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We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans and , then .
Signal reconstruction is an important problem in engineering and has a wide variety of applications. Recovering signals when there is partial loss of information is a significant challenge. Partial loss of phase information occurs in application areas such as speech recognition [1,2,3], and optics applications such as X-ray crystallography [4,5,6], and there is a need to do phase retrieval efficiently. The concept of phase retrieval for Hilbert space frames was introduced in 2006 by Balan, Casazza, and Edidin , and since then it has become an active area of research in signal processing and harmonic analysis.
Phase retrieval has been defined for vectors as well as for projections and in general deals with recovering the phase of a signal given its intensity measurements from a redundant linear system. Phase retrieval by projections, where the signal is projected onto some lower dimensional subspaces and has to be recovered from the norms of the projections of the vectors onto the subspaces, appears in real life problems such as crystal twinning . We refer the reader to  for a detailed study of phase retrieval by projections.
Another related problem is that of phaseless reconstruction, where the unknown signal is reconstructed from the intensity measurements. Recently, the two terms phase retrieval and phaseless reconstruction were used interchangeably. However, it is not clear from their respective Definitions how these two are equivalent. Recently, in  the authors proved the equivalence of phase retrieval and phaseless reconstruction in real as well as in complex case. Due to this equivalence, in this paper, we restrict ourselves to proving results regarding phase retrieval. Further, a weaker notion of phase retrieval and phaseless reconstruction was introduced in .
In this work, we consider the notion of norm retrieval which was recently introduced by Bahmanpour et al. in , and is the problem of retrieving the norm of a vector given the absolute value of its intensity measurements. Norm retrieval arises naturally from phase retrieval when one utilizes both a collection of subspaces and their orthogonal complements. Here we study norm retrieval and certain classifications of it. We use projections to do norm retrieval and to extend certain results from  for frames. We provide a complete classification of subspaces of which do norm retrieval. Various examples for phase and norm retrieval by projections are given. Further, a classification of norm retrieval using Naimark’s theorem is also obtained.
We organize the rest of the paper as follows. In Section 2, we include basic Definitions and results of phase retrieval. Section 3 introduces the norm retrieval and properties. Section 4 provides the relationship between phase and norm retrieval and related results. Detailed classifications of vectors and subspaces which do norm retrieval are provided in Section 5.
We denote by a N dimensional real or complex Hilbert space, and we write or when it is necessary to differentiate between the two explicitly. Below, we give the Definition of a frame in .
A family of vectors in is aframeif there are constants so that for all ,
The following Definitions and terms are useful in the sequel.
The constants A and B are called the lower and upper frame bounds of the frame, respectively.
If , the frame is called an A-tight frame (or a tight frame). In particular, if , the frame is called a Parseval frame.
Φ is an equal norm frame if for all and is called a unit norm frame if for all .
If, only the right hand side inequality holds in (1), the frame is called a B-Bessel family with Bessel bound B.
Note that in a finite dimensional setting, a frame is a spanning set of vectors in the Hilbert space. We refer to  for an introduction to Hilbert space frame theory and applications.
Let be a frame in . The analysis operator associated with Φ is defined as the operator to be
Here, is understood to be the natural orthonormal basis for . The adjoint of the analysis operator T is called the synthesis operator of the frame Φ. It can be shown that
The frame operator for the frame Φ is defined as That is,
Note that the frame operator S is a positive, self-adjoint and invertible operator satisfying the operator inequality , where A and B are the frame bounds and I denotes the identity on . Frame operators play an important role since they are used to reconstruct the vectors in the space. To be precise, any can be written as
The analysis operator of a Parseval frame is an isometry and the frame operator is the identity operator. Thus, if is a Parseval frame, it follows from Equation (2) that
We concentrate on norm retrieval and its classifications in this paper. We now see the basic Definitions of phase retrieval formally, starting with phase retrieval by projections. Throughout the paper, the term projection is used to describe orthogonal projection (orthogonal idempotent operator) onto subspaces.
Let be a collection of subspaces in and let be the projections onto each of these subspaces. We say that (or ) yieldsphase retrievalif for all satisfying for all then for some scalar c such that
Phase retrieval by vectors is a particular case of the above.
Let be such that for
Φ yields phase retrieval with respect to an orthonormal basis if there is a such that , for all , where .
Orthonormal bases fail to do phase retrieval, since in any given orthonormal basis, the corresponding coefficients of a vector are unique. One of the fundamental properties to identify the minimum number of vectors required to do phase retrieval is the complement property.
