Phase-retrieval is a nonlinear problem that seeks to recover a signal x, up to a global phase ambiguity, from the magnitudes of its linear measurements
Besides phase-retrieval, the sampling theory for a union of subspaces (UoS for short) is another important sampling problem (c.f. [8, 9, 10]). In signal processing, while traditionally we work on signals in a single linear space or subspace, there are practical demands requiring us to deal with the signals that lie in a UoS. A typical example is the sparse signal recovering or compressed sensing (c.f. ) where the signals are sitting in the finite union of (small dimensional) subspaces. M. Mishali, Y. Eldar and A. Elron  established Xampling for recovering signals in the UoS of . By Xampling, the target subspace where the signal sits is detected before recovery. As mention in , the detection can considerably reduce the computational complexity and measurement cost (the sampling rate). Note that the phase information of the measurements in  is assumed known. Motivated by  we will study the phase-retrieval problem for the union of cones (UoC for short). In order to introduce the main problems and discuss our main contributions, we need to recall and establish some notations and definitions.
I-a Notations and definitions
We use boldface letters to denote column vectors, e.g.,calligraphic and upper-case letters to denote matrices (operator), e.g.,
, and underlined letters to denote a random variable, e.g.,. For a matrix , its Hermitian transpose and transpose are denoted by and , respectively. For a linear operator from a vector space to another vector space , we denote by and the range and null spaces of , respectively. Moreover, for a set , denote by the inverse image of For a vector , implies every coordinate of x is strictly larger (smaller) than . Denote . The standard orthornormal basis for is denoted by .
For a matrix , we denote by the cone generated from its column vectors, i.e.,
A finite set of vectors is called a frame for if there exist two constants such that
holds for every . Equivalently, a finite set is a frame for if and only if it is a spanning set of . The cone in (1.1) is called a frame cone if is a frame for .
Based on the above denotations, in what follows we propose the definition of a detectable UoC.
We say that is detectable, if there exists a so-called detector such that for any nonzero target signal , the unique index can be determined by the detection measurements so that .
I-B Our goals, schemes and problems in the present paper
The union of cones (UoC) is an important type of subset of which has been widely considered in many areas of research such as operations research (c.f. [12, 13, 14]), signal processing (c.f. [11, 15, 16, 17]), and representation theory (c.f. ). Incidentally, since a linear space is a special type of cone (e.g. can be regarded as the cone generated from ), a union of linear spaces can be regarded as a UoC.
It is well-known that the computational complexity and the amount of measurements are two important considerations for the performance of any phase-retrieval method (c.f. [4, 19, 20]). The goal of this paper is to establish a phase retrieval method in a UoC having low computational complexity and requiring very few measurements. Motivated by , this goal will be achieved by establishing the following two-step PR-scheme:
Naturally, we need to address the issues in the following problem:
Under what conditions, is a UoC detectable, namely, the target cone can be detected by magnitude measurements? Is it possible to utilize the detectability to reduce the amount of phase-retrievable measurement vectors and computational complexity (e.g. it can be or the FFT complexity )?
I-C Existing results and our contributions
In what follows we introduce our main contribution in this paper from the aspects of cost of measurements and computational complexity (computational cost).
I-C1 Cost of measurements
For the detection, we establish the necessary and sufficient condition on the detectability of . Based on the condition we design a detector and the detection algorithm to detect the target cone. The detection can be achieved by using only -number of measurements. Once the detection is completed, we then perform the phase-retrieval on a single cone. As will be discussed in Remark II.2, there are at least cones, e.g. in the detectable which satisfy the following overlap property
For the target cone in (1.4), we will design -number of measurement vectors for the phase retrieval. Our contribution on the amount of measurements is that if all the cones satisfy (1.4), then ()-number of measurement vectors are sufficient for the two-step PR-scheme (1.3), where .
We emphasize two features of this approach. (i) By the complement property for phase retrievable frames (c.f. [4, 5, 6]), we know that any phase-retrieval method that applies to the signals in requires at least measurement vectors. Obviously, for many detectable UoCs, the scheme (1.3) requires much fewer than measurements (ii) It is well known that the amount of measurement vectors can be significantly reduced for sparse signals (e.g. [21, 22]). In our case, being small does not necessarily imply that the signals in are sparse. So the reduction strategy for the amount of measurements by scheme (1.3) is different from the treatment of sparse signals.
