Optimal dual frames and frame completions for majorization
aa r X i v : . [ m a t h . F A ] J un Optimal dual frames and frame completions forma jorization
P. G. Massey, M. A. Ruiz and D. Stojanoff
Depto. de Matem´atica, FCE-UNLP, La Plata, Argentina and IAM-CONICET
Dedicated to the memory of “el flaco” L. A. Spinetta.
Abstract
In this paper we consider two problems in frame theory. On the one hand, given a set ofvectors F we describe the spectral and geometrical structure of optimal completions of F bya finite family of vectors with prescribed norms, where optimality is measured with respect tomajorization. In particular, these optimal completions are the minimizers of a family of con-vex functionals that include the mean square error and the Benedetto-Fickus’ frame potential.On the other hand, given a fixed frame F we describe explicitly the spectral and geometricalstructure of optimal frames G that are in duality with F and such that the Frobenius norms oftheir analysis operators is bounded from below by a fixed constant. In this case, optimality ismeasured with respect to submajorization of the frames operators. Our approach relies on thedescription of the spectral and geometrical structure of matrices that minimize submajorizationon sets that are naturally associated with the problems above. AMS subject classification: 42C15, 15A60.Keywords: frames, dual frames, frame completions, majorization, Schur-Horn
Contents D t ( F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Introduction
Finite frame theory is a well established research field that has attracted the attention of manyresearchers (see [9, 16, 22] for general references to frame theory). On the one hand, finite framesprovide redundant linear encoding-decoding schemes that are useful when dealing with transmissionof signals through noisy channels . Indeed, the redundancy of frames allows for reconstruction ofa signal, even when some frame coefficients are lost. Moreover, frames have also shown to berobust under erasures of the frame coefficients when a blind reconstruction strategy is considered(see [4, 5, 6, 23, 26, 28, 33]). On the other hand, there are several problems in frame theory thathave deep relations with problems in other areas of mathematics (such as matrix analysis, operatortheory and operator algebras) which constitute a strong motivation for research. For example,we can mention the relation between the Feichtinger conjecture in frame theory and some majoropen problems in operator algebra theory such as the Kadison-Singer problem (see [12, 13]). Otherexamples of this phenomenon are the design problem in frame theory, the so-called Paulsen problemin frame theory and frame completion problems ([1, 8, 10, 11, 14, 17, 18, 19, 20, 25, 29]) which areknown to be equivalent to different aspects of the Schur-Horn theorem. Recently, matrix analysishas served as a tool to show some structural properties of minimizers of the Benedetto-Fickus framepotential ([2, 15]) and other convex functionals in the finite setting ([30, 31, 32]).Following [1, 17, 29, 30, 31, 32], in this paper we explore new connections of problems thatarise naturally in frame theory with some results in matrix theory related with the notion of(sub)majorization between vectors and positive matrices. Indeed, one of the main problems inframe theory is the design of frames with some prescribed parameters and such that they areoptimal in some sense. Optimal frames F are usually the minimizers of a tracial convex functionali.e., a functional of the form P f ( F ) = tr( f ( S F )) for some convex function f ( x ), where S F isthe frame operator of F . For example, we mention the Benedetto-Fickus’ frame potential (i.e. f ( x ) = x ) or the mean square error (i.e. f ( x ) = x − ) or the negative of von Neumann’s entropy(i.e. f ( x ) = x log( x )). Thus, in many situations it is natural to ask whether the optimal framescorresponding to different convex potentials coincide: that is, whether optimality with respect tothese potentials is an structural property. One powerful tool to deal with this type of problemsis the notion of (sub)-majorization between positive operators, because of its relation with tracialinequalities with respect to convex functions as above (see Section 2.3). Hence, a (sub)-majorizationbased strategy can reveal structural properties of optimal frames. It is worth pointing out that(sub)-majorization is not a total preorder and therefore the task of computing minimizers of thisrelation within a given set of positive operators - if such minimizers exist - is usually a non trivialproblem.In this paper we consider the following two optimality problems in frame theory in terms of(sub)-majorization (see Section 2 for the notation and terminology). Given a finite sequence ofvectors F ⊆ H ∼ = C d and a finite sequence of positive numbers b we are interested in computingoptimal frame completions of F , denoted by F , obtained by adding vectors with norms prescribedby the entries of b (see Section 3.1 for the motivation and a detailed description of this problem).In this context we show the existence of minimizers of majorization in the set of frame completionsof F with prescribed norms, under certain hypothesis on b ; we also compute the spectral andgeometrical structure of these optimal completions. Our results can be considered as a further stepin the classical frame completion and frame design problems considered in [1, 8, 10, 14, 17, 18, 20,25]. In particular, we solve the frame completion problem recently posed in [20], where optimalityis measured with respect to the mean square error of the completed frame.On the other hand, given a fixed frame F for a finite dimensional Hilbert space H ∼ = C d ,let D ( F ) denote the set of all frames G that are in duality with F . It is well known that thecanonical dual of F , denoted F , has some optimality properties among the elements in D ( F ).Nevertheless, although optimal in some senses, there might be alternate duals that are more suitablefor applications (see [7, 21, 26, 28, 33, 34]). In order to search for optimal alternative duals for2 we restrict attention the set D t ( F ) which consists of frames G that are in duality with F andsuch that the Frobenius norm of their frame operators is bounded from below by a constant t .Therefore, in this paper we show the existence of minimizers of submajorization in D t ( F ) and weexplicitly describe their spectral and geometrical structure (see Section 3.2 for the motivation anda detailed description of this problem).Both problems above are related with the minimizers of (sub)majorization in certain sets S of positive semidefinite matrices that arise naturally. We show that these sets S that we considerhave minimal elements with respect to (sub)-majorization, a fact that is of independent interest (seeTheorems 5.12, 5.16 and 3.12). Notably, the existence of such minimizers is essentially obtainedwith insights coming from frame theory.The paper is organized as follows: In Section 2 we establish the notation and terminology usedthroughout the paper, and we state some basic facts from frame theory and majorization theory.In Sections 3.1 and 3.2 we give a detailed description of the two main problems of frame theorymentioned above, including motivations, related results and specific notations. Section 3 ends withthe definitions and statements of the matrix theory results of the paper, which give a unified matrixmodel for the frame problems; in order to avoid some technical aspects of these results, their proofsare presented in an Appendix (Section 5). In Section 4 we apply the previous analysis of thematrix model to obtain the solutions of the frame problems, including algorithmic implementationsand several examples. With respect to the problem of optimal completions, we obtain a completedescription in several cases, that include the case of uniform norms for the added vectors. Withrespect to the problem of minimal duals, we completely describe their spectral and geometricalstructure. The Appendix, Section 5, contains the proofs of the matrix theory results of Section 3.3;it is divided in three subsections in which we develop the following steps: the characterization of theset of vectors of eigenvalues of elements in the matrix model, the description of the minimizers forsub-majorization in this set, and the description of the geometric structure of the matrices whichare minimizers for sub-majorization in the matrix model. In this section we describe the basic notions that we shall consider throughout the paper. We firstestablish the general notations and then we recall the basic facts from frame theory that are relatedwith our main results. Finally, we describe submajorization which is a notion from matrix analysis,that will play a major role in this note.
Given m ∈ N we denote by I m = { , . . . , m } ⊆ N and = m ∈ R m denotes the vector with allits entries equal to 1. For a vector x ∈ R m we denote by x ↓ the rearrangement of x in decreasingorder, and R m ↓ = { x ∈ R m : x = x ↓ } the set of ordered vectors.Given H ∼ = C d and K ∼ = C n , we denote by L ( H , K ) the space of linear operators T : H → K .Given an operator T ∈ L ( H , K ), R ( T ) ⊆ K denotes the image of T , ker T ⊆ H the null spaceof T and T ∗ ∈ L ( K , H ) the adjoint of T . If d ≤ n we say that U ∈ L ( H , K ) is an isometry if U ∗ U = I H . In this case, U ∗ is called a coisometry. If K = H we denote by L ( H ) = L ( H , H ), by G l ( H ) the group of all invertible operators in L ( H ), by L ( H ) + the cone of positive operators and by G l ( H ) + = G l ( H ) ∩ L ( H ) + . If T ∈ L ( H ), we denote by σ ( T ) the spectrum of T , by rk T = dim R ( T )the rank of T , and by tr T the trace of T . By fixing an orthonormal basis (onb) of the Hilbertspaces involved, we shall identify operators with matrices, using the following notations:By M n,d ( C ) ∼ = L ( C d , C n ) we denote the space of complex n × d matrices. If n = d we write M n ( C ) = M n,n ( C ). H ( n ) is the R -subspace of selfadjoint matrices, G l ( n ) the group of all invertibleelements of M n ( C ), U ( n ) the group of unitary matrices, M n ( C ) + the set of positive semidefinite3atrices, and G l ( n ) + = M n ( C ) + ∩ G l ( n ). If d ≤ n , we denote by I ( d , n ) ⊆ M n , d ( C ) the setof isometries, i.e. those U ∈ M n , d ( C ) such that U ∗ U = I d . Given S ∈ M n ( C ) + , we write λ ( S ) ∈ R n + ↓ the vector of eigenvalues of S - counting multiplicities - arranged in decreasing order.If λ ( S ) = λ = ( λ , . . . , λ n ) ∈ R n + ↓ , a system { h i } i ∈ I n ⊆ C n is a “ONB of eigenvectors for S , λ ”if it is an orthonormal basis for C n such that S h i = λ i h i for every i ∈ I n .If W ⊆ H is a subspace we denote by P W ∈ L ( H ) + the orthogonal projection onto W , i.e. R ( P W ) = W and ker P W = W ⊥ . Given x , y ∈ H we denote by x ⊗ y ∈ L ( H ) the rank oneoperator given by x ⊗ y ( z ) = h z , y i x for every z ∈ H . Note that if k x k = 1 then x ⊗ x = P span { x } .For vectors in C n we shall use the euclidean norm. On the other hand, for T ∈ M n , d ( C ) we shalluse both the spectral norm, denoted k T k , and the Frobenius norm, denoted k T k , given by k T k = max k x k =1 k T x k and k T k = (tr T ∗ T ) / = (cid:0) X i ∈ I n , j ∈ I d | T ij | (cid:1) / . In what follows we consider ( n, d )-frames. See [2, 9, 16, 22, 30] for detailed expositions of severalaspects of this notion.Let d, n ∈ N , with d ≤ n . Fix a Hilbert space H ∼ = C d . A family F = { f i } i ∈ I n ∈ H n is an( n, d )-frame for H if there exist constants A, B > A k x k ≤ n X i =1 | h x , f i i | ≤ B k x k for every x ∈ H . (1)The frame bounds , denoted by A F , B F are the optimal constants in (1). If A F = B F we call F a tight frame. Since dim H < ∞ , a family F = { f i } i ∈ I n is an ( n, d )-frame if and only ifspan { f i : i ∈ I n } = H . We shall denote by F = F ( n , d ) the set of all ( n, d )-frames for H .Given F = { f i } i ∈ I n ∈ H n , the operator T F ∈ L ( H , C n ) defined by T F x = (cid:0) h x , f i i (cid:1) i ∈ I n , for every x ∈ H (2)is the analysis operator of F . Its adjoint T ∗F is called the synthesis operator: T ∗F ∈ L ( C n , H ) given by T ∗F v = X i ∈ I m v i f i for every v = ( v , . . . , v n ) ∈ C n . Finally, we define the frame operator of F as S F = T ∗F T F = P i ∈ I n f i ⊗ f i ∈ L ( H ) + . Noticethat, if F ∈ F ( n , d ), then h S F x , x i = P i ∈ I n (cid:12)(cid:12) h x , f i i (cid:12)(cid:12) for every x ∈ H , so S F ∈ G l ( H ) + and A F k x k ≤ h S F x , x i ≤ B F k x k for every x ∈ H . (3)In particular, A F = λ min ( S F ) = k S − F k − and λ max ( S F ) = k S F k = B F . Moreover, F is tight ifand only if S F = τd I H , where τ = tr S F = P i ∈ I n k f i k .The frame operator plays an important role in the reconstruction of a vector x using its framecoefficients {h x , f i i } i ∈ I n . This leads to the definition of the canonical dual frame associated to F : for every F = { f i } i ∈ I n ∈ F ( n , d ), the canonical dual frame associated to F is the sequence F ∈ F defined by F def = S − F · F = { S − F f i } i ∈ I m ∈ F ( n , d ) . Therefore, we obtain the reconstruction formulas x = X i ∈ I n h x , f i i S − F f i = X i ∈ I n h x , S − F f i i f i for every x ∈ H . (4)4bserve that the canonical dual F satisfies that given x ∈ H , then T F x = (cid:0) h x , S − F f i i (cid:1) i ∈ I n = (cid:0) h S − F x , f i i (cid:1) i ∈ I n for x ∈ H = ⇒ T F = T F S − F . (5)Hence T ∗F T F = I H and S F = S − F T ∗F T F S − F = S − F .In their seminal work [2], Benedetto and Fickus introduced a functional defined (on unit normframes), the so-called frame potential, given byFP ( { f i } i ∈ I n ) = X i, j ∈ I n |h f i , f j i| . One of their major results shows that tight unit norm frames - which form an important class offrames because of their simple reconstruction formulas - can be characterized as (local) minimizersof this functional among unit norm frames. Since then, there has been interest in (local) minimizersof the frame potential within certain classes of frames, since such minimizers can be considered asnatural substitutes of tight frames (see [15, 30, 31]). Notice that, given F = { f i } i ∈ I n ∈ H n thenFP ( F ) = tr( S F ) = P i ∈ I d λ i ( S F ) . These remarks have motivated the definition of general convexpotentials as follows: Definition 2.1.