A frame in satisfies thecomplement propertyif for all subsets , either or spans the whole space .
It is proved in  that phase retrieval is equivalent to the complement property in . Further, it is proven that a generic family of -vectors in does phase retrieval, however no set of -vectors can. Here, generic refers to an open dense set in the set of -element frames in . Full spark is another important notion of vectors in frame theory. A formal Definition is given below:
Given a family of vectors in , thesparkof Φ is defined as the cardinality of the smallest linearly dependent subset of Φ. When spark, every subset of size N is linearly independent, and in that case, Φ is said to befull spark.
Note from the Definitions that full spark frames with have the complement property and hence do phase retrieval. Moreover, if then the complement property clearly implies full spark.
Next result, known as Naimark’s theorem, characterizes Parseval frames in a finite dimensional Hilbert space. This theorem facilitates a way to construct Parseval frames, and crucially it is the only way to obtain Parseval frames. Later, we use this to obtain a classification of frames which do norm retrieval. The notation is used throughout the paper.
Theorem 1 (Naimark’s Theorem).
 A frame is a Parseval frame for if and only if with orthonormal basis so that the orthogonal projection P onto satisfies: for every .
3. Beginnings of Norm Retrieval
In this section, we provide the Definition of norm retrieval along with certain related results, and pertinent examples.
Let be a collection of subspaces in and let be the orthogonal projections onto each of these subspaces. We say that (or ) yields norm retrieval if for all satisfying for all then .
In particular, a set of vectors in does norm retrieval, if for satisfying for all then .
It is immediate that a family of vectors doing phase retrieval does norm retrieval.
An obvious choice of vectors which do norm retrieval are orthonormal bases. For, let be an orthonormal basis in . Now, for , Thus
The following theorem provides a sufficient condition under which the subspaces spanned by the canonical basis vectors do norm retrieval.
Let be an orthonormal basis in . Let be subspaces of where each , . If there exists m such that for all j, , then does norm retrieval.
Let be orthogonal projections onto , for all j. Now, by assumption, we have
It is easy to see that tight frames do norm retrieval.
Tight frames do norm retrieval.
et in be an A-tight frame. Now, if
Observe that if does norm retrieval so does for any . This is generalized in the following proposition.
If does norm retrieval, then so does for any projections . In particular, if a frame contains an orthonormal basis, then it does norm retrieval. Moreover, in this case, does norm retrieval.
Let be an orthonormal basis for and let be the projections onto , for each i. Given , we have
The above proposition does not hold if the number of hyperplanes is strictly less than N. This is proved in the next theorem.
If is an orthonormal basis for then where cannot do norm retrieval for .
Without loss of generality consider the collection (for ). Now, let and so that and . Thus, . However and which proves the theorem. ☐
Now, we strengthen the above result by not requiring the vectors to be orthogonal. To prove this, we need the following lemma.
If are independent vectors in , then there is a vector satisfying:
We do this by induction on N with the case obvious. So assume this holds for . Given , we can find a and satisfying
Choose and note that linear independence of the implies
Consider . For ,
As λ varies from to , the right hand side varies from to and for some λ, we have
If are independent and unit norm and , for all , then cannot do norm retrieval.
Let be the projection onto and choose
By the assumption, pick any , there is a vector and In particular, we may scale ϕ and c simultaneously such that and it follows that . Let , where Note that for all i implies that , and so
Now, for all ,
But while , and so norm retrieval fails. ☐
However, in the following theorem, we show that three proper subspaces of codimension one can do norm retrieval in .
In three proper subspaces of codimension one can do norm retrieval.
Let be an orthonormal basis for . Let
We claim does norm retrieval. Let be the orthogonal projection onto . Let . We then have that
Case 1: If or , we know that or respectively.
Case 2: Assume both and . We then know both of the equalities below:
If either c or d is nonzero, then the proof is complete as in that case, we can express as a linear combination of , , and .
Now, suppose that . If , then and if , then . This implies that
which holds only if either or or both is zero which contradicts our assumption. ☐
It follows that in , two 2-dimensional subspaces cannot do norm retrieval but three 2-dimensional subspaces can do norm retrieval.
For every , there exist subspaces of which do norm retrieval and span a K dimensional space.
Choose an orthonormal basis of , say . Let and for all . If , then and for . Therefore . Since for all i, it’s clear that is spanned by , which has dimension K. ☐
The following proposition shows a relationship between subspaces doing norm retrieval and the sum of the dimensions of the subspaces. The importance of this proposition is that we are looking for conditions on subspaces to do norm retrieval. To do so, the dimension of the subspaces is one of the tools we have.