I-C2 Computational complexity
By using the i.i.d Gaussian measurement vectors, E. Candes, Y. Eldar, T. Strohmer and V. Voroninski  proposed the well-known PhaseLift method to recover z in (or ). Since then, based on the random measurements, many other efficient phase-retrieval methods such as Wirtinger Flow , Alternating Minimization , PhaseCut  and BlockPR  have been proposed. Among the above methods, the BlockPR, which holds for flat signals, has the lowest computational complexity .The signals in a cone may not be necessarily flat, and so they do not necessarily satisfy the condition required for the BlockPR method. However, by exploiting the structure of the detectable UoC, the goal of significantly reducing the computational complexity can also be achieved. Our Algorithm 1 for detection costs -number of operations. Theorem II.5 shows that if the target signal lies in the cone satisfying (1.4), then after detection, the phase-retrieval of the target signal can be completed by -operations, where . Our contribution on the computational complexity is that the proposed phase-retrieval scheme (1.3) for a detectable union of -cones all satisfying (1.4) has the computational complexity , which can be or for many cases of and .
Ii Two-step scheme for recovering signals in detectable union of cones
Our PR-scheme (1.3) consists of detection and recovery. In Subsection II-A we establish the sufficient and necessary condition for the detectability of a UoC. The algorithm for this detection is presented in Algorithm 1. Following this we discuss in Subsection II-B (Remark II.2) the cone structure derived from the above condition that is crucial to help achieve our goal. We also found that a union of linear subspaces (or spaces) is not detectable (Remark II.3). The main results on the recovery will be presented in Subsection II-C.
This subsection aims at establishing the sufficient and necessary condition for the detectability of a UoC, and presenting a detection algorithm for the target cone.
A UoC , where , is detectable if and only if for every we have either
Proof: The proof is given in Subsection V-A. Suppose that . If, for example, the first equation in (2.5) holds, pick , then we determine that when , and when . It is easy to see that based on (2.5), we can use the exclusions similar to the above to detect the target cone. The detection can be completed by using Algorithm 1.
Ii-B Remarks on the detectable union of cones
(i) By Algorithm 1, the source of any can be detected through exclusions if condition in (2.5) is satisfied. Only one measurement vector is required for every exclusion. Therefore we need -number of measurement vectors for the target cone detection. Moreover the detection requires -number of operations. (ii) The condition (2.5) implies that the overlap property
holds for at leat number of cones.
As mentioned in Section I, a linear space (subspace) is a special type of cone. An interesting problem is: can the union of linear spaces (subspaces) be detectable? The following remark tells us that a detectable UoC has at most one of the cones that is a linear subspace (This can be easily proved by Remark II.2 (ii) and the fact that a cone satisfying (2.6) is not a linear subspace). That is for any signal in the union of linear spaces (subspaces), the target cone where the signal is residing can not be detected by magnitude measurements.
Suppose that the UoC is detectable. Consequently, there exist at least cones satisfying the overlap property (2.6), and none of the cones is a linear space (subspace). If there exists a linear space (subspace) among the cones, then it is the unique one and does not have the overlap property (2.6). In other words, a union of linear spaces (subspaces) is not detectable, and it does not satisfy the requirement for the proposed approach.
The condition (2.6) is equivalent to that the system of linearly inequalities
has a solution. There exist many methods (e.g. in [27, 28, 29]) in the liturature that can be used to determine whether the (2.7) has a solution. The condition in (2.5) is equivalent to that the optimum of the following quadratic programming problem
After the detection by the procedures outlined in Algorithm 1, we can detect the cone that contains the target signal. What left is to perform phase retrieval on the target cone but not on the entire set UoC. As discussed in Section I, applying some of the existing methods to a finitely generated cone is either too expensive in terms of computational complexity and measurements or not even applicable due to the restriction of the methods. For example, the recently proposed fast method BlockPR by M. A. Iwen, A. Viswanathan, and Y. Wang  applies to flat vectors, but does not necessarily applies to vectors in a cone. In this subsection we establish a fast PR method for the cone in a detectable UoC with relatively fewer measurements and low computational complexity. The main results are outlined in Theorem II.5, Theorem II.6 and Proposition II.7.
Let be a cone with such that the overlap property (2.6) holds. Then there exist -vectors such that determines z (up to a unimodular scalar) for any , where . Moreover, can be designed in such a way that the recovery of z requires only -number of operations, i.e., the computational cost is FFT-time.
Proof: The proof is given in Section III.
Theorem II.5 implies that the property (2.6) is crucial for reducing the amount of measurements and computational complexity for the PR in a cone. By Remark II.2 (ii) there are at least cones in the detectable UoC which satisfy (2.6). We have the following result for the case when all the cones in satisfy (2.6).
Proof: By Remark II.2(i), the detection strategy in Algorithm 1 needs magnitude measurements. After the detection step, the phase-retrieval is performed on the target cone. Since all the cones satisfy the overlap property (2.6), by Theorem II.5 the phase-retrieval on the target cone needs at most magnitude measurements. Then measurements are sufficient for the two-step PR-scheme. The rest of the proof can be concluded by Remark II.2(i) and Theorem II.5.