Let f : [0 , ∞ ) → [0 , ∞ ) be a convex function. Following [30] we consider the(generalized) frame potential associated to f , denoted P f , given by P f ( F ) = tr( f ( S F )) for F = { f i } i ∈ I n ∈ H n . △ Of course, one of the most important generalized potential is the Benedetto-Fickus’ (BF) framepotential. As shown in [30, Sec. 4] these convex functionals (which are related with the so-calledentropic measures of frames) share many properties with the BF-frame potential. Indeed, undercertain restrictions both the spectral and geometric structures of minimizers of these potentialscoincide (see [30]).
Next we briefly describe submajorization, a notion from matrix analysis theory that will be usedthroughout the paper. For a detailed exposition of submajorization see [3].Given x, y ∈ R d we say that x is submajorized by y , and write x ≺ w y , if k X i =1 x ↓ i ≤ k X i =1 y ↓ i for every k ∈ I d . If x ≺ w y and tr x = d P i =1 x i = d P i =1 y i = tr y , then we say that x is majorized by y , and write x ≺ y .On the other hand we write x y if x i ≤ y i for every i ∈ I d . It is a standard exercise to show that x y = ⇒ x ↓ y ↓ = ⇒ x ≺ w y . Majorization is usually considered because of its relation withtracial inequalities for convex functions. Indeed, given x, y ∈ R d and f : I → R a convex functiondefined on an interval I ⊆ R such that x, y ∈ I d , then (see for example [3]):1. If one assumes that x ≺ y , then tr f ( x ) def = d P i =1 f ( x i ) ≤ d P i =1 f ( y i ) = tr f ( y ) .
2. If only x ≺ w y , but the map f is also increasing, then still tr f ( x ) ≤ tr f ( y ).3. If x ≺ w y and f is an strictly convex function such that tr( f ( x )) = tr( f ( y )) then there existsa permutation σ of I d such that y i = x σ ( i ) for i ∈ I d .5he notion of submajorization can be extended to the context of self-adjoint matrices as follows:given S , S ∈ H ( d ) we say that S is submajorized by S , denoted S ≺ w S , if λ ( S ) ≺ w λ ( S ).If S ≺ w S and tr( S ) = tr( S ) we say that S is majorized by S and write S ≺ S . Thus, S ≺ S if and only if λ ( S ) ≺ λ ( S ). Notice that (sub)majorization is an spectral relation betweenself-adjoint operators. We begin this section with a detailed description of our two main problems together with theirmotivations. In both cases we search for optimal frame designs (frame completions and duals),that are of potential interest in applied situations. In order to tackle these problems we obtain (seeSections 3.1 and 3.2) equivalent versions of them in a matrix analysis context. In section 3.3 wepresent a unified matrix model and develop some notions and results that allow us to solve the twoproblems in frame theory (see Section 4).
We begin by describing the following frame completion problem posed in [20]. Let
H ∼ = C d and let F = { f i } i ∈ I n o ∈ H n o be a fixed (finite) sequence of vectors. Let n > n o be an integer; denote by k = n − n o and assume that rk S F ≥ d − k . Consider a sequence a = { α i } i ∈ I n ∈ R n> such that k f i k = α i for every i ∈ I n o .With the fixed data from above, the problem posed in [20] is to find a sequence F = { f i } ni = n o +1 ∈H k with k f i k = α i , for n o + 1 ≤ i ≤ n , such that the the mean square error of the resulting com-pleted frame F = ( F , F ) = { f i } i ∈ I n ∈ F ( n , d ), namely tr( S − F ), is minimal among all possiblesuch completions. It is worth pointing out that the mean square error of F = ( F , F ) dependson F through the eigenvalues λ ( S F ) of its frame operator.Note that there are other possible ways to measure robustness of the completed frame F asabove. For example, we can consider optimal (minimizing) completions, with prescribed norms,for the Benedetto-Fickus’ potential. In this case we search for a frame F = ( F , F ) = { f i } i ∈ I n ∈ F ( n , d ), with k f i k = α i for n o + 1 ≤ i ≤ n , and such that its frame potential FP ( F ) = tr( S F ) isminimal among all possible such completions. As before, we point out that the frame potential ofthe resulting completed frame F = ( F , F ) depends on F through the eigenvalues λ ( S F ) of theframe operator of F .Hence, in order to solve both problems above we need to give a step further in the classicalframe completion problem (i.e. decide whether F can be completed to a frame F = ( F , F ) withprescribed norms and frame operator S ∈ M d ( C ) + ) and search for optimal (e.g. minimizers of themean square error or Benedetto-Fickus’ frame potential) frame completions with prescribed norms.At this point a natural question arises as whether the minimizers corresponding to the meansquare error and to the Benedetto-Fickus’ potential, or even more general convex potentials, co-incide (see [2, 15, 30]). As we shall see, the solutions of these problems are independent of theparticular choice of convex potential considered. Indeed, we show that under certain hypothesison the final sequence b = { α i } ni = n o +1 (which includes the uniform case) we can explicitly computethe completing sequences F = { f i } ni = n o +1 ∈ H k such that the frame operators of the completedsequences F = ( F , F ) are minimal with respect to majorization (within the set of frame operatorsof all completions with norms prescribed by the sequence a ). In order to do this, we begin by fixingsome notations. Definition 3.1.
Let F = { f i } i ∈ I n o ∈ H n o and a = { α i } i ∈ I n ∈ R n> such that d − rk S F ≤ n − n o and k f i k = α i , i ∈ I n o . We consider the sets C a ( F ) = (cid:8) { f i } i ∈ I n ∈ F ( n , d ) : { f i } i ∈ I n o = F and k f i k = α i for i ≥ n o + 1 (cid:9) , C a ( F ) = { S F : F ∈ C a ( F ) } . △ In what follows we shall need the following solution of the classical frame completion problem.
Proposition 3.2 ([1, 29]) . Let B ∈ M d ( C ) + with λ ( B ) ∈ R d + ↓ and let b = ( β i ) i ∈ I k ∈ R k> . Thenthere exists a sequence G = { g i } i ∈ I k ∈ H k with frame operator S G = B and such that k g i k = β i for every i ∈ I k if and only if b ≺ λ ( B ) (completing with zeros if k = d ). (cid:3) Since our criteria for optimality of frame completions will be based on majorization, our analysis ofthe completed frame F = ( F , F ) will depend on F through S F . Hence, the following descriptionof SC a ( F ) plays a central role in our approach. Proposition 3.3.
Let F = { f i } i ∈ I n o ∈ H n o and a = { α i } i ∈ I n ∈ R n> such that k f i k = α i , i ∈ I n o .Then, we have that SC a ( F ) = (cid:8) S ∈ G l ( d ) + : S ≥ S and ( α i ) ni = n o +1 ≺ λ ( S − S ) (cid:9) . In particular, if we let k = n − n o then we get the inclusion SC a ( F ) ⊆ { S F + B : B ∈ M d ( C ) + , rk B ≤ k , tr( S F + B ) = n X i =1 α i } . (6) Proof.
Observe that if F = ( F , F ) ∈ F ( n , d ), then S F = S F + S F . Denote by S = S F and B = S − S , for any S ∈ G l ( d ) + . Applying Proposition 3.2 to the matrix B (which must benonnegative if S ∈ SC a ( F ) ), we get the first equality.The inclusion in Eq. (6) follows using that, if F = ( F , F ) ∈ F ( n , d ), then rk B = rk S F ≤ k = d − ( d − k ). On the other hand, recall that tr( S F ) = P ni =1 k f i k . (cid:3) Let F = { f i } i ∈ I n ∈ F ( n , d ). Then F induces and encoding-decoding scheme as described in Eq.(4), in terms of the canonical dual F . But, in case that F has nonzero redundancy then we geta family of reconstruction formulas in terms of different frames that play the role of the canonicaldual. In what follows we say that G = { g i } i ∈ I n ∈ F ( n , d ) is a dual frame for F if T ∗G T F = I H (and hence T ∗F T G = I H ), or equivalently if the following reconstruction formulas hold: x = X i ∈ I n h x , f i i g i = X i ∈ I n h x , g i i f i for every x ∈ H . We denote by D ( F ) def = {G ∈ F ( n , d ) : T ∗G T F = I H } the set of all dual frames for F . Observe that D ( F ) = ∅ since F ∈ D ( F ).Notice that the fact that F = { f i } i ∈ I n ∈ F ( n , d ) implies that T ∗F is surjective. In this case, asequence G ∈ D ( F ) if and only if its synthesis operator T ∗G is a pseudo-inverse of T F . Moreover,the synthesis operator T ∗F of the canonical dual F corresponds to the Moore-Penrose pseudo-inverse of T F . Indeed, notice that T F T ∗F = T F S − F T ∗F ∈ L ( C n ) + , so that it is an orthogonalprojection. From this point of view, the canonical dual F has some optimal properties that comefrom the theory of pseudo-inverses. Nevertheless, the canonical dual frame might not be the optimalchoice for a dual frame from an applied point of view. For example, it is well known that thereare classes of structured frames that admit alternate duals that share this structure but for whichtheir canonical duals are not structured ([7, 34]); in the theory of signal transmission through noisychannels, it is well known that there are alternate duals that perform better than F ([26, 28, 33])7hen we assume that the frame coefficients can be corrupted by the noise in the channel. Thereare other cases in which F may be ill-conditioned or simply too difficult to compute: for example,it is known (see [21]) that under certain hypothesis we can find Parseval dual frames G ∈ D ( F )(i.e. such that S G = I H ), which lead to more stable reconstruction formulas for vectors in H .In the general case, we can measure the stability of the reconstruction formula induced by adual frame G ∈ D ( F ) in terms of the spread of the eigenvalues of the frame operator S G ; this canbe seen if we consider, as it is usual in applied situations, the condition number of S G as a measureof stability of linear processes that depend on S G . There are finer measures of the dispersionwhich take into account all the eigenvalues of S G , if one restricts to the case of fixed trace. As anexample of such a measure we can mention the Benedetto-Fickus’ potential. Our approach basedon majorization - which is the structural measure of the spread of eigenvalues for matrices with afixed trace - allows us to show that minimizers with respect to a large class of convex potentialscoincide. The main advantages of considering the partition of D ( F ) into slices determined by thetrace condition tr( S G ) = t are: • There exists a unique vector ν ( t ) of eigenvalues which is minimal for majorization among thevectors λ ( S G ), for dual frames G ∈ D ( F ) with with tr S G = t . • Moreover, the vector ν ( t ) is also submajorized by the vectors λ ( S G ) for every G ∈ D ( F ) withwith tr S G ≥ t . • The map t ν ( t ) is increasing (in each entry) and continuous. • Continuous sections t
7→ G t ∈ D ( F ) such that λ ( G t ) = ν ( t ) can be computed. • In addition, the condition number of ν ( t ) decreases when t grows until a critical point (whichis easy to compute).We point out that both the vector ν ( t ) and the duals G t can be computed explicitly in terms ofimplementable algorithms. In order to obtain a convenient formulation of the problem we considerthe following notions and simple facts. Definition 3.4.
