Lifted Sets: A Dictionary

Proposition 8.2.10(tangent space, (Vandereycken, 2013, Proposition 2.1)). Let S≤rbe the set of matrices inR^m×nwith rank at most r, let X ∈S≤rwithrank(X) =r, and let UΣV^>be

Example 8.2.12. Now we consider a simple example with the assumption that n > m > 2r:

Take x¯ = at most 2r while the only matrix in C with rank less than 2r is x itself. This is an example¯ of an affine subspace that intersects S≤r at only one point and which is tangential to S≤r. By (Hesse, 2014, Theorem 5.19 c)), since max

hu,viu∈T_S≤r(x¯)∩_B,v ∈T_C(x¯)∩_B = 1, we cannot have a locally linearly regular intersection C∩S≤r. See Figure 8.3 for an example of the resulting slow convergence of alternating projections between C and S≤r.

8.3 Lifted Sets: A Dictionary

In Chapter 10, we will study the behavior of the method of alternating projections where one of the sets is the set of matrices of rank one. As a preparation of this, we establish a link between constraint sets inCⁿ and constraint sets of matrices in C^n×n. The motivation in our case is the fact that we can reformulate the physical problem of phase retrieval inCⁿin terms of minimizing the rank of a matrix over an affine subspace of matrices inC^n×n.

8 The Set of Low Rank Matrices

Figure 8.3: Convergence of alternating projections between the tangent space ofS≤rand S≤ritself.

1. for all Hermitian X∈ C ∩S≤1there exists x∈ C such that xx^∗ =X;

2. xx^∗ ∈ Cfor all x ∈C.

Remark 8.3.2. For a given set C ⊂ _Cⁿ, Definition 8.3.1 does not determine the lift of C uniquely. The idea is more totranslateconstraints, formulated inCⁿ, into well-described con-straints in the matrix space. Then an analysis of the intersection of the lift with the set of rank one matrices also gives further insight on C. In particular, for a subset C⊂_Cⁿ, it would be fully sufficient to takeC _B{xx^∗|x∈C}. However, in order to benefit from additional structure of lifts of C, we can also take for example an affine hull of{xx^∗|x∈C}.

Proposition 8.3.3(uniqueness up to a global phase). For all Hermitian X∈ C ∩S≤1, the vector x∈C such that xx^∗ = X is determined up to global phase. This means that from some rank one matrixxx^∗ the vector xis unique at least up to a global phaseθ. Assume now that there existsy∈_Cⁿsuch thatyy^∗ = xx^∗andy ,e^iθx for allθ ∈ [0, 2π]. Becauseyy^∗ = xx^∗, we know thatx_jx_j = y_jy_j for allj. Hence, there existθ₁, . . . ,θ_n such thaty_j = _e^iθj_{xj for all j. Ify , e}^iθ_{x for all}θ ∈ [_{0, 2π}], then there exists at least oneksuch thatθ_k , θ_jfor allj, k. We look at one of those entries ofyy^∗

88

8.3 Lifted Sets: A Dictionary

indexed bykandjwithθ_j ,θ_k:

yy^∗_jk =y_jy_k =e^iθj x_je^−iθkx_k =e^i(θj−θk)xx^∗_jk ,xx^∗_jk. (8.14) This is a contradiction. Hence, there exists someθ ∈[0, 2π]such thaty=e^iθx.

8.3.1 Lifts of Linear Spaces and Cones

We focus now on affine subspaces and cones ofCⁿ. The first and easiest example would be the lift of an affine subspace ofCⁿ. Additionally, we consider two examples moti-vated by the physical application of phase retrieval that will be revisited in Definition 9.2.2.

Proposition 8.3.4(lifts of affine spaces). Let A : C^m → _Cⁿ be a linear mapping and let b∈_C^m. Define the affine subspace

BB{x∈_Cⁿ|Ax=b}. (8.15)

Then the set

B _BX∈_C^n×nAXA^∗ = bb^∗ (8.16)

is an affine subspace, and it is a lift of B.

Proof. If there is some x ∈ B, then the rank one matrix xx^∗ satisfies the equation Axx^∗A^∗ =bb^∗.

The setBis an affine subspace ofC^n×nif for everyλ∈_Cwe haveλx+ (1−λ)y∈ B for allx,y∈ B. Choose some arbitraryλand someX,Y∈ B_{. Then}

A(λX+ (1−λ)Y)A^∗ = AλXA^∗+A(1−λ)YA^∗

=λAXA^∗+ (1−λ)AYA^∗ =bb^∗.