If in does norm retrieval then . Moreover, if and then there exist doing norm retrieval where for each .
If then we may pick non-zero for each i so that for all i and therefore fails norm retrieval.
For the moreover part, let be an orthonormal basis. We represent this basis L-times as a multiset:
and index it as: . We may pick a partition of in the following manner:
Now define span with projection . Then if then
Hence the result.
As we have seen, the above proposition may fail if .
4. Phase Retrieval and Norm Retrieval
In this section, we provide results relating phase retrieval and norm retrieval. The following theorem of Edidin  is significant in phase retrieval as it gives a necessary and sufficient condition for subspaces to do phase retrieval.
(). A family of projections in does phase retrieval if and only if for every , the vectors span the space.
If in does phase retrieval, then spans the space.
If does not span, then there exists . So for all , and so does not span. Thus, by Theorem 6, does not do phase retrieval. ☐
Let be a collection of subspaces of with denoting the projection onto for each . If does phase retrieval in then for every with , the collection spans .
If not, pick non-zero for all . This implies and therefore contains at most distinct vectors and can not span . This contradicts the Theorem 6. ☐
The following example shows that it is possible for subspaces to do norm retrieval even if do not span the space which we see as one of main differences between phase retrieval and norm retrieval.
Let be a orthonormal basis for , then let
Then, does norm retrieval since . But do not span .
Note that if , then itself does norm retrieval while .
Any collection of subspaces which does phase retrieval yields norm retrieval, which follows from the Definitions. However, the converse need not hold true always. For instance, any orthonormal basis does norm retrieval in . But it has too few vectors to do phase retrieval as it requires at least vectors to do phase retrieval in .
Given subspaces of which yield phase retrieval, it is not necessarily true that do phase retrieval. The following result proves that norm retrieval is the condition needed to pass phase retrieval to orthogonal complements. Though the result is already proved in , we include it here for completeness.
Suppose subspaces , with respective projections , does phase retrieval. Then does phase retrieval if and only if does norm retrieval.
Assume that for all and does norm retrieval i.e., . Then
Since , we have
Since does phase retrieval, it follows that for some .
The other direction of the theorem is clear. ☐
Next is an example of a family of subspaces which does phase retrieval but complements fail phase retrieval and hence fail norm retrieval .
Let and be orthonormal bases for such that is full spark. Consider the subspaces
Then allow phase retrieval for while the orthogonal complements do not.
If does phase retrieval and contains an orthonormal basis, then does phase retrieval.
If is an orthonormal basis, then does norm retrieval. Hence so does the larger set . Since does phase retrieval, and does norm retrieval, we can conclude the latter does phase retrieval as well which follows from Lemma 2. ☐
The next result gives us a sufficient condition for the subspaces to do norm retrieval. It is enough to check if the identity is in the linear span of the projections in order for the subspaces to do norm retrieval. A similar result in the case of phase retrieval is proved in .
(). Let be subspaces of with corresponding projections . If there exist such that , then does norm retrieval.
Given , then
Since for each i the coefficients and are known, the collection does norm retrieval. ☐
A counter example for the converse of the above proposition is given in  where the authors construct a collection of projections, , which do phase retrieval but . Here, we provide another example for the same. We give a set of five vectors in which does phase retrieval; however the identity operator is not in the span of these vectors. We need the following theorem that provides a necessary and sufficient condition for a frame to be not scalable in . Recall that a frame is said to be scalable if there exists scalars such that is a Parseval frame . Later in the next section, we prove that scalable frames always do norm retrieval.
 A frame ϕ in for is not scalabale iff all frame vectors of ϕ are contained in an interior of an elliptical conical surface with vertex 0 and intersecting the corners of a rotated unit cube.
A frame in which does phase retrieval but
Choose five full spark vectors in the cone referred in the previous Theorem 7. These vectors do phase retrieval and hence norm retrieval in . Now, given , for . But, is still inside the cone for each i. Therefore
The next proposition gives a sufficient condition for the complements to do norm retrieval when the subspaces do.
If are subspaces of with corresponding projections such that and . Then does norm retrieval.
Observe the following
Let then a short calculation shows . By the previous proposition this shows does norm retrieval. ☐
It is possible that with but , as we will see in the following example.
Let be an orthonormal basis for . Now let
Both and do norm retrieval. Let denote the projections on to , then and . However, .