Let
F ∈ F ( n , d ). We denote by SD ( F ) = { S G : G ∈ D ( F ) } the set of frame operators of all dual frames for F . △ Proposition 3.5.
Let
F ∈ F ( n , d ) . Then SD ( F ) = { S F + B : B ∈ M d ( C ) + and rk B ≤ n − d } . (7) Proof.
Given
G ∈ F ( n , d ), then G ∈ D ( F ) ⇐⇒ Z = T G − T F ∈ L ( H , C n ) satisfies Z ∗ T F = 0.In this case, by Eq. (5), we know that T F = T F S − F = ⇒ Z ∗ T F = 0, and S G = ( T F + Z ) ∗ ( T F + Z ) = S F + B = S − F + B , where B = Z ∗ Z ∈ M d ( C ) + . Moreover, S F = T ∗F T F ∈ G l ( d ) + = ⇒ rk T F = d , and the equation T ∗F Z = 0 implies that R ( Z ) ⊆ ker T ∗F = R ( T F ) ⊥ = ⇒ rk B = rk( Z ∗ Z ) = rk Z ≤ n − d . Since any B ∈ M d ( C ) + with rk B ≤ n − d can be represented as B = Z ∗ Z for some Z ∈ L ( H , R ( T F ) ⊥ ), we have proved Eq. (7). (cid:3) Fix a system F = { f i } i ∈ I n ∈ F ( n , d ). Notice that Proposition 3.5 shows that if G ∈ D ( F ) then S F ≤ S G , which is a strong minimality property of the frame operator of the canonical dual F .8s we said befire, we are interested in considering alternate duals that are more stable than F .In order to do this, we consider the set D t ( F ) of dual frames G ∈ D ( F ) with a further restriction,namely that tr( S G ) ≥ t for some t ≥ tr( S F ). Therefore, the problem we focus in is to find dualframes G t ∈ D t ( F ) such that their frame operators S G t are minimal with respect to submajorizationwithin the set SD t ( F ) def = { S G : G ∈ D t ( F ) } . (8)Notice that as an immediate consequence of Proposition 3.5 we get the identity SD t ( F ) = { S F + B : B ∈ M d ( C ) + , rk B ≤ n − d , tr( S F + B ) ≥ t } . (9)As we shall see, these optimal duals G t decrease the condition number and, in some cases are eventight frames. Moreover, because of the relation between submajorization and increasing convexfunctions, our optimal dual frames G t ∈ D t ( F ) are also minimizers of a family of convex framepotentials (see Definition 2.1 below) that include the Benedetto-Fickus’ frame potential. In this section we introduce and develop some aspects of a set U t ( S , m ) ⊆ M d ( C ) + that will playan essential role in our approach to the frame problems described above (see Remark 3.7). Ourmain results related with U t ( S , m ) are Theorem 3.12 and Proposition 3.14. In order to avoidsome technicalities, we postpone their proofs to the Appendix (Section 5). Definition 3.6.
Let S ∈ M d ( C ) + with λ ( S ) = λ ∈ R d + ↓ , t = tr S , and t ≥ t . For any integer m < d we consider the following subset of M d ( C ) + : U t ( S , m ) = { S + B : B ∈ M d ( C ) + , rk B ≤ d − m , tr( S + B ) ≥ t } . (10)Observe that if m ≤ U t ( S , m ) = { S ∈ M d ( C ) + : S ≥ S , tr( S ) ≥ t } . △ Remark 3.7.
As a consequence of Eq. (6) and Eq. (9) we see that the two main problems areintimately related with the structure of the set U t ( S , m ) for suitable choices of the parameters S ∈ M d ( C ) + , m < d and t ≥ tr S :1. Note that Eq. (6) shows that SC a ( F ) ⊆ U t ( S F , m ), where t = tr a and m = d − n + n o .2. Similarly, Eq. (9) shows that identity SD t ( F ) = U t ( S F , m ) where m = 2 d − n . △ Remark 3.8.
Given λ = λ ( S ) ∈ R d + ↓ and m < d we look for a ≺ w -minimizer on the setΛ( U t ( S , m ) ) def = { λ ( S ) : S ∈ U t ( S , m ) } ⊆ R d + ↓ . (11)Heuristic computations suggest that in some cases such a minimizer should have the form ν = ( λ , . . . , λ r , c , . . . c ) ∈ R d> ↓ with tr ν = t for some r ∈ I d − and c ∈ R > . Observe that if ν ∈ Λ( U t ( S , m ) ) then λ ν = ν ↓ . Hence we need that c = t − P rj =1 λ j d − r and λ r +1 ≤ c ≤ λ r . These restrictions on the numbers r and c suggest the following definitions: △ efinition 3.9. Let λ ∈ R d + ↓ and t ∈ R such that tr λ ≤ t < d λ . Consider the set A λ ( t ) def = (cid:8) r ∈ I d − : p λ ( r , t ) def = t − P rj =1 λ j d − r ≥ λ r +1 (cid:9) . Observe that t ≥ tr λ = ⇒ t − d − P j =1 λ j ≥ λ d , so that d − ∈ A λ ( t ) = ∅ . The t -irregularity of theordered vector λ , denoted r λ ( t ), is defined by r λ ( t ) def = min A λ ( t ) = min { r ∈ I d − : p λ ( r , t ) ≥ λ r +1 } . (12)If t ≥ d λ , we set r λ ( t ) def = 0 and p λ (0 , t ) = t/d . △ For example, if t = tr λ then for every r ∈ I d − we have that p λ ( r , t ) = t − P rj =1 λ j d − r = P dj = r +1 λ j d − r ≥ λ r +1 ⇐⇒ λ r +1 = λ d . Therefore in this case • If λ = c d for some c ∈ R > , then r λ ( t ) = 0. • If λ > λ d , then r λ ( t ) + 1 = min { i ∈ I d : λ i = λ d } and r λ ( t ) = max { r ∈ I d − : λ r > λ d } . (13) Definition 3.10.
Let λ ∈ R d + ↓ and t = tr λ . We define the functions r λ : [ t , + ∞ ) → { , . . . , d − } given by r λ ( s ) (12) = the s -irregularity of λ (14) c λ : [ t , + ∞ ) → R ≥ given by c λ ( s ) = p λ ( r λ ( s ) , s ) = s − P r λ ( s ) i =1 λ i d − r λ ( s ) , (15)for every s ∈ [ t , + ∞ ), where we set P i =1 λ i = 0. △ Fix λ ∈ R d + ↓ . As we shall show in Lemma 5.8, the vector ν = ( λ , . . . , λ r λ ( t ) , c λ ( t ) d − r λ ( t ) ) ∈ R d> ↓ for every t ≥ t , and the map c λ is piece-wise linear, strictly increasing and continuous. Thislast claim allows us to introduce the following parameter: given m ∈ I d − we denote by s ∗ = s ∗ ( λ , m ) def = c − λ ( λ m ) = m X i =1 λ i + ( d − m ) λ m (16)that is, the unique s ∈ [ t , + ∞ ) such that c λ ( s ) = λ m . These facts and other results of Section 5give consistency to the following definitions: Definition 3.11.
Let λ ∈ R d + ↓ , t = tr λ . Take an integer m < d . If m > t ∈ [ t , + ∞ ) let c λ , m ( t ) def = c λ ( t ) if t ≤ s ∗ λ m + t − s ∗ d − m if t > s ∗ and r λ , m ( t ) def = min { r ∈ I d − ∪ { } : c λ , m ( t ) ≥ λ r +1 } . If m ≤ t ∈ [ t , + ∞ ) we define c λ , m ( t ) = c λ ( t ) and r λ , m ( t ) = r λ ( t ). △ r λ , m ( t ) = r λ ( t ) for every t ≤ s ∗ . The following results willbe used throughout Section 4; see the Appendix (Section 5) for their proofs. Theorem 3.12.
Let S ∈ M d ( C ) + with λ = λ ( S ) and m < d be an integer. For t ≥ tr S , let usdenote by r ′ = max { r λ , m ( t ) , m } and c = c λ , m ( t ) . Then, there exists ν ∈ Λ( U t ( S , m ) ) such that1. The vector ν is ≺ w -minimal in Λ( U t ( S , m ) ) , i.e. ν ≺ w µ for every µ ∈ Λ( U t ( S , m ) ) .2. For every matrix S ∈ U t ( S , m ) the following conditions are equivalent: (a) λ ( S ) = ν (i.e. S is ≺ w -minimal in U t ( S , m ) ). (b) There exists { v i } i ∈ I d , an ONB of eigenvectors for S , λ such that B = S − S = d − r ′ X i =1 ( c − λ r ′ + i ) v r ′ + i ⊗ v r ′ + i . (17)3. If we further assume any of the following conditions: • m ≤ , • m ≥ and λ m > λ m +1 , or • m ≥ and λ m = λ m +1 but t ≤ s ∗ ( λ , m ) (see Eq. 16),then B and S are unique. Moreover, in these cases Eq. (17) holds for any ONB of eigenvectorsof S as above. (cid:3) Remark 3.13.
Suppose that m ≥
1. In this case the map c λ , m ( · ) is continuous and strictlyincreasing. Indeed, by Lemma 5.8 we know that s ∗ = P mi =1 λ i + ( d − m ) λ m . Hence c λ , m ( t ) = λ m + t − s ∗ d − m = t − P mj =1 λ j d − m for every t > s ∗ . (18)The fact that the map c λ is continuous and strictly increasing will be also proved in Lemma 5.8.Let us abbreviate by r = r λ , m ( t ) for any fixed t > s ∗ . Then, if r > r < m and λ r ≥ c λ , m ( t ) = λ m + t − s ∗ d − m ≥ λ r +1 . (19)Finally, notice that the previous remarks allow to define s ∗∗ = c − λ , m ( λ ) (18) = ( d − m ) λ + m X j =1 λ j ≥ s ∗ (with equality ⇐⇒ λ = λ m ) . (20)Then c λ , m ( t ) ≥ λ and r = r λ , m ( t ) = 0 for every t > s ∗∗ (by Definition 3.11). These remarks arenecessary to characterize the vector ν of Theorem 3.12: △ Proposition 3.14.
