Canonical Correlation Decomposition Research Articles

The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

Submatrix constrained least-squares inverse problem for symmetric matrices from the design of vibrating structures

Given a full column rank matrix X ∈ R n × m , a matrix B ∈ R m × m and a symmetric matrix A 0 ∈ R p × p . In structural dynamic model updating, Yuan and Dai (2007) considered the matrix equation X T A X = B with a leading principal submatrix A 0 constraint. For a given matrix A ∗ ∈ R n × n , they updated the mass and stiffness matrix in the Frobenius norm sense such that the corrected matrices satisfy the generalized eigenvalue equation and orthogonality conditions. But due to measurement errors, the measured mass and stiffness matrices will not always satisfy these requirements. Since they still contain some useful information, we would like to retrieve their least-squares approximations to correct these matrices. Then we obtain the least-squares symmetric solutions of the equation X T A X = B with a trailing principal submatrix A 0 constraint by using the matrix differential calculus and canonical correlation decomposition. Furthermore, by applying the generalized singular value decomposition and projection theorem we get the best Frobenius norm approximate symmetric solution of this equation according to a given matrix A ∗ ∈ R n × n with A 0 as its trailing principal submatrix. Finally, a numerical algorithm for computing the best approximate solution is established. Some illustrated numerical examples are also presented.

Analytical best approximate Hermitian and generalized skew-Hamiltonian solution of matrix equation [formula omitted

In this paper, we consider the matrix equation AXAH+CYCH=F, where A, C and F are given matrices with appropriate sizes and [X,Y] is an unknown Hermitian and generalized skew-Hamiltonian matrix pair. Based on matrix differential calculus and projection theorem in inner product spaces, we exploit the best approximate solution [X̂,Ŷ] in the set S to a given matrix pair [X∗,Y∗], where S signifies the least-squares Hermitian and generalized skew-Hamiltonian solution set of the matrix equation AXAH+CYCH=F. The analytical expression of the best approximate solution is presented by applying the canonical correlation decomposition and the generalized singular value decomposition. Finally, a numerical algorithm and an illustrated example are given.

Generalized inverse eigenvalue problem for (P,Q)-conjugate matrices and the associated approximation problem

In this paper, the generalized inverse eigenvalue problem for the (P,Q)-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition (GSVD). Moreover, the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition (CCD). The solutions to these problems are derived. Some numerical examples are given to illustrate the main results.

Inverse problems for [formula omitted]-symmetric matrices in structural dynamic model updating

Let R,S∈Cn×n be nontrivial involutions, i.e., R=R−1≠±In and S=S−1≠±In. A matrix A∈Cn×n is referred to as (R,S)-symmetric if and only if RAS=A. The set of all (R,S)-symmetric matrices of order n is denoted by Csn×n(R,S). Given a full column rank matrix X∈Cn×m, a matrix B∈Cm×m and a matrix A∗∈Cn×n. In structural dynamic model updating, we usually consider the sets S1={