Lemma 8.3.5(support constraints). Let I be an index set in{1, . . . ,n}and define

C_sB

x∈_Cⁿx_j =₀if j<I . (8.17) Then the set

C_sBX∈_C^n×n X_jk =0if j< I or if k< I is a lift of Cs.

Proof. LetX∈ Cand rank(X) =1. ThenXis Hermitian, and it can be written as some xx^∗forx∈_Cⁿ. Suppose there existsj< I. ThenX_jk=0 for all 1≤k ≤n. If there exists ksuch thatX_kk,0, then this means thatx_j =0, and thusx∈Cs.

On the other hand, ifx ∈C_s, then, for allj<I, we havex_j =0. This gives usxx^∗_jk=0 for all 1≤k≤ n, which is all that is required to show thatxx^∗ ∈ C_s.

8 The Set of Low Rank Matrices

Proposition 8.3.6(lift of a convex cone). Let A:C^m →_Cⁿbe a linear mapping. Define the convex cone

K={x ∈_Cⁿ|Ax≥0}. (8.18)

The set

K_BX∈_C^n×n AXA^∗ ≥0 (8.19)

is a lift of K, and it is a convex cone.

Proof. Note that for every convex coneKwe have 0∈ K. This means that for everyx∈ Kwe haveAxx^∗A^∗ ≥ 0. On the other hand, if a positive semidefinite rank one matrix xx^∗ satisfies Axx^∗A^∗ ≥ 0, then either x ∈ Kor −x ∈ K. LetX ∈ K, then^˜ AXA^∗ ≥ 0, and naturally, for allλ ≥ 0, we have AλXA^∗ = λAXA^∗ ≥ 0. On the other hand, let λ∈[0, 1]andX,Y. ThenA(λX+ (1−λ)Y)A^∗ = λAXA^∗+ (1−λ)AYA^∗≥0.

Lemma 8.3.7(positive real cone). For

CBRⁿ≥0=x∈_Cⁿx ∈_Rⁿ,x_j ≥0for all j , (8.20) the setC _BⁿX∈_C^n×nX∈ _Rⁿ_≥^×₀ⁿ,X^∗ = Xo

is a lift of C.

Proof. LetX∈ C_sand rank(X) =1. ThenXis Hermitian, and it can be written as some xx^∗ for x ∈ _Cⁿ. Let j be such that X_jj , 0. If such a j does not exist, then X is the zero matrix. Choosex_j = ^p_Xjj > _{0. Because xjxk} ∈ _R≥0 for allk, we conclude that

x_k ∈_R≥0.

8.3.2 Quadratic Constraints

The examples so far gave subspaces as lifts of subspaces. An interesting example of the benefits of lifted sets is the following: suppose we are given an unitary mapping A:Cⁿ →_Cⁿand, for someb∈_Rⁿ_≥0, the set

MB n

x

|ha_k,xi|²=b_kfor all 1≤k≤ no

, (8.21)

where a_k are the complex conjugates of the rows of A. This is due to the fact that ha_k,xi = a^∗_kx. We reformulate this problem. We define for a vectorx ∈ _Cⁿthe matrix X B xx^∗, which is of rank one and positive semidefinite. Similarly, define for every k=1, . . . ,mthe matrixA_kBa_ka^∗_k.

Proposition 8.3.8. Define the mappingA_kviaA_k(X) =Tr A^∗_kX

. The set

B_BX∈_C^n×nA_k(X) =b for all1≤k≤n (8.22) is an affine subspace ofC^n×n, andBis a lift of M.

Proof. The restrictions can be written as

|ha_k,xi|² =Tr(x^∗a_ka^∗_kx) =Tr(a_ka^∗_kxx^∗) =Tr(A_kX) =A_k(X). (8.23)

90

8.4 Second-Order Subdifferentials at the Set of Rank-Constrained Matrices In other words, for every x ∈ M there existsX = xx^∗ ∈ B, and for everyX ∈ B ∩S≤1

there existsx∈_Cⁿsuch thatxx^∗ = X.

If A is the discrete Fourier transform, then the above model can be applied to the phase retrieval problem (Chapter 9). The idea of reformulating quadratic constraints as linear equations in the matrix space comes from semidefinite programming. Its appli-cation to phase retrieval goes back to the article by Cand`es et. al in (Cand`es et al., 2011, Section 2.2). We discuss this matter in Section 10.1.

8.4 Second-Order Subdifferentials at the Set of