The following proposition states that for many cases of and , the scheme (1.3) requires very few measurements and has very low computational complexity.
(i) The smaller , the fewer measurements we need for our PR scheme (1.3). In particular, when we can use less then measurements (the critical amount related to complement property) to complete our PR scheme.
(ii) As for the computational complexity, if and is a constant independent of , then our scheme can be performed by -number of operations. If , then our scheme can be done by -number of operations, the FFT time.
Proof: We first prove Part (i). By Theorem II.5, there exist phase retrievable vectors for . If the unit ball , then the vectors above can also do PR for B and for . By the complement property in , however, it requires at least vectors to do PR for and also for the unit ball. This is a contradiction, and the proof is concluded. Part (ii) can be proved similarly by Theorem II.6 and the complement property.
Iii Proof of Theorem ii.5, algorithm for the phase-retrievable measurement vectors, and the recovery formula
Before proving Theorem II.5 and presenting an algorithm for therein, we need some preparations. Recall that in Theorem II.5 may not be a frame cone. However, the cone in Lemma III.1 or Lemma III.2 will be required to be a frame-type. In order to avoid notation confusions, we will use instead of before we present the proof of Theorem II.5, where .
is the unique solution to the following equation with respect to the variable ,
Since the measurements are all positive, we will call p an anchor vector.
Iii-a Two auxiliary lemmas and design of special anchor vector
Proof: If , then the -column vectors of are a basis of . Naturally, in this case, and the result holds. We next prove the lemma for the case of Without losing generality, we assume that the first -column vectors of form a basis of . Let Denote
By (3.9), is the solution to (3.10) with z being replaced by . Recall that of is a basis of . Then can be also expressed as . Since the set of all the invertible matrices is dense in , there exist for such that
is invertible and
For , define
That is, . Using (3.13
) again, the invertible matrixconsist of the first rows of . Thus , and the proof is concluded.
We also need circulant matrices that ensure fast computation (More details about this topic can be referred to ). For a vector
, its discrete Fourier transform (DFT)is defined by . For the row vector , we denote its generating circulant matrix by , namely,
The circulant matrix can be decomposed by DFT via
where is the scaled DFT matrix
with . For any , by the fast Fourier transform (FFT), the computation of only costs -number of operations. The -norm of any vector is defined as the number of its nonzero coordinates. By (3.15), the circulant matrix is invertible if and only if
The following lemma tells us how to explicitly construct a special anchor vector p of in Lemma III.1 such that It will be seen in the proof of Theorem II.5 that such an anchor vector is crucial for explicitly constructing a special class of measurement vectors that will satisfy the requirements of Theorem II.5.
Let the frame cone of be as in Lemma III.1 such that . Then there exists an anchor vector such that .
Proof: As in the proof of Lemma III.1, we assume that the first -column vectors of form a basis of . For convenient narration, denote . By Lemma III.1, there exist -linearly independent vectors , where Define , and as in (3.13), Then Moreover, by (3.9), . Therefore, Now for any fixed , there exists a column vector of
If not, then it is easy to conclude that where is the -th row of . From the invertibility of , we deduce that , which is a contradiction with the invertibility of .
Pick a vector . If , then the proof is completed by letting . Otherwise, by the property (3.16), there exists such that , where is the support of , and . It is easy to prove that , where
On the other hand, it is obvious that Thus by at most -procedures discussed above, we will be able to get a vector such that Therefore
is an anchor vector satisfying .
Iii-B Proof of Theorem ii.5
The proof will be concluded from two cases: frame cone and non-frame cone.
Iii-B1 is a frame cone
Obviously, By Algorithm 2, we can construct an anchor vector such that
Thus the circulant matrix is invertible. Denote . Let and design by
where is selected in such a way that any satisfies
It follows from (3.20) that for any and On the other hand, it is easy to follow from
that is a basis of . Thus the target signal can be determined, up to a global sign, by the following linear system of equations
By (3.15), the above system can be rewritten as
That is, up to a global sign, z can be recovered by
It is easy to see that the computational complexity of (3.22) is .
Iii-B2 is not a frame cone
Denote Then Define an isometry . Specifically,
where and are the orthornormal basis and the standard orthornormal basis of and , respectively. Denote . By the linear and isometry property, , and also satisfies the overlap property (2.6). By Algorithm 2, we can design an anchor vector of such that and
Denote Invoking Case III-B1 for , we can additionally design vectors such that are phase retrievable for . That is, any signal can be determined by the magnitude measurements , and the corresponding complexity is . Particularly, for the target z, its projection can be recovered by invoking (3.22), namely,