# New Formulation and Computation for Generalized Singular Values of Grassman Matrix Pair

In this paper, we derive new model formulations for computing generalized singular values of a Grassman matrix pair. These new formulations make use of truncated filter matrices to locate the i-th generalized singular value of a Grassman matrix pair. The resulting matrix optimization problems can be solved by using numerical methods involving Newton's method on Grassmann manifold. Numerical examples on synthetic data sets and gene expression data sets are reported to demonstrate the high accuracy and the fast computation of the proposed new ormulations for computing arbitrary generalized singular value of Grassman matrix pair.

## Authors

• 1 publication
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07/24/2019

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## 1 Introduction

The generalized singular value decomposition (GSVD) used in mathematics and numerical computations

[10, 11, 22] is a very useful and versatile tool. The GSVD of two matrices having the same number of columns was first proposed by Van Loan [22]. It is very useful in many matrix computation problems and practical applications, such as the Kronecker canonical form of a general matrix pencil, the linearly constrained least-squares problem, the general Gauss-Markov linear model, the generalized total least-squares problem, real-time signal processing, comparative analysis of DNA microarrays and so on; see for example [2, 3, 4, 5, 6, 7, 8, 10, 11, 14, 18, 20, 22, 23, 24, 26].

Numerical methods and perturbation analysis of GSVD have been well developed; see for instance [4, 7, 8, 14, 18, 20, 21, 22, 23, 24, 26]. The GSVD of two matrices having the same number of columns was first proposed by Van Loan [22]. Van Loan [22] and Paige [18] provided algorithms for computing the generalized singular value decomposition. Bai and Demmel [4] described a variation of Paige’s algorithm for computing the GSVD introduced by Van Loan [22] and Paige and Saunders [19]. Ewerbring and Luk [8] and Zha [26] proposed a GSVD for matrix triplets. Stewart [20] and Van Loan [23] proposed two algorithms for computing the GSVD. On perturbation analysis of the GSVD, Sun [21] and Li [14] presented several perturbation bounds of generalized singular values (GSVs) of a Grassman matrix pair (GMP) and their associated subspaces. Xu et al. [24] provided the explicit expression and sharper bounds of the chordal metric between GSVs of a GMP.

Recently, the GSVD plays an important role in the analysis of DNA microarrays and gene data. For arbitrary generalized singular value of a GMP, the ratio can be used in comparison analysis of gene data as an indicator; see for example [2, 3]

. Moreover, the formulation for arbitrary GSV of a GMP has not been studied. Usually, the GSVD are computed by making decompositions (e.g., QR decomposition and CS decomposition). In these cases, computing the whole decomposition makes computational cost higher. The main goal of this paper is to propose new model formulations for computing arbitrary GSV of a GMP. We first derive new formulations for computing arbitrary GSV of the GMP. By using truncated filter matrices, the matrix optimization problems can be reformulated to locate the

-th GSV of the GMP. The resulting optimization problems can be solved by using Newton’s method on Grassmann manifold. Numerical examples on synthetic data sets and gene expression data sets are reported to demonstrate the high accuracy and the fast computation of the proposed method for computing arbitrary GSV of a GMP.

### 1.1 Organization

The rest of this paper is organized as follows. In Section 2, we provide mathematical preliminaries and derive new formulations for computing GSVs of a GMP. In Section 3, we present the numerical method for solving the resulting optimization models. In Section 4, we provide numerical examples to show the efficiency of the theoretical results. Finally, some concluding remarks are given in Section 5.

### 1.2 Notation

Throughout this paper we always use the following notation and definitions. denotes imaginary unit and , , , and are the sets of real numbers, complex numbers,

-dimensional real vectors,

complex matrices and unitary matrices accordingly. stands for the absolute value of a complex number. and

stand for the identity matrix of order

and the zero matrix, respectively. , , , denote the conjugate, transpose, conjugate transpose, inverse, determinant and trace of a matrix accordingly. By we denote the spectral norm of a matrix. The singular value set of is denoted by . For given matrices , means is a positive (semi-)definite matrix. For a matrix , we denote by its singular values, arranged in decreasing order. We denote the largest integer less than or equal to a real number . The symbol “” means the Kronecher product and creates a column vector from a matrix by stacking its column vectors below one another. Finally, denotes a block diagonal matrix in which the diagonal blocks are square matrices and .

###### Definition 1.1

[10] Let and . A matrix pair is an -GMP if rank .

