minimal measurements.For the real frames, they gave the following result.Proposition 3 (see [4,Theorem 2.2]).Let Φ = {  } 1≤≤ be a frame for R  .If  ≥ 2 − 1, then a generic frame Φ gives phase retrieval.
Here a generic frame Φ means a member of some nonempty Zariski open subset of the set of all -element frames in R  .We refer to [5, Section 2.2] for details.
Recently, people have studied more properties of such frames.Ohlsson and Eldar [7] studied the minimal measurements of sparse signals.If sparse signals contain noise, Eldar and Mendelson [8] also studied the minimal measurements and gave a recovery algorithm.By convex programming, Candès, Strohmer, and Vladislav gave a method which is called phaselift to recover a signal from its magnitude measurements.Then Gross et al. [9] and Alexeev et al. [10] developed other methods with phaselift.Chen et al. [11] estimated the error of phaselift.Schniter and Rangan [12] used compressed sensing to recover sparse signals via generalized approximate message passing.
If we do not require that Φ gives phase retrieval, for all  ∈ K  , we may need fewer measurements.Fickus et al. [13] studied when Φ gives phase retrieval for almost all vectors of K  .In this paper, we first introduce some preliminary results of frames which give phase retrieval.Then, we show our characterization and some properties of such frames.We give a method to test if a frame gives almost phase retrieval.Finally, we provide some examples to illustrate our results.

Main Results
First, we introduce some preliminary results.
A set of finitely many vectors Φ = {  } 1≤≤ in K  is called a frame if there are two positive constants  ≤  such that, for every  ∈ K  , A frame is said to be tight if  =  and Parseval if  =  = 1.
If the right-hand side of (2) holds, it is said to be a Bessel sequence.Define the analysis and synthesis operators of a frame by respectively.We call {⟨,   ⟩} 1≤≤ the frame coefficients of .It can be shown that  *  is invertible on K  and, for each We call that  =  *  is the frame operator and { φ = ( * ) −1   } 1≤≤ the canonical dual frame for Φ.It is easy to see that  −1 is the frame operator of { φ }.Generally, if for every  ∈ K  , we call that {  } 1≤≤ is a dual frame for Φ.
For more details on frame theory and applications, we refer the reader to [14,15] and so forth.Fickus et al. gave the following definition.
Definition 6 (see [13,Definition 7]).A set of vectors {  } 1≤≤ in K  gives almost phase retrieval if it gives phase retrieval for almost every  ∈ K  .
Next, we study real frames which give almost phase retrieval.Characterizing the complex frames which give almost phase retrieval remains an open problem.
Denote the th row and the th column of a matrix  by row   and col  , respectively.They gave a characterization of such frames.
Proposition 7 (see [13,Theorem 12]).Let  be an  ×  matrix on R. Suppose each column of  is nonzero.Then  gives almost phase retrieval if and only if  is of rank  and, for each nonempty proper subset Γ ⊂ {1, 2, . . ., }, where  and  are the matrices consisting of {  } ∈Γ and {  } ∈Γ  , respectively.
Here we give another characterization which shows more properties of the almost phase retrieval.Let Γ be a subset of {1, 2, . . ., }.Denote For a matrix , denote Denote the range and null space of an operator or matrix  by R() and N(), respectively.Our first main result is the following.
Sufficiency.Let Γ be a nonempty proper subset of {1, 2, . . ., }.Then N() ∩ N( Γ ) is a proper subspace of N() = R().Let C 0 be the union of all such subspaces and S 0 =  −1 C 0 .Since S 0 is the union of finitely many proper subspaces of R  , its Lebesgue measure in R  is 0. And for every  ∈ R  \ S 0 ,  ∉ C 0 .That means Φ gives phase retrieval on R  \ S 0 .Hence Φ gives almost phase retrieval.
Proposition 7 can be considered as a consequence of Theorem 8.In fact, let  and  be defined as in Proposition 7. We have By Theorem 8, we get Proposition 7. Now we have characterizations of frames which give almost phase retrieval.But it is tedious to test if a frame satisfies (6) or (10).