Let S ∈ M d ( C ) + with λ = λ ( S ) , t = tr( S ) and m ∈ Z such that m < d .Fix t ∈ [ t , + ∞ ) and denote by r = r λ , m ( t ) . Then, the minimal vector ν = ν ( λ , m , t ) ∈ R d + ↓ ofTheorem 3.12 has tr ν = t and it is given by the following rule: • If m ≤ then ν = (cid:0) λ , . . . , λ r , c λ , m ( t ) d − r (cid:1) = (cid:0) λ , . . . , λ r , c λ ( t ) d − r (cid:1) .If m ≥ we have that • ν = (cid:0) λ , . . . , λ r , c λ , m ( t ) d − r (cid:1) for t ≤ s ∗ (so that r ≥ m and c λ , m ( t ) ≤ λ m ). • ν = (cid:16) λ , . . . , λ r , c λ , m ( t ) d − m , λ r +1 , . . . , λ m (cid:17) for t ∈ ( s ∗ , s ∗∗ ) , and • ν = (cid:0) c λ , m ( t ) d − m , λ , . . . , λ m (cid:1) for t ≥ s ∗∗ .If λ = λ m , the second case above disappears. (cid:3) Solutions of the main problems
In this section we present the solutions of the problems in frame theory described in Section 3. Ourstrategy is to apply Theorem 3.12 and Proposition 3.14 to the matrix-theoretic reformulations ofthese problems obtained in Sections 3.1 and 3.2. We point out that our arguments are not onlyconstructive but also algorithmically implementable. This last fact together with recent progressin algorithmic constructions of solutions to the classical frame design problem allow us to effec-tively compute the optimal frames from Theorems 4.3 and 4.12 below (for optimal completions seeRemarks 4.5, 4.6 and Section 4.2; for optimal duals see Remark 4.14 and Example 4.15).
Next we show how our previous results and techniques allow us to partially solve the frame com-pletion problem described in Section 3.1 (which includes the problem posed in [20]). We begin byextracting the relevant data for the problem:
In what follows, we fix the following data: A space
H ∼ = C d .D1. A sequence of vectors F = { f i } i ∈ I n o ∈ H n o .D2. An integer n > n o . We denote by k = n − n o . We assume that rk S F ≥ d − k .D3. A sequence a = { α i } i ∈ I n ∈ R n> such that k f i k = α i for every i ∈ I n o .D4. We shall denote by t = tr a and by b = { α i } ni = n o +1 ∈ R k> .D5. The vector λ = λ ( S F ) ∈ R d + ↓ .D6. The integer m = d − k = ( d + n o ) − n . Observe that d − m = k = n − n o . △ In order to apply the results of Section 3.3 to this problem, we need to recall and restate someobjects and notations:
Definition 4.2.
Fix the data F = { f i } i ∈ I n o and a = { α i } i ∈ I n as in 4.1. Recall that t = tr a , λ = λ ( S F ) and m = d − k . We rename some notions of previous sections:1. The vector ν ( F , a ) = ν ∈ R d ≥ of Theorem 3.12 (see also Proposition 3.14).2. The number c = c ( F , a ) def = c λ , m ( t ) (see Definition 3.11).3. The integer r = r ( F , a ) def = max { r λ , m ( t ) , m } (see Definition 3.11). Note that d − r ≤ k .4. Now we consider the vector µ = µ ( F , a ) def = (cid:16) c ( F , a ) − λ r + j (cid:17) j ∈ I d − r ∈ ( R d − r ≥ ) ↑ . Observe that tr µ = tr ν − tr λ = t − P i ∈ I n o k f i k = tr a − P i ∈ I n o α i = tr b . △ Throughout the rest of this section we shall denote by S = S F the frame operator of F . Recallthe following notations of Section 3.1: C a ( F ) = (cid:8) { f i } i ∈ I n ∈ F ( n , d ) : { f i } i ∈ I n o = F and k f i k = α i for i ≥ n o + 1 (cid:9) , SC a ( F ) = { S F : F ∈ C a ( F ) } and Λ a ( F ) def = { λ ( S ) : S ∈ SC a ( F ) } . Theorem 4.3.
Fix the data of 4.1 and 4.2. If we assume that b ≺ µ ( F , a ) then . The vector ν = ν ( F , a ) ∈ Λ a ( F ) .2. We have that ν ≺ β for every other β ∈ Λ a ( F ) .3. Let r = r ( F , a ) . Given F = { f i } ni = n o +1 ∈ H k such that F = ( F , F ) ∈ C a ( F ) , thefollowing conditions are equivalent: (a) λ ( S F ) = ν (i.e. S F is ≺ -minimal in SC a ( F ) ). (b) There exists { h i } i ∈ I d , an ONB of eigenvectors for S , λ ( S ) such that S F = B = d − r X i =1 µ i h r + i ⊗ h r + i . (21)Since, by the hypothesis, b ≺ µ = µ ( F , a ) then such an F exists.4. Moreover, if any of the conditions in item 3 of Theorem 3.12 holds, then(a) Any ONB of eigenvectors for S , λ produces the same operator B via (21).(b) Any F = ( F , F ) ∈ C a ( F ) satisfies that λ ( S F ) = ν ( F , a ) ⇐⇒ S F = B . Proof.
Since the elements of C a ( F ) must be frames, we have first to show that ν ( F , a ) >
0. Bythe description of ν = ν ( F , a ) ∈ R d + ↓ given in Proposition 3.14, there are two possibilities: Inone case ν d = λ m which is positive because we know from the data given in 4.1 that rk S ≥ m .Otherwise t ≤ s ∗ so that ν d = c ( F , a ) = c λ ( t ) by Proposition 3.14 and Definition 3.11. But c λ ( t ) > b > ⇒ t > tr S (see Lemma 3.13 and Definition 3.10).By Proposition 3.2, we know that the hypothesis b ≺ µ = µ ( F , a ) assures that there exists asequence F = { f i } ni = n o +1 ∈ H k such that F = ( F , F ) ∈ C a ( F ) and S F = B . Then λ ( S F ) = λ ( S F + S F ) = λ ( S F + B ) = ν ( F , a ) ∈ Λ a ( F ) , by Theorem 3.12. Observe that Λ a ( F ) ⊆ Λ( U t ( S F , m ) ), by Remark 3.7. Hence the majorizationof item 2, the equivalence of item 3 and the uniqueness results of item 4 follow from Theorem 3.12.Note that all the vectors of Λ a ( F ) have the same trace. So we have ≺ instead of ≺ w . (cid:3) Theorem 4.4.
Fix the data of 4.1 and 4.2. If we assume that b ≺ µ ( F , a ) then1. Any F ∈ C a ( F ) such that λ ( S F ) = ν ( F , a ) satisfies that X i ∈ I d f ( ν ( F , a ) i ) = P f ( F ) ≤ P f ( G ) for every G ∈ C a ( F ) , and every (not necessarily increasing) convex function f : [0 , ∞ ) → [0 , ∞ ) .2. If f is strictly convex then, for every global minimizer F ′ of P f ( · ) on C a ( F ) we get that λ ( S F ′ ) = ν ( F , a ) .In particular the previous items holds for the Benedetto-Fickus’ potential and the mean square error.Proof. It follows from Theorem 3.12 and the majorization facts described in Section 2.3. (cid:3)
Fix the data F and b of 4.1 and 4.2. We shall say that “the completion problem is feasible ” ifthe condition b ≺ µ ( F , a ) of Theorem 4.3 is satisfied. Remark 4.5.
The data ν ( F , a ), r ( F , a ), c ( F , a ) and µ ( F , a ) are essential for Theorem 4.3,both for checking the feasibility hypothesis b ≺ µ ( F , a ) and for the construction of the matrix B of (21), which is the frame operator of the optimal extensions of F . Notice that the vector µ ( F , a ) measures how restrictive is the feasibility condition. Fortunately, this condition can beeasily computed according to the following algorithm:13. The numbers t = tr a and m = d − k are included in the data 4.1.2. The main point is to compute the irregularity r = r ( F , a ) = max { r λ , m ( t ) , m } . If m ≤ r λ ( t ). If m ≥
1, the number s ∗ = s ∗ ( λ , m ) of Eq. (16) allowsus to compute r : If t > s ∗ then r = m by Eq. (19), and if t ≤ s ∗ then r = r λ ( t ) by theremark which follows Definition 3.11.3. Once r is obtained, we can see that the wideness of the allowed weights b depends on thedispersion of the eigenvalues ( λ r +1 , . . . , λ d ) of S F .4. Indeed, the number t def = tr b = t − tr S F is known data. Also tr µ ( F , a ) = t . Hence c ( F , a ) and µ ( F , a ) can be directly computed: Let s = P di = r +1 λ i . Then t = tr µ = ( d − r ) c ( F , a ) − s = ⇒ c ( F , a ) = t + sd − r = tr b + P di = r +1 λ i d − r . And we have the vector µ = µ ( F , a ) = (cid:16) c ( F , a ) − λ r + j (cid:17) j ∈ I d − r ∈ ( R d − r ≥ ) ↑ . Then b ≺ µ ⇐⇒ p X i =1 b ↓ i + λ d − i +1 ≤ pd − r (cid:0) tr b + d X i = r +1 λ i (cid:1) for 1 ≤ p < d − r , since the last inequalities s + P pi =1 b ↓ i ≤ s + tr b (for d − r ≤ p ≤ k ) clearly hold.It is interesting to note that the closer F is to be tight (at least in the last r entries of λ ), themore restrictive Theorem 4.3 becomes; but in this case F and F are already “good”.On the other hand, if F is far from being tight then the sequence ( λ r +1 , . . . , λ d ) has more dis-persion and the feasibility condition b ≺ µ ( F , a ) becomes less restrictive. It is worth mentioningthat in the uniform case b = b k is always feasible and Theorem 4.3 can be applied.Observe that as the number k of vectors increases (or as the weights α i increase) the trace t growsand the numbers r and m become smaller, taking into account more entries λ i of λ ( F ) . This factoffers a criterion for choosing a convenient data k and b for the completing process. We remarkthat the vector µ (and therefore the feasibility) only depends on λ , k and the trace of b , so thefeasibility can also be obtained by changing b maintaining its length (size) and its trace.The above algorithm (which tests the feasibility of our method for fixed data F and b ) can beeasily implemented in M ATLAB with low complexity (see Section 4.2). △ Remark 4.6 (Construction of optimal completions for the mean square error) . Consider the data in4.1. Apply the algorithm described in Remark 4.5 and assume that b ≺ µ ( F , a ). Then construct B as in Eq. (21). In order to obtain an optimal completion of F with prescribed norms we haveto construct a sequence F ∈ H k with frame operator B and norms given by the sequence b (whichis minimal for the mean square error by Theorem 4.4). But once we know B and the weights b wecan apply the results in [8] in order to concretely construct the sequence F . In fact, in [8] theygive a M ATLAB implementation which works fast, with low complexity.We also implemented a M
ATLAB program which compute the matrix B as in Eq. (21). Thisprocess is direct, but it is more complex because it depends on finding a ONB of eigenvectors forthe matrix S F . In Section 4.2 we shall present several examples which use these programs forcomputing explicit solutions. △ f : [0 , ∞ ) → [0 , ∞ ) be an strictly convex function. ByRemark 3.7, Theorem 3.12 and the remarks in Section 2.3, in general we have that X i ∈ I d f ( ν ( F , a ) i ) ≤ P f ( F ) for every F ∈ C a ( F ) . (22)Notice that although the left-hand side of Eq.(22) can be effectively computed, the inequality mightnot be sharp. Indeed, Eq.(22) is sharp if and only if the completion problem is feasible and, in thiscase, the lower bound is attained if and only if λ ( S F ) = ν ( F , a ). Nevertheless, Eq.(22) providesa general lower bound that can be of interest for optimization problems in C a ( F ). In this section we show several examples obtained by implementing the algorithms described inRemarks 4.5 and 4.6 in M
ATLAB , for different choices of F = { f i } i ∈ I n o and a = { α i } i ∈ I n (as in4.1). Indeed, we have implemented the computation of r , c , µ and ν by a fast algorithm using b = { α i } ni = n o +1 ∈ R k> and the vector λ = λ ( S F ) as data. Then, after computing the eigenvectorsof S F with the function ’eig’ in M ATLAB we computed the matrix B , and we apply the one-sidedBendel-Mickey algorithm (see [17] for details) to construct the vectors of F satisfying the desiredproperties. The corresponding M-files that compute all the previous objects are freely distributedby the authors. Example 4.7.