For any -GMP , one may see as a point of the complex projective space of all -dimensional subspaces of the -dimensional complex space , i.e., one can identify with the linear subspace (see e.g. [15, 17, 21]). Clearly, if is an -GMP, then is a definite matrix pair, i.e., for all nonzero . The definite pair has

generalized eigenvalues, and thus the GMP

has generalized singular values. A well-developed perturbation theory for the generalized eigenvalue problem of definite pencils is available. Perturbation bounds for the generalized singular value problem can be obtained with the help of the close relation between the two problems. However, the bounds obtained in this way are often unsatisfactory, just like the perturbation bounds for the singular values (SVs) of a single matrix A obtained through the perturbation bounds for the eigenvalues of the Hermitian matrix . Therefore, special attention deserves to be paid to perturbations for the generalized singular value problem. Therefore, research on numerical methods and perturbation analysis of the GSVD of a GMP is an important topic (see e.g. [4, 14, 21, 22, 24]).

###### Definition 1.2

[10] Let be an -GMP. A nonnegative number pair is a GSV of the GMP if

 (α,β)≠(0,0),det(β2AHA−α2BHB)=0,α,β≥0.

The set of GSV of is denoted by Evidently,

 σ{A,B}={(α,β)≠(0,0)|det(β2AHA−α2BHB)=0,α,β≥0}.

In the literature [22, 23], there are several formulations of the GSVD. Here we adopt the following form.

###### Definition 1.3

Let {A,B} be an (m,p,n)-GMP. Then there exist unitary matrices and a nonsingular matrix such that

 UHAR−1=ΣA, VHBR−1=ΣB, (1.1)
 ΣA=(ΛO(m−r−s)×(n−r−s)), ΣB=(O(p+r−n)×rΩ), (1.2)

where

 Λ=diag(α1,…,αr+s),Ω=diag(βr+1,…,βn),

with

 1=α1=⋯=αr>αr+1≥⋯≥αr+s>αr+s+1=⋯=αn=0,
 0=β1=⋯=βr<βr+1≤⋯≤βr+s<βr+s+1=⋯=βn=1,

and

 α2i+β2i=1,1≤i≤n.

## 2 Trace function optimization model formulation

In this section we give new model formulation of any GSV of a GMP by trace function under one variable.

###### Lemma 2.1

[25] Let and with for . Then we have

 maxU1,…,Um∈UnRe[tr(cIn±m∏j=1UjAj)]=nc+n∑i=1m∏j=1σi(Aj), maxU1,…,Um∈Un|tr(cIn±m∏j=1UjAj)|=n∣c∣+n∑i=1m∏j=1σi(Aj), minU1,…,Um∈UnRe[tr(cIn±m∏j=1UjAj)]=nc−n∑i=1m∏j=1σi(Aj),
 minU1,⋯,Um∈Un|tr(cIn±m∏j=1UjAj)|=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩n∣c∣−n∑i=1m∏j=1σi(Aj),1nn∑i=1m∏j=1σi(Aj)≤∣c∣,0,1nn∑i=1m∏j=1σi(Aj)≥∣c∣.
###### Lemma 2.2

[12] Let be an analytical function of several complex variables on the domain . Then attains its maximum modulus on the characteristic manifold .

We now give new formula model of any GSV of a GMP by trace function optimization under one variable.

###### Lemma 2.3

Let be an (m,p,n)-GMP and the GSV of be given in (1.1) and (1.2). The following conclusions hold true.

• If then for ,

 α2i = maxΦ1∈Umtr(AHΦH1QiΦ1A(AHA+BHB)−1) (2.1) −maxΦ1∈Umtr(AHΦH1Qi−1Φ1A(AHA+BHB)−1)≡ϕi,

where

 Qi=diag(Ti,O(m−n)×(m−n))withTi=diag(Ii,O(n−i)×(n−i)). (2.2)
• If , then for ,

 β2i = maxΦ2∈Uptr(BHΦH2PiΦ2B(AHA+BHB)−1) (2.3) −maxΦ2∈Uptr(BHΦH2Pi−1Φ1B(AHA+BHB)−1)≡ψi,

where

 Pi=diag(O(p−n)×(p−n),Fi)withFi=diag(O(i−1)×(i−1),In−i+1). (2.4)
• If , then and for ,