Next we give a method to test if a general frame gives almost phase retrieval.By Theorem 8, (11) holds if Φ does not give almost phase retrieval.First we give a condition for  satisfying (11).
Lemma 10.Let  be an  ×  matrix on R. Suppose that its columns are nonzero.Denote by  the space spanned by rows of .If the rows of  are orthogonal basis of , then, for every nonempty proper subset Γ ⊂ {1, 2, . . ., }, holds if and only if, for every  ∈ Γ,  ∈ Γ  , Proof.First we prove the necessity.We assume that ( 16) holds.
Then each row of  Γ is a linear combination of rows of .Therefore there exists   ∈ R such that That is, where  is an  ×  matrix on R. Since the rows of  form an orthonormal basis of , the rows of  Γ form also an orthonormal basis of .Hence  is an orthogonal matrix.Hence, for every  ∈ Γ,  ∈ Γ  , Therefore, Sufficiency.Denote by  and  the matrices consisting of {col  } ∈Γ and {col  } ∈Γ  , respectively.Let   and   be the spaces spanned by columns of  and , respectively.By ( 17), we have Since each column of  is nonzero,   and   are proper subspaces of R  .Therefore, there exists an  ×  matrix  on R such that, for every  1 ∈   and  2 ∈   , Then we have Hence we get (16).
Lemma 10 shows that if the matrix  can not be split into two orthogonal parts, then the corresponding frame Φ (see Theorem 8) gives almost phase retrieval.Here we need the fact that the rows of  are orthogonal.If the columns of  contain a natural basis, the result still holds.Lemma 11.Let  be an  ×  matrix on R. Suppose that the columns of  contain a natural basis {  } 1≤≤ of R  .For every nonempty proper subset Γ ⊂ {1, 2, . . ., }, Since N() = R(),  gives almost phase retrieval on R − , thanks to Theorem 8.
From the above results, we give a method to test if an  ×  matrix  gives almost phase retrieval on R  or not.
Denote by  1 and  2 the matrices consisting of the first  columns and the last  −  columns of , respectively.Without loss of generality, we assume that  1 is of rank .And  −1 1  has the following form: ) . (34) Then Ã gives almost phase retrieval if and only if  does.Without loss of generality, we assume that each column of Ã is nonzero; otherwise we can drop the columns which are zero.Denote where col   Ã is the th element of col  Ã.Next we do the following steps.
After finitely many steps, say  steps, we get Γ and ⋃  1 supp(col   Ã) ≜ Λ finally.There are two cases.
Remark 14. Balan et al. [4, Theorem 2.9] proved that Ã gives almost phase retrieval while Ã has a column col   such that supp col   = {1, 2, . . ., }.By the above method, it is easy to see that Ã gives almost phase retrieval.And we find more general matrices which give almost phase retrieval in some sense.
Remark 15.In our method, we need to compute the inverse of  1 .But rank() =  is necessary; the processes of transforming  to Ã and computing rank() are conducted at the same time.Next it is very easy to compute Γ.Therefore, the computation complexity ( − ) is acceptable.
Here we give a method to test if a frame gives almost phase retrieval or not.With this method, we can construct frames which have this property.For the unit norm tight frames, Fickus et al. [13] gave a sufficient condition.
Different from Proposition 16, for any  > , we can construct a tight frame Φ = {  } 1≤≤ which gives almost phase retrieval from the above methods.
First, we can construct an × matrix Ã of the form (34) such that it gives almost phase retrieval.For example, we can choose Ã such that one of the last  −  columns does not contain zero element.Second, by the Gram-Schmidt process or other methods, we get a matrix Â from Ã such that the rows of Â are orthogonal.
where  ∈ R is a constant.Then the columns of  form a tight frame with frame bound  by the following proposition.
Let  =  1 B. Then  gives almost phase retrieval.On the other hand, the canonical dual frame is always used to reconstruct the original vector.Sun [17] gave the following stability result of canonical dual frame.
For R  ,  ≥ 2 − 1 is necessary.But for C  , Bandeira et al. conjectured the following.