Consider the frame F ∈ F (7 ,
5) whose analysis operator is T ∗F = . − . − . . . . − . . . . . − . . . − . − . − . . − . − . . . − . − . − . . . . . . − . − . − . . . . The spectrum of it frame operator is λ = λ ( S F ) = (9 , , , ,
1) and t = tr S F = 21. As in 4.1,fix the data k = 2 and b = { α i } i =8 = (3 , . ∈ R > , so that m = d − k = 3. We compute:1. The number r λ , m (26 .
5) = 2 and the vector µ = (2 . , . b = (3 , . ≺ (2 . , .
25) = µ . Therefore the completion problem is feasible.2. The optimal spectrum is ν λ , m (26 .
5) = (9 , , . , . , F of F , with squared norms given by b is given by: T ∗F = − . − . . . − . . . . − . − . .
4. If we take b = (3 . ,
2) then the number t = t + tr b (and so also r and µ ) are the same asbefore but the problem is not feasible, because in this case b µ . Example 4.8.
We want to complete frame F of Example 4.7 with 4 vectors in R , whose normsare given by b = { α i } i =8 = (1 , , , ) ∈ R > . We can compute that1. m = d − k = 1 and r = r λ , m (23 .
75) = 3.2. The vector µ = (0 . , . b ≺ µ and the problem is feasible.3. The optimal spectrum is ν = (9 , , , . , . F of F , with squared norms given by b is given by: T ∗F = − . − . − . − . . . . . . − . . . . . . . − . − . − . − . .
5. If we take b = (2 , , , ) ∈ R > the problem becomes not feasible. △ Example 4.9.
Suppose now that H = C and that our original set of vectors F = { f i } i ∈ I ∈ H is such that the spectrum of S F is given by λ = (7 , , , , t = tr S F = 19. Let b = (2 , , k = 3, m = d − k = 2 and t = 24.With these initial data, we obtain the values r λ , m (24) = 1 and c λ , m (24) = 4 .
33. The spectrumof the completion B is µ = (0 . , . , .
33) (notice that b ≺ µ ) and the optimal spectrum is ν λ , m (24) = (7 , . , . , . , B of the optimal completion isnot unique, since λ m = λ m +1 and t = t + tr b = 24 >
23 = s ∗ (see Theorem 3.12). △ D t ( F ) In this section we show how our previous results and techniques allow us to solve the problem ofcomputing optimal duals in D t ( F ) = {G ∈ F ( n , d ) : T ∗G T F = I and tr S G ≥ t } for a given frame F ,described in Section 3.2. In order to state our main results we introduce the set Λ t ( D ( F ) ), calledthe spectral picture of the set SD ( F ) (see Eq. (8)), given byΛ t ( D ( F ) ) = { λ ( S G ) : G ∈ D t ( F ) } . Remark 4.10.
Recall from Remark 3.7 that if
F ∈ F ( n , d ) with λ = λ ( S − F ) , m = 2 d − n and t ≥ tr λ , then SD t ( F ) = U t ( S F , m ). Hence, by Theorem 3.12, there exists a unique ν ∈ Λ t ( D ( F ) )that is ≺ w -minimizer on this set. Moreover, recall that such vector ν is explicitly described inProposition 3.14. △ Theorem 4.11 (Spectral structure of Global minima in D t ( F )) . Let F = { f i } i ∈ I n ∈ F ( n , d ) with λ = λ ( S − F ) , m = 2 d − n and t ≥ tr λ . Let ν = ν ( λ, m, t ) ∈ R d + ↓ be as in Proposition 3.14. Then, ν ∈ Λ t ( D ( F ) ) and we have that:1. If G t ∈ D t ( F ) is such that λ ( S G t ) = ν then X i ∈ I d f ( ν i ) = P f ( G t ) ≤ P f ( G ) for every G ∈ D t ( F ) , and every increasing convex function f : [0 , ∞ ) → [0 , ∞ ) .2. If we assume further that f is strictly convex then, for every global minimizer G ′ t of P f ( · ) on D t ( F ) we get that λ ( G ′ t ) = ν .Proof. As explained in Remark 4.10 we see that ν ∈ Λ t ( D ( F ) ) is such that ν ≺ w µ for every µ ∈ Λ t ( D ( F ) ). By the remarks in Section 2.3 we conclude that, if G t is as above and G ∈ D t ( F )then P f ( G t ) = tr( f ( S G t )) = tr f ( ν ) ≤ tr f ( λ ( S G )) = P f ( G ) , since λ ( S G ) ∈ Λ t ( D ( F ) ). Assume further that f is strictly convex and let G ′ t be a global minimizerof P f ( · ) on D t ( F ). Then, we have that ν ≺ w λ ( S G ′ t ) but tr f ( λ ( S G ′ t )) = P f ( G ′ t ) ≤ P f ( G t ) = tr f ( ν ) . These last facts imply (see Section 2.3) that λ ( S G ′ t ) = ν as desired.16ext we describe the geometric structure of the global minimizers of the (generalized) frame po-tential P f ( · ) in D t ( F ), in terms of their frame operators. Theorem 4.12 (Geometric Structure of global minima in D t ( F )) . Let
F ∈ F ( n , d ) , m = 2 d − n ,let t ≥ tr S − F and denote by λ = λ ( S − F ) . Let f : [0 , ∞ ) → [0 , ∞ ) an increasing and strictly convexfunction.1. If G ∈ D t ( F ) is a global minimum of P f in D t ( F ) then there exists { h i } i ∈ I d , an ONB ofeigenvectors for S − F , λ such that S G = S − F + d − r ′ X i =1 (cid:16) c λ , m ( t ) − λ r ′ + i (cid:17) h r ′ + i ⊗ h r ′ + i , where r ′ = max { r λ , m ( t ) , m } .2. If we further assume any of the conditions of item 3 of Theorem 3.12, there exists a unique S t ∈ SD t ( F ) such that if G is a global minimum of P f in D t ( F ) then S G = S t .Proof. It is a consequence of Theorems 3.12 and 4.11 together with Proposition 3.14. (cid:3)
Remark 4.13.
Fix F = { f i } i ∈ I n ∈ F ( n , d ) and m = 2 d − n . Denote by λ = λ ( S − F ). If m > t ∈ R > and a constant vector c d ∈ Λ t ( D ( F ) ) ⇐⇒ λ = λ m . (23)In this case c = λ and t = d λ . The proof uses the characterization of Λ t ( D ( F ) ) given in Remark3.7 and Corollary 5.7 (see also Definition 5.6). Indeed, if ν = c d ∈ Λ t ( D ( F ) ) then, by Eq. (29), c = ν d ≤ λ m ≤ λ ≤ ν = c = ⇒ λ = λ m = c . Conversely, if λ = λ m and t = d λ , then by Corollary 5.7 it is easy to see that the vector λ d ∈ Λ t ( D ( F ) ). Therefore the frame F has a dual frame which is tight if and only if • m = 2 d − n ≤
0. Recall that in this case ν λ , m ( t ) = td · d for every t ≥ d λ . • m ∈ I d − and λ d − m +1 ( S F ) = λ d ( S F ) i.e., the multiplicity of the smaller eigenvalue λ d ( S F ) of S F is greater or equal than m . This is a consequence of Eq. (23).In particular, if m ∈ I d − then there is a Parseval dual frame for F if and only if λ d − m +1 ( S F ) = λ d ( S F ) = 1 ⇐⇒ S F ≥ I d and rk ( I d − S F ) ≤ d − m = dim ker T ∗F . Observe that the equivalence also holds if 2 d ≤ n . In this case there is a Parseval dual frame for F ⇐⇒ S F ≥ I d , because the restriction dim ker T ∗F = n − d ≥ d ≥ rk ( I d − S F ) is irrelevant. Thischaracterization was already proved by Han in [21], even for the infinite dimensional case. △ Remark 4.14.
Using the characterization of D ( F ) given in the proof of Proposition 3.5 everyoptimal dual frame G = { g i } i ∈ I n is constructed from the canonical dual of F : each g i = S − F f i + h i ,for a family F = { h i } i ∈ I n which satisfies T ∗F T F = B and T ∗F T F = 0. △ As it was done with the completion problem, the previous results can be implemented in M
ATLAB in order to construct optimal dual frames for a given one when a a tracial condition is imposed.It turns out that in this case, once we have calculated the optimal B , we must improve a differenttype of factorization of B . Now B = X ∗ X should satisfy R ( T F ) ⊆ ker X ∗ . In the algorithm devel-oped, X ∗ = B / W ∗ where B / has no cost of construction since we already have the eigenvectorsof S − F . In addition W is constructed using the first d − r vectors of the ONB of ker T ∗F (computedwith the ’null’ function) and adding r zero vectors in order to obtain an n × d partial isometry.17 xample 4.15. The frame operator of the following frame
F ∈ F (8 ,
5) has eigenvalues listed by λ = ( , , , , ): T ∗F = − . . . − . − . . . − . − . . . − . − . − . − . . . . − . . . − . . − . − . − . − . − . . − . − . . . − . − . − . − . . − . . . Therefore, λ = λ ( S − F ) = (4 , , , , ) and tr S − F = 9 .
4. We also have that m = 2 d − n = 2.Consider t = 16 .
5, then an optimal dual
G ∈ D t ( F ) for F is given by T ∗G = − . . − . − . . . . − . − . . . . . − . . . . . − . − . . − . . − . − . . − . . . − . − . − . . . − . − . − . . − . . . Here, the optimal spectrum ν = ν λ , m (16 .
5) is given by ν = (4 , . , . , . , △ In this section we obtain the proofs of Theorem 3.12 and Proposition 3.14 stated in Section 3.3 ina series of steps. In the first step we introduce the set U ( S , m ) := U tr( S ) ( S , m ) and characterizeits spectral picture Λ( U ( S , m )) - i.e. the subset of R d + ↓ of eigenvalues λ ( S ), for S ∈ U ( S , m )- in terms of the so-called Fan-Pall inequalities. In the second step we show the existence of a ≺ w -minimizer within the set Λ( U t ( S , m )) and give an explicit (algorithmic) expression for thisvector. Finally, in the third step we characterize the geometrical structure of the positive operators S ∈ U t ( S , m ) such that λ ( S ) are ≺ w -minimizers within the set Λ( U t ( S , m )), in terms of therelation between the eigenspaces of S and the eigenspaces of S . It is worth pointing out that thearguments in this section are constructive, and lead to algorithms that allow to effectively computeall the parameters involved. Step 1: spectral picture of U ( S , m ) Recall that R d + ↓ is the set of vectors µ ∈ R d + with non negative and decreasing entries (i.e. µ ∈ R d + with µ ↓ = µ ); also, given S ∈ M d ( C ) + , λ ( S ) ∈ R d + ↓ denotes the vector of eigenvalues of S -counting multiplicities - and arranged in decreasing order.Given S ∈ M d ( C ) + , m < d and integer and t ≥ tr( S ) then in Eq. (10) we introduced U t ( S , m ) = { S + B : B ∈ M d ( C ) + , rk B ≤ d − m , tr( S + B ) ≥ t } . In this section we consider U ( S , m ) := U tr( S ) ( S , m ) = { S + B : B ∈ M d ( C ) + , rk B ≤ d − m } together with its spectral picture Λ( U ( S , m )) := Λ( U tr( S ) ( S , m )) (see Eq. (11) in Remark 3.8).We shall also use the following notations:1. Given x ∈ C d then D ( x ) ∈ M d ( C ) denotes the diagonal matrix with main diagonal x .2. If d ≤ n and y ∈ C d , we write ( y , n − d ) ∈ C n , where 0 n − d is the zero vector of C n − d . In thiscase, we denote by D n ( y ) = D (cid:0) ( y , n − d ) (cid:1) ∈ M n ( C ). Theorem 5.1.