 α2i = maxΨ1∈Umtr(AHΨH1QiΨ1A(AHA+BHB)−1) (2.5) −maxΨ1∈Umtr(AHΨH1Qi−1Ψ1A(AHA+BHB)−1)≡φi,

where

 Qi=diag(Ii,O(m−i)×(m−i)). (2.6)
• If , then and for ,

 β2i = maxΨ2∈Uptr(BHΨH2PiΨ2B(AHA+BHB)−1) (2.7) −maxΨ2∈Uptr(BHΨH2Pi−1Ψ2B(AHA+BHB)−1)≡χi.

where

 Pi=diag(O(p−n+i−1)×(p−n+i−1),In−i+1). (2.8)

Proof: a) If then for , we let be defined by (2.2). We set and for any , we let with . Using the GSVD of in (1.1) and (1.2) we have

 maxΦ1∈Umtr(AHΦH1QiΦ1A(AHA+BHB)−1) = maxΦ1∈Umtr((AHA+BHB)−1/2AHΦH1QiΦ1A(AHA+BHB)−1/2) = maxΦ1∈Umtr(ΣHAUHΦH1QiΦ1UΣA) = maxϕ11ϕH11≤Intr(^ΣHAϕH11Tiϕ11^ΣA).

Observe is an analytical function of several complex variables on the domain . By Lemma 2.2 we have attains its maximum modulus on the characteristic manifold . Therefore,

 maxϕ11ϕH11≤Intr(^ΣHAϕH11Tiϕ11^ΣA)=maxϕ11ϕH11=Intr(^ΣHAϕH11Tiϕ11^ΣA).

Thus, by Lemma 2.1 we have

 maxΦ1∈Umtr(AHΦH1QiΦ1A(AHA+BHB)−1)=maxϕ11ϕH11=Intr(^ΣHAϕH11Tiϕ11^ΣA)=α21+⋯+α2i. (2.9)

Similarly, we have

 maxΦ1∈Umtr(AHΦH1Qi−1Φ1A(AHA+BHB)−1)=α21+⋯+α2i−1. (2.10)

From (2.9) and (2.10) it follows that (2.1) holds for .

b) If , then for , let be defined by (2.4). Let . For any , let with . Using the GSVD of in (1.1) and (1.2) we have

 maxΦ2∈Uptr(BHΦH2PiΦ2B(AHA+BHB)−1) maxΦ2∈Uptr((AHA+BHB)−1/2BHΦH2PiΦ2B(AHA+BHB)−1/2) = maxΦ2∈Uptr(ΣHBVHΦH2PiΦ2VΣB) = maxψ22ψH22≤Intr(^ΣHBψH22Fiψ22^ΣB),

which attains its maximum modulus on the characteristic manifold by Lemma 2.2. Then by Lemma 2.1 we have

 maxΦ2∈Uptr(BHΦH2PiΦ2B(AHA+BHB)−1)=maxψ22ψH22=Intr(^ΣHBψH22Fiψ22^ΣB)=β2i+⋯+β2n. (2.11)

Similarly, we have

 maxΨ1∈Uptr(BHΨH1Pi+1Ψ1B(AHA+BHB)−1)=β2i+1+⋯+β2n. (2.12)

Using (2.11) and (2.12) we can conclude that (2.3) holds for .

c) If , then it is easy to check that . For , let be defined by (2.6). We set . Using the GSVD of in (1.1) and (1.2) and Lemma 2.1 we have

 maxΨ1∈Umtr(AHΨH1QiΨ1A(AHA+BHB)−1) = maxΨ1∈Umtr(ΣHAUHΨH1QiΨ1UΣA) = maxΨ1∈Umtr(diag(^ΣHAUHΨH1QiΨ1U^ΣA,O(n−m)×(n−m))) = maxΨ1∈Umtr(^ΣHAUHΨH1QiΨ1U^ΣA) = α21+⋯+α2i. (2.13)

Similarly, we have

 maxΨ1∈Umtr(AHΨH1Qi−1Ψ1A(AHA+BHB)−1)=α21+⋯+α2i−1. (2.14)

From (2) and (2.14) it follows that (2.5) holds for .

d) If , then . For , let be defined by (2.8). We set . From the GSVD of in (1.1) and (1.2) and Lemma 2.1 we have

 maxΨ2∈Uptr(BHΨH2PiΨ2B(AHA+BHB)−1) = maxΨ2∈Uptr((AHA+BHB)−1/2BHΨH2PiΨ2B(AHA+BHB)−1/2) = maxΨ2∈Uptr(ΣHBVHΨH2PiΨ2VΣB) = maxΨ2∈Uptr(diag(O(n−p)×(n−p),^ΣHBVHΨH2PiΨ2V^ΣB)) = maxΨ2∈Uptr(^ΣHBVHΨH2PiΨ2V^ΣB) = β2i+⋯+β2n. (2.15)

Similarly, we have

 maxΨ2∈Uptr(BHΨH2Pi+1Ψ2B(AHA+BHB)−1)=β2i+1+⋯+β2n. (2.16)

Using (2) and (2.16) we have (2.7) holds for . The proof is complete.