5. Classification of Norm Retrieval
In this section, we give classifications of norm retrieval by projections. The following theorem in  uses the span of the frame elements to classify norm retrievable frames in .
() A frame does norm retrieval if and only if for any partition of , .
Next, we prove one of the main results of this paper. This is an extension of the previous Theorem 8 and it fully classifies the subspaces of which do norm retrieval.
Let be projections onto subspaces of . Then the following are equivalent:
does norm retrieval,
Given any orthonormal bases of and any subcollection then
For any orthonormal basis of , then the collection of vectors do norm retrieval.
: Suppose , and and let then,
Since does norm retrieval, we have
and so .
: Assume that for all Then, by () we can find an orthonormal basis for such that
Denote and . Now we can see that
By (2), we must have that which implies that x and y have the same norm. The third equivalence is immediate from the result in Theorem (8). ☐
If does norm retrieval then does norm retrieval. Hence all scalable frames do norm retrieval.
This is an immediate result of Theorem 9. Observe the conditions in Theorem 9 do not depend on the norm of each vector . ☐
For the complex case we have:
If does norm retrieval, then whenever we choose orthonormal bases of and any subcollection then
Given as above,
Since our vectors do norm retrieval, we have
and so . ☐
We use Theorem 9 to give a simple proof of a result in  which has a very complicated proof in that paper.
If do norm retrieval in , then the vectors are orthogonal.
Assume and that is not orthogonal to span . Choose a unit vector for all . Let . Now,
Let . Then
contradicting the theorem. ☐
Consider a frame . The followings are equivalent:
Φ does norm retrieval.
For if and then, .
By Theorem 9, it follows that Φ does norm retrieval if and only if . This happens if and only if . Hence the proof. ☐
Both phase retrieval and norm retrieval are preserved when applying projections to the vectors. Also, phase retrieval is preserved under the application of any invertible operator (refer to  for details). This is not the case with norm retrieval, in general. We prove this in the next corollary.
Norm retrieval is not preserved under the application of an invertible operator, in general.
Let be linearly independent vectors in which are not orthogonal. Then by Corollary 5, Φ cannot do norm retrieval. But there exists an invertible operator T on so that is an orthonormal basis and so does norm retrieval. ☐
However, we note that unitary operators, which are invertible, do preserve norm retrieval.
The following corollary about Parseval frames also holds in the infinite dimensional case with the same proof.
If Φ is a Parseval frame, it does norm retrieval. Hence, if we partition Φ into two disjoint sets, and choose a vector orthogonal to each set, then these vectors are orthogonal.
Let be a Parseval frame and let . Let T be its analysis operator. If and . Then and do not have any nonzero coordinates in common. So . Since, the analysis operator of a Parseval frame is an isometry, we have . ☐
A classic result in frame theory is that a Parseval frame has the property that for , if then It turns out that a much more general result holds.
Let be a Parseval frame in . For , let and . If , then .
If is a frame for with frame operator S which does norm retrieval, then for every , if then . In particular, if Φ is a Parseval frame then .
Given x as in the corollary,
We next provide a classification of norm retrieval using Naimark’s theorem. It turns out that every frame can be scaled to look similar to Naimark’s theorem.
If is a frame with Bessel bound 1 on , then there is an isometry with orthonormal basis so that the orthogonal projection onto satisfies: for every .
Let be the eigenbasis for the frame with respective eigenvalues . For let
Now, for any we have
So is a Parseval frame. The analysis operator of this Parseval frame is then an isometry where where is the unit vector basis of . Let P be the orthogonal projection of onto . Then given , we have for all :
It follows that for all . ☐
We can now prove one of the main results in this section.
Let be a frame for . The following are equivalent:
Φ does norm retrieval.
By Proposition 8 if is an isometry and is the unit vector basis for then for every with for , we have
: We have for ,
By (1), we know that . Hence,
Since , the result follows.
: If for all then applying (2) as above we have:
so and Φ does norm retrieval. ☐
We thank the referees for a careful reading of the paper which substantially improved the presentation. The first, second and fourth authors were supported by NSF DMS 1609760; NSF ATD 1321779; and ARO W911NF-16-1-0008. Part of this research was carried out while the first and fourth authors were visiting the Hong Kong University of Science and Technology on a grant from (ICERM) Institute for computational and experimental research in Mathematics.
Peter G. Casazza, Dorsa Ghoreishi, Shani Jose and Janet C. Tremain contributed equally to this manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
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