Let S ∈ M d ( C ) + , m < d be an integer and µ ∈ R d + ↓ . Then the followingconditions are equivalent:1. There exists S ∈ U ( S , m ) such that λ ( S ) = µ . . There exists an orthogonal projection P ∈ M d − m ( C ) such that rk P = d and λ ( P D d − m ( µ ) P ) = (cid:0) λ ( S ) , d − m (cid:1) . (24) Proof. ⇒
2. Let B ∈ M d ( C ) + be such that rk( B ) ≤ d − m and λ ( S + B ) = µ . Thus, B can befactorized as B = V ∗ V for some V ∈ M d − m, d ( C ). If T = S / V ! ∈ M d − m, d ( C ) ⇒ T ∗ T = S + B and T T ∗ = S S / V ∗ V S / V V ∗ ! . (25)Let U ∈ U (2 d − m ) be such that U ( T T ∗ ) U ∗ = D ( λ ( T T ∗ )) = D d − m ( µ ) and let P ∈ M d − m ( C ) begiven by P = U P U ∗ , where P = I d ⊕ d − m . Notice that, by construction, P is an orthogonalprojection with rk P = d and, by the previous facts, P D d − m ( µ ) P = U P ( T T ∗ ) P U ∗ = U (cid:18) S
00 0 (cid:19) U ∗ , which shows that Eq. (24) holds in this case.2 ⇒
1. Let P ∈ M d − m ( C ) be a projection as in item 2. Then, there exists U ∈ U (2 d − m ) suchthat U ∗ P U = P , where P = I d ⊕ d − m as before. Hence, we get that λ ( P ( U ∗ D d − m ( µ ) U ) P ) = λ ( S , d − m ) . (26)Since rk( U ∗ D d − m ( µ ) U ) ≤ d then we see that there exist T ∈ M d − m, d ( C ) such that U ∗ D d − m ( µ ) U = T T ∗ . Let T ∈ M d ( C ) + and T ∈ M d − m,d ( C ) such that T = (cid:18) T T (cid:19) = ⇒ U ∗ D d − m ( µ ) U = T T ∗ = (cid:18) T T ∗ T T ∗ T T ∗ T T ∗ (cid:19) . Then λ ( T T ∗ ) = λ ( S ) by Eq. (25). On the other hand, notice that λ ( T ∗ T ) = µ and T ∗ T = T ∗ T + T ∗ T def = S + B with λ ( S ) = λ ( T T ∗ ) = λ ( S ) and rk( B ) ≤ d − m . Let W ∈ U ( d ) such that W ∗ S W = S . Then S def = W ∗ ( T ∗ T ) W = S + B satisfies that λ ( S ) = µ and rk( B ) = rk( W ∗ B W ) ≤ d − m . Then µ = λ ( T ∗ T ) = λ ( S ) ∈ Λ( U ( S , m ) ). (cid:3) Remark 5.2.
Let S ∈ M d ( C ) + , m < d be an integer and µ ∈ R d + ↓ as in Theorem 5.1. It turnsout that condition (24) can be characterized in terms of interlacing inequalities.More explicitly, given µ ∈ R d + ↓ , by the Fan-Pall inequalities (see [27]), the existence of a projection P ∈ M d − m ( C ) satisfying (24) for µ is equivalent to the following inequalities:1. µ > λ ( S ), i.e. µ i ≥ λ i ( S ) for every i ∈ I d .2. If m ≥ µ also satisfies µ d − m + i ≤ λ i ( S ) for every i ∈ I m , where the last inequalities compare the first m entries of λ ( S ) with the last m of µ .These facts together with Theorem 5.1 give a complete description of the spectral picture of theset U ( S , m ), which we write as follows. △ Corollary 5.3.
Let S ∈ M d ( C ) + and m < d be an integer. Then, the set Λ( U ( S , m )) can becharacterized as follows: . If m ≤ , we have that µ ∈ Λ( U ( S , m )) ⇐⇒ µ > λ ( S ) . (27)
2. If m ≥ , then µ ∈ Λ( U ( S , m )) ⇐⇒ µ > λ ( S ) and µ d − m + i ≤ λ i ( S ) for i ∈ I m . (28) Proof.
It follows from Theorem 5.1 and the Fan-Pall inequalities of Remark 5.2. (cid:3)
Corollary 5.4.
Let S ∈ M d ( C ) + and m < d be an integer. Then Λ( U ( S , m )) is convex.Proof. It is clear that the inequalities given in Eqs. (27) and (28) are preserved by convex combi-nations. Observe that also the set R d + ↓ is convex. (cid:3) Remark 5.5.
Let S ∈ M d ( C ) + , m < d be an integer and S ∈ M d ( C ) + . The reader should notethat the fact that λ ( S ) ∈ Λ( U ( S , m )) does not imply that S ∈ U ( S , m ). Indeed, it is fairly easyto produce examples of this phenomenon. Therefore, the spectral picture of Λ( U ( S , m )) doesnot determine the set U ( S , m ). This last assertion is a consequence of the fact that U ( S , m ) isnot saturated by unitary equivalence. Nevertheless, Λ( U ( S , m )) allows to compute minimizers ofsubmajorization in U ( S , m ), since submajorization is an spectral preorder. Step 2: minimizers for submajorization in Λ( U t ( S , m )) The spectral picture of U ( S , m ) studied in the previous section motivates the definition of thefollowing sets. Definition 5.6.
Let λ ∈ R d + ↓ and take an integer m < d . We consider the setΛ( λ , m ) = (cid:8) µ ∈ R d + ↓ : µ > λ (cid:9) if m ≤ (cid:8) µ ∈ Λ( λ ,
0) : µ d − m + i ≤ λ i for every i ∈ I m (cid:9) if m ≥ . (29)Denote by t = tr λ . For t ≥ t , we also consider the setΛ t ( λ , m ) = (cid:8) µ ∈ Λ( λ , m ) : tr µ ≥ t } . △ Now Corollary 5.3 can be rewritten as
Corollary 5.7.
Let S ∈ M d ( C ) + with λ ( S ) = λ , m < d be an integer and t ≥ tr( λ ). Then wehave the identities Λ( U ( S , m )) = Λ( λ , m ) and Λ( U t ( S , m )) = Λ t ( λ , m ) . (cid:3) In this section, as a second step towards the proof of Theorem 3.12, we show that the sets Λ t ( λ , m )have minimal elements with respect to submajorization and we describe explicitly these elements.Let λ ∈ R d + ↓ and t = tr λ . We recall the maps r λ ( · ) and c λ ( · ) introduced in 3.3. Fix t ≥ t . Then1. Given r ∈ I d − ∪ { } we denote by p λ ( r , t ) = t − P rj =1 λ j d − r , where we set P j =1 λ j = 0.2. The maps r λ : [ t , + ∞ ) → I d − ∪ { } and c λ : [ t , + ∞ ) → R ≥ given by r λ ( t ) = min { r ∈ I d − ∪ { } : p λ ( r , t ) ≥ λ r +1 } and c λ ( t ) = t − P r λ ( t ) i =1 λ i d − r λ ( t ) . (30)In the following Lemma we state several properties of these maps, which we shall use below. Theproofs are technical but elementary, so that we only sketch the essential arguments.20 emma 5.8. Let λ ∈ R d + ↓ and t = tr λ .1. The function r λ is non-increasing and right-continuous, with λ r λ ( t )+1 = λ d .2. The image of r λ is the set B = { k ∈ I d − : λ k > λ k +1 } ∪ { } .3. The map c λ is piece-wise linear, strictly increasing and continuous.4. We have that c λ ( t ) = λ d and c λ ( t ) = t/d for t ≥ d λ .5. For every t ∈ [ t , d λ ) , if r = r λ ( t ) then λ r +1 ≤ c λ ( t ) < λ r . In other words r λ ( t ) = min { r ∈ I d − ∪ { } : λ r +1 ≤ c λ ( t ) } . (31)
6. For any k ∈ B let s k = k P i =1 λ i + ( d − k ) λ k +1 . Then r λ ( s k ) = k and c λ ( s k ) = λ k +1 . Moreover,the set A of discontinuity points of r λ satisfies that A = { t ∈ ( t , + ∞ ) : c λ ( t ) = λ r λ ( t )+1 } = c − λ { λ i : λ i = λ d } = { s k : k ∈ B} .
7. Given t ∈ [ t , + ∞ ) , such that c λ ( t ) = λ m (even if m / ∈ B ), then • t ∈ A ⇐⇒ λ m = λ d . • r λ ( t ) = 0 ⇐⇒ c λ ( t ) = λ ⇐⇒ t = d λ . • If λ m = λ , then r λ ( t ) = max { j ∈ I d : λ j > λ m } and t = m X i =1 λ i + ( d − m ) λ m = r λ ( t ) X i =1 λ i + ( d − r λ ( t ) ) λ m . (32) Proof.