Next, we give new formula model of any GSV of a GMP by trace function optimization under two variables.

###### Lemma 2.4

Let be an (m,p,n)-GMP and the GSV of be given in (1.1) and (1.2). The following conclusions hold true.

• If then for ,

 αi = maxΠ1∈Um,Π2∈Un|tr(Π1A(AHA+BHB)−1/2Π2Gi)| (2.17) −maxΠ1∈Um,Π2∈Un|tr(Π1A(AHA+BHB)−1/2Π2Gi−1)|≡~ϕi,

where

 Gi=(Gi,On×(m−n)),Gi=diag(Ii,O(n−i)×(n−i)). (2.18)
• If , then for ,

 βi = maxΞ1∈Up,Ξ2∈Un|tr(Ξ1B(AHA+BHB)−1/2Ξ2Hi)| (2.19) −maxΞ1∈Up,Ξ2∈Un|tr(Ξ1B(AHA+BHB)−1/2Ξ2Hi−1)|≡~ψi,

where

 Hi=(On×(p−n),Hi),Hi=diag(O(i−1)×(i−1),In−i+1). (2.20)
• If , then and for ,

 αi = maxΠ3∈Un,Π4∈Um|tr(Π3(AHA+BHB)−1/2AHΠ4Si)| (2.21) −maxΠ3∈Un,Π4∈Um|tr(Π3(AHA+BHB)−1/2AHΠ4Si−1)|≡~φi,
 Si=(Si,Om×(n−m)),Si=diag(Ii,O(m−i)×(m−i)). (2.22)
• If , then and for ,

 βi = maxΞ3∈Un,Ξ4∈Up|tr(Ξ3(AHA+BHB)−1/2BHΞ4Wi)| (2.23) −maxΞ3∈Un,Ξ4∈Up|tr(Ξ3(AHA+BHB)−1/2BHΞ4Wi−1)|≡~χi,

where

 Wi=(Op×(n−p),Wi),Wi=diag(O(p−n+i−1)×(p−n+i−1),In−i+1). (2.24)

Proof: a) If then for , we let be defined by (2.18). We set . For any , let with . By the GSVD of in (1.1) and (1.2) we have

 maxΠ1∈Um,Π2∈Un|tr(Π1A(AHA+BHB)−1/2Π2Gi)| = maxΠ1∈Um,Π2∈Un|tr(Π1UΣAR(RHR)−1/2Π2Gi)| = maxΠ11ΠH11≤In,Π2∈Un|tr(Π11^ΣAR(RHR)−1/2Π2Gi)|.

Observe is an analytical function of several complex variables on the domain . By Lemma 2.2 we know that attains its maximum modulus on the characteristic manifold . Therefore,

 maxΠ11ΠH11≤In,Π2∈Un|tr(Π11^ΣAR(RHR)−1/2Π2Gi)| = maxΠ11ΠH11=In,Π2∈Un|tr(Π11^ΣAR(RHR)−1/2Π2Gi)|

and it follows from Lemma 2.1 that

 maxΠ1∈Um,Π2∈Un|tr(Π1A(AHA+BHB)−1/2Π2Gi)| =maxΠ11ΠH11=In,Π2∈Un|tr(Π11^ΣAR(RHR)−1/2Π2Gi)| =α1+⋯+αi. (2.25)

Similarly, we have

 maxΠ1∈Um,Π2∈Un|tr(Π1A(AHA+BHB)−1/2Π2Gi−1)|=α1+⋯+αi−1. (2.26)

From (2) and (2.26) we know that (2.17) holds for .

b) If , then for , let be defined by (2.20). We set . For any , let with . By the GSVD of in (1.1) and (1.2), Lemma 2.2, and Lemma 2.1 we have

 maxΞ1∈Up,Ξ2∈Un|tr(Ξ1B(AHA+BHB)−1