Given t ∈ [ t , d λ ) and 1 ≤ r ≤ d −
1, then r = r λ ( t ) if and only if c λ ( t ) = p λ ( r , t ) ≥ λ r +1 and p λ ( r − , t ) < λ r . (33)On the other hand the map t p λ ( r, t ) is linear, continuous and increasing for any r fixed. Fromthese facts one easily deduces the right continuity of the map r λ , and that the map c λ is continuousat the points where r λ is. We can also deduce that if c λ ( t ) = λ r λ ( t )+1 then r λ is continuous (i.e.constant) near the point t . Observe that, if r = r λ ( t ), then λ r (33) > p λ ( r − , t ) = ( d − r ) p λ ( r , t ) + λ r d − r + 1 = ⇒ λ r > p λ ( r , t ) ≥ λ r +1 = ⇒ r ∈ B . (34)Using that r λ ( t ) = 0 for t ≥ d λ , that c λ ( t ) = λ d , and the right continuity of the map r λ , wehave that A = { t ∈ ( t , + ∞ ) : c λ ( t ) = λ r λ ( t )+1 } = c − λ { λ i : λ i = λ d } .Hence, in order to check the continuity of c λ we have to verify the continuity of c λ from the left atthe points t > t for which c λ ( t ) = λ r λ ( t )+1 . Note that, if r = r λ ( t ), then r ∈ B and c λ ( t ) = p λ ( r , t ) = t − P rj =1 λ j d − r = λ r +1 = ⇒ t = r X j =1 λ j + ( d − r ) λ r +1 . (35)If c λ ( t ) = λ d then t = t and there is nothing to prove. Assume that c λ ( t ) = λ r λ ( t )+1 > λ d . Thenˆ r = max { j ∈ I d − : λ j = λ r +1 } is the first element of B after r . Note that λ ˆ r +1 < λ ˆ r = λ r +1 . Weshall see that if s < t near t , then r λ ( s ) = ˆ r . Indeed, as in Eq. (35), p λ (ˆ r , t + x ) = ( d − r ) λ r +1 − P ˆ rj = r +1 λ j + xd − ˆ r = λ r +1 + xd − ˆ r > λ ˆ r +1 and21 λ (ˆ r − , t + x ) = ( d − r ) λ r +1 − P ˆ r − j = r +1 λ j + xd − ˆ r + 1 = λ r +1 + xd − ˆ r + 1 < λ r +1 = λ ˆ r . for x ∈ ( − ε,
0] if ε > r λ ( t + x ) = ˆ r = r λ ( t ) forsuch an x , so that t ∈ A ( r λ is discontinuous at t ). On the other hand, c λ ( t + x ) = p λ (ˆ r, t + x ) = λ r λ ( t )+1 + xd − ˆ r = ⇒ lim x → − c λ ( t + x ) = λ r λ ( t )+1 = c λ ( t ) . This last fact implies that c λ is continuous and, since r λ is right-continuous, that c λ is a piece-wiselinear and strictly increasing function. With the previous remarks, the proof of all other statementsof the lemma becomes now straightforward. Fix λ ∈ R d + ↓ . Take an integer m < d . Recall that if m > s ∗ = s ∗ ( λ , m ) = c − λ ( λ m ) = m X i =1 λ i + ( d − m ) λ m . Now we rewrite the definition of the maps r λ , m and c λ , m : If m > t ∈ [ t , + ∞ ) let c λ , m ( t ) def = ( c λ ( t ) if t ≤ s ∗ λ m + t − s ∗ d − m if t > s ∗ and r λ , m ( t ) def = min { r ∈ I d − ∪ { } : c λ , m ( t ) ≥ λ r +1 } . If m ≤ t ∈ [ t , + ∞ ) we define c λ , m ( t ) = c λ ( t ) and r λ , m ( t ) = r λ ( t ). Note that, by Eq. (31), r λ , m ( t ) = r λ ( t ) for every t ≤ s ∗ . △ Corollary 5.10.
Let λ ∈ R d + ↓ and fix an integer m < d . Then the map r λ , m is not increasingand right continuous and the map c λ , m is strictly increasing and continuous on [tr λ , + ∞ ) .Proof. The mentioned properties of the map c λ , m were proved in Remark 3.13 (whose proof usesLemma 5.8). With respect to the map r λ , m , the statement follows from Lemma 5.8 and 5.9. Minimizers for submajorization in Λ t ( λ , m ) for m ≤ . The following Lemma is a standard fact in majorization theory. We include a short proof of it forthe sake of completeness.
Lemma 5.11.
Let α , γ ∈ R p , β ∈ R q and x ∈ R such that x ≤ min k ∈ I p γ k . Then,tr ( γ , b q ) ≤ tr ( α , β ) and γ ≺ w α = ⇒ ( γ , x q ) ≺ w ( α , β ) . Observe that we are not assuming that ( α , β ) = ( α , β ) ↓ . Proof.
Let h = tr β and ρ = hq q . Then it is easy to see that P i ∈ I k ( γ ↓ , x q ) i ≤ P i ∈ I k ( α ↓ , ρ ) i ≤ P i ∈ I k ( α ↓ , β ↓ ) i for every k ∈ I p + q . Since ( γ ↓ , x q ) = ( γ , x q ) ↓ , we can conclude that ( γ , x q ) ≺ w ( α , β ). (cid:3) In the following statement we shall use the maps r λ and c λ defined in Eq. (30) (or Definition 3.10). Theorem 5.12.
Fix m ≤ . Let λ ∈ R d + ↓ , t = tr λ and t ∈ [ t , + ∞ ) . Consider the vector ν = ν λ ( t ) def = (cid:0) λ , . . . , λ r λ ( t ) , c λ ( t ) , . . . , c λ ( t ) (cid:1) if r λ ( t ) > , (36) or ν = td d = c t ( λ ) d ∈ Λ t ( λ , m ) if r λ ( t ) = 0 . Then ν satisfies that ν ∈ Λ t ( λ , m ) , tr ν = t and ν ≺ w µ for every µ ∈ Λ t ( λ , m ) . (37)22 roof. Given t ∈ [ t , + ∞ ), we denote by r = r λ ( t ). If r = 0 then, t ≥ d λ and λ = λ ↓ = ⇒ c λ ( t ) = td ≥ λ = ⇒ ν = c d ∈ Λ t ( λ , m ) . It is clear that such a vector must satisfy that ν ≺ w µ for every µ ∈ Λ t ( λ , m ).Suppose now that r ≥
1, so that t < d λ . Recall from Lemma 5.8 that in this case we have that λ r +1 ≤ c λ ( t ) < λ r . Hence ν > λ and ν = ν ↓ . It is clear from Eq. (15) that tr( ν ) = t . From thesefacts we can conclude that ν ∈ Λ t ( λ , m ) as claimed.Now let µ ∈ Λ t ( λ , m ) and notice that, since µ > λ , we get that k X i =1 µ i ≥ k X i =1 λ i = k X i =1 ν i for every 1 ≤ k ≤ r λ ( t ) . Now we can apply Lemma 5.11 (with p = r λ ( t ) and x = c λ ( t ) ) and deduce that ν ≺ w µ . (cid:3) Minimizers for submajorization in Λ t ( λ , m ) . The general case. Recall that Λ t ( λ , m ) = (cid:8) µ ∈ R d + ↓ : µ > λ , tr µ ≥ t and µ d − m + i ≤ λ i for every i ∈ I m (cid:9) , foreach m ∈ I d − . In what follows we shall compute a minimal element in Λ t ( λ , m ) with respect tosubmajorization in terms of the number s ∗ = s ∗ ( λ , m ) def = c − λ ( λ m ) and the maps r λ , m and c λ , m described in Definition 3.10 (see also 5.9). Proposition 5.13.
Let λ ∈ R d + ↓ , t = tr λ , m ∈ I d . If t ∈ [ t , s ∗ ( λ , m )] , then the vector ν = (cid:0) λ , . . . , λ r λ ( t ) , c λ ( t ) , . . . , c λ ( t ) (cid:1) of Eq. (36) satisfies that ν ∈ Λ t ( λ , m ) . Hence tr ν = t , ν d = c λ ( t ) and ν ≺ w µ for every µ ∈ Λ t ( λ , m ) . Proof.
We already know by Theorem 5.12 that ν ∈ Λ t ( λ ,
0) and tr ν = t . Using the inequality c λ ( t ) ≤ c λ ( s ∗ ) = λ m , the verification of the fact that ν ∈ Λ t ( λ , m ) is direct. By Theorem 5.12, weconclude that ν ≺ w µ for every µ ∈ Λ t ( λ , m ) ⊆ Λ t ( λ , (cid:3) Recall the number s ∗∗ = c − λ , m ( λ ) = ( d − m ) λ + m P j =1 λ j ≥ s ∗ (with equality ⇐⇒ λ = λ m ) definedin Eq. (20) (see also Remark 3.13). Definition 5.14.
Let λ ∈ R d + ↓ , t = tr λ and m ∈ Z such that m < d . Fix t ∈ [ t , + ∞ ) anddenote by r = r λ , m ( t ) . Consider the vector ν λ , m ( t ) ∈ R d + given by the following rule: • If m ≤ ν λ , m ( t ) = ν λ ( t ) (36) = (cid:0) λ , . . . , λ r , c λ , m ( t ) d − r (cid:1) .If m ≥ • ν λ , m ( t ) = (cid:0) λ , . . . , λ r , c λ , m ( t ) d − r (cid:1) for t ≤ s ∗ (so that r ≥ m and c λ , m ( t ) ≤ λ m ). • ν λ , m ( t ) = (cid:16) λ , . . . , λ r , c λ , m ( t ) d − m , λ r +1 , . . . , λ m (cid:17) for t ∈ ( s ∗ , s ∗∗ ), and • ν λ , m ( t ) = (cid:0) c λ , m ( t ) d − m , λ , . . . , λ m (cid:1) for t ≥ s ∗∗ .If λ = λ m , the second case of the definition of ν λ , m ( t ) disappears. △ In the following Lemma we state several properties of the map ν λ , m ( · ), which are easy to see: Lemma 5.15.
Let λ ∈ R d + ↓ , and m ∈ Z such that m < d . The map ν λ , m ( · ) of Definition 5.14 hasthe following properties: 23. By Remark 3.13 the vector ν λ , m ( t ) ∈ R d + ↓ (i.e. it is decreasing) for every t .2. The map ν λ , m ( · ) is continuous.3. It is increasing in the sense that t < t = ⇒ ν λ , m ( t ) ν λ , m ( t ) .4. More precisely, for any fixed k ∈ I d , the k -th entry ν ( k ) λ , m ( t ) of ν λ , m ( t ) is given by ν ( k ) λ , m ( t ) = max { λ k , c λ , m ( t ) } if k ≤ d − m , min n max { λ k , c λ , m ( t ) } , λ i o if k = d − m + i , i ∈ I m .
5. The vector ν λ , m ( t ) ∈ Λ t ( λ , m ) and tr ν λ , m ( t ) = t for every t ∈ [ t , + ∞ ). (cid:3) We can now state the main result of this section.
Theorem 5.16.
Let λ ∈ R d + ↓ , t = tr λ and t ∈ [ t , + ∞ ) . Fix m ∈ Z such that m < d . Then thevector ν λ , m ( t ) defined in 5.14 is the unique element of Λ t ( λ , m ) such that ν λ , m ( t ) ≺ w µ for every µ ∈ Λ t ( λ , m ) . (38) Proof. If m ≤ m ≥
1. By Lemma 5.15,the vector ν λ , m ( t ) ∈ Λ t ( λ , m ) and tr ν λ , m ( t ) = t for t ∈ [ t , + ∞ ). In Proposition 5.13 we haveshown that ν λ , m ( t ) satisfies (38) for every t ∈ [ t , s ∗ ( λ , m ) ]. Hence we check the other two cases:Case t ∈ ( s ∗ , s ∗∗ ): fix µ ∈ Λ t ( λ , m ) such that tr µ = t . Let us denote by r = r λ , m ( t ), α = ( µ , . . . , µ r ) , β = ( µ r +1 , . . . , µ r + d − m ) , γ = ( µ r + d − m +1 , . . . , µ d ) ,ρ = ( λ , . . . , λ r ) and ω = ( λ r +1 , . . . , λ m ). Then µ = ( α , β , γ ) and ν λ , m ( t ) = ( ρ , c λ , m ( t ) d − m , ω ) . Since µ ∈ Λ t ( λ , m ) and tr ν λ , m ( t ) = tr µ = t , then ρ α , γ ω and tr ( α , β ) ≥ tr( ρ , c λ , m ( t ) d − m ) . Then we can apply Lemma 5.11 to deduce that ( ρ , c λ , m ( t ) d − m ) ≺ w ( α , β ). Using this factjointly with γ ω one easily deduces that ν λ , m ( t ) ≺ µ (because tr µ = tr ν λ , m ( t ) = t ).The case t ≥ s ∗∗ for vectors µ ∈ Λ t ( λ , m ) such that tr µ = t follows similarly.If we have that µ ∈ Λ t ( λ , m ) with tr µ = a > t , then µ ∈ Λ a ( λ , m ) = ⇒ ν λ , m ( t ) ν λ , m ( a ) ≺ µ = ⇒ ν λ , m ( t ) ≺ w µ , where the first inequality follows from Lemma 5.15. (cid:3) Step 3: minimizers for submajorization in U t ( S , m ) Let S ∈ M d ( C ) + and let t ≥ t = tr( S ). Notice that Corollary 5.7 together with Theorem5.16 show that the sets U t ( S , m ) have minimal elements with respect to submajorization. We shalldescribe the geometrical structure of minimal elements in U t ( S , m ) with respect to submajoriza-tion for any m < d in terms of the geometry of S . We shall see that, under some mild assumptions,24here exists a unique S t ∈ U t ( S , m ) such that λ ( S t ) = ν λ , m ( t ) (the vector of Theorem 5.16 definedin 5.14). In order to do this we recall a series of preliminary results and we fix some notations. Interlacing inequalities . Let A ∈ H ( d ) with λ ( A ) ∈ R d ↓ and let P = P = P ∗ ∈ M d ( C ) + be aprojection with rk P = k . The interlacing inequalities (see [3]) relate the eigenvalues of A with theeigenvalues of P AP ∈ H ( d ) as follows: λ d − k + i ( A ) ≤ λ i ( P AP ) ≤ λ i ( A ) for every i ∈ I k . (39)On the other hand, if we have the equalities λ i ( P AP ) = λ i ( A ) for every i ∈ I k then P A = AP , (40)and that R ( P ) has an ONB { h i } i ∈ I k such that A h i = λ i h i for every i ∈ I k . Indeed, if Q = I − P ,then tr QAQ = d P i = k +1 λ i ( A ). The interlacing inequalities applied to QAQ imply that λ k + j ( A ) ≤ λ j ( QAQ ) for j ∈ I d − k = ⇒ λ j ( QAQ ) = λ k + j ( A ) for j ∈ I d − k . Taking Frobenius norms, we get that k A k = d X i =1 λ i ( A ) = k P AP k + k QAQ k = ⇒ P AQ = QAP = 0 , so that A = P AP + QAQ . The Ky-Fan inequalities (see [3]) assure that k X i =1 λ i ( A ) = max n tr P AP : P ∈ M d ( C ) + , P = P = P ∗ and rk P = k o . (41)As before, given an orthogonal projection P with rk P = k such thattr P AP = k X i =1 λ i ( A ) (39) = ⇒ λ i ( P AP ) = λ i ( A ) for i ∈ I k (40) = ⇒ P A = AP , (42)and R ( P ) has an ONB of eigenvectors for A associated to λ ( A ) , . . . , λ k ( A ). If we further assumethat λ k ( A ) > λ k +1 ( A ) then in both cases (40) and (42) the projection P is unique, since theeigenvectors associated to the first k eigenvalues of A generate a unique subspace of C d . Notations.
We fix a matrix S ∈ M d ( C ) + with λ ( S ) = λ = ( λ , . . . , λ d ) ∈ R d + ↓ . We shall alsofix an orthonormal basis { h i } i ∈ I d of C d such that S h i = λ i h i for every i ∈ I d . Any other such basis will be denoted as a “ONB of eigenvectors for
S , λ ”. Lemma 5.17.
Let B ∈ M d ( C ) + and r ∈ I d − such that λ ( S + B ) = ( λ , . . . , λ r , α ) , for some α ∈ R d − r + ↓ such that α ≤ λ r . Let M r def = span { h i : i ∈ I r } and P = P M r . Then P B = BP = P BP = 0 . Proof.
Since rk P = r and tr( P SP ) = r P i =1 λ i , then the Ky Fan theorem (41) assures that0 ≤ tr( P BP ) = tr( P ( S + B ) P ) − tr( P SP ) ≤ r X i =1 λ i ( S + B ) − r X i =1 λ i = 0 . Since B ≥
0, we have that tr(
P BP ) = 0 = ⇒ P BP = 0 = ⇒ BP = P B = 0.25 roposition 5.18.
Let r ∈ I d − , then for each c ∈ [ λ r +1 , λ r ] there is a unique B ∈ M d ( C ) + suchthat λ ( S + B ) = ( λ , . . . , λ r , c d − r ) . Moreover, it is given by B = d − r X i =1 ( c − λ r + i ) h r + i ⊗ h r + i and S + B = r X i =1 λ i · h i ⊗ h i + c · d X i = r +1 h i ⊗ h i . (43) Proof.
Let M r def = span { h i : i ∈ I r } and P = P M r . Suppose that B ∈ M d ( C ) + is such that λ ( S + B ) = ( λ , . . . , λ r , c d − r ). Then, by Lemma 5.17, BP = P B = 0. Hence P ( S + B ) P = ( S + B ) P = SP = r X i =1 λ i h i ⊗ h i Eq . (42) = ⇒ ( S + B ) Q = c Q , where Q = I − P . Hence B = BQ = c Q − S Q = d − r P i =1 ( c − λ r + i ) h r + i ⊗ h r + i . Remark 5.19.
In Lemma 5.17, we allow the case where λ r = λ r +1 = α . In this case we couldchange h r by h r +1 (or any other eigenvector for λ r ) as a generator for M r . The proof of the Lemmaassures that we get another projector P ′ which also satisfies that BP ′ = 0.Similarly, in Proposition 5.18 we allow the case where λ r = λ r +1 = c . By the previous comments,the projection P in the proof of Proposition 5.18 is not unique. Nevertheless, in this case thepositive perturbation B is unique, because we have that rk B < d − m (this follows from the factthat ( c − λ r +1 ) h r +1 ⊗ h r +1 = 0). In fact B = c Q − S Q , where Q is the orthogonal projector ontothe sum of the eigenspaces of S for the eigenvalues λ i < c . △ Lemma 5.20.
Let m ∈ I d − and B ∈ M d ( C ) + with rk B ≤ d − m . Assume that λ ( S + B ) = ( c d − m , λ , . . . , λ m ) , for some c ≥ λ . Then there exists an ONB { v i } i ∈ I d of eigenvectors for S , λ such that B = d − m X i =1 ( c − λ m + i ) v m + i ⊗ v m + i so that S + B = m X i =1 λ i · v i ⊗ v i + c · d X i = m +1 v i ⊗ v i . (44) If we assume further that λ m > λ m +1 then B is unique, and Eq. (44) holds for any ONB ofeigenvectors for S , λ .Proof.
Note that, since rk B ≤ d − m , then d − m X i =1 λ i ( B ) = tr B = tr( B + S ) − tr S = c ( d − m ) − d X j = m +1 λ j . (45)Take a subspace M ⊆ C n such that R ( B ) ⊆ M and dim M = d − m . Denote by Q = P M . Then QBQ = B , and the Ky-Fan inequalities (41) for S + B assure thattr( QSQ ) = tr( Q ( S + B ) Q ) − tr B ≤ d − m P i =1 λ i ( S + B ) − tr B = c ( d − m ) − tr B (45) = d P j = m +1 λ j . The equality in Ky-Fan inequalities (for − S ) force that M = span { v m +1 , . . . , v d } , for someONB { v i } i ∈ I d of eigenvectors for S , λ (see the remark following Eq. (42) ). Thus, we get that
Q S = S Q = d − m P i =1 λ m + i v m + i ⊗ v m + i . Since R ( B ) ⊆ M then P def = I − Q ≤ P ker B , and B P = 0 = ⇒ P ( S + B ) P = S P = m X i =1 λ i v i ⊗ v i Eq. (42) = ⇒ ( S + B ) Q = c Q . B = B Q = ( S + B ) Q − SQ = d − m X i =1 ( c − λ m + i ) v m + i ⊗ v m + i . (46)Finally, if we further assume that λ m > λ m +1 then the subspace M = span { v m +1 , . . . , v d } isindependent of the choice of the ONB of eigenvectors for S , λ . Thus, in this case B is uniquelydetermined by (46). Proposition 5.21.
Let m ∈ I d − and B ∈ M d ( C ) + with rk B ≤ d − m . Let c ∈ R such that λ r +1 ≤ c < λ r , for some r < m . Assume that λ ( S + B ) = (cid:16) λ , . . . , λ r , c d − m , λ r +1 , . . . , λ m (cid:17) . Then there exists an ONB { v i } i ∈ I d of eigenvectors for S , λ such that B = d − m X i =1 ( c − λ m + i ) v m + i ⊗ v m + i so that S + B = m X i =1 λ i · v i ⊗ v i + c · d X i = m +1 v i ⊗ v i . If we further assume that λ m > λ m +1 then B is unique.Proof. Consider the subspace M r = span { h , . . . , h r } and P = P M r . By Lemma 5.17, we knowthat P B = B P = 0. Let S = S (cid:12)(cid:12) M ⊥ r and B = B (cid:12)(cid:12) M ⊥ r (= B ) considered as operators in L ( M ⊥ r ).Then S and B are in the conditions of Lemma 5.20, so that there exists an ONB { w i } i ∈ I d − r of M ⊥ r of eigenvectors for S , ( λ r +1 , . . . , λ d ) such that B = B = d − m X i =1 ( c − λ m + i ) w m + i ⊗ w m + i . Finally, let { v i } i ∈ I d be given by v i = h i for 1 ≤ i ≤ r and v r + i = w i for r + 1 ≤ i ≤ d . Then { v i } i ∈ I d has the desired properties. Notice that if we further assume that λ m > λ m +1 then Lemma 5.20implies that B is unique and therefore B is unique, too. Remark 5.22.
With the notations of Lemma 5.20 assume that λ m = λ m +1 . In this case B is notuniquely determined. Next we obtain a parametrization of the set of all operators B ∈ M d ( C ) + such that λ ( S + B ) = ( c d − m , λ , . . . , λ m ). Consider p = ( d − m ) − { i : λ i < λ m +1 } andnotice that in this case we have that 1 ≤ p < { i : λ i = λ m +1 } = dim ker( S − λ m +1 I ). Then, forevery B ∈ M d ( C ) + as above there corresponds a subspace N = span { h i : m + 1 ≤ i ≤ m + p } ⊂ ker( S − λ m I ) with dim N = p such that B = ( c − λ m ) P N + d − m X i = p +1 ( c − λ m + i ) h m + i ⊗ h m + i . (47)Conversely, for every subspace N ⊂ ker( S − λ m I ) with dim N = p then the operator B ∈ M d ( C ) + given by (47) satisfies that λ ( S + B ) = ( c d − m , λ , . . . , λ m ). Since the previous map B P N is bijective, we see that the set of all such operators B is parametrized by the set of projections P N such that N ⊂ ker( S − λ m I ) is a p -dimensional subspace. Moreover, this map is actually anhomeomorphism between these sets, with their usual metric structures.Finally, if we let k = { i : λ i > λ m } then the set of operators S + B such that B ∈ M d ( C ) + withrk B ≤ m − d and such that λ ( S + B ) = ( c d − m , λ , . . . , λ m ) is given by S + B = k X i =1 λ i · h i ⊗ h i + λ m · P N ′ + c · ( P N + d − m X i = p +1 h i ⊗ h i ) , N ⊂ ker( S − λ m I ) is a subspace with dim N = p and N ′ = ker( S − λ m +1 I ) ∩ N ⊥ .As a consequence of the proof of Proposition 5.21, we have a similar description of the operators B of its statement. △ Proofs of the main results
Proof of Theorem 3.12.
It is a consequence of Corollary 5.7, Theorem 5.16, and the results of thissection (Lemma 5.20 and Propositions 5.18, 5.21). The arrow (b) = ⇒ (a) in Item 2 follows byDefinition 5.14 and the fact that both matrices S and B are diagonal on the same basis (as, forexample, in Eq. (43)). (cid:3) Proof of Proposition 3.14.
It is a consequence of Corollary 5.7, Definition 5.14, Lemma 5.15 andTheorem 5.16. (cid:3)
Acknowledgment.
The authors would like to thank the reviewers of the manuscript for severaluseful suggestions that improved the exposition of the results contained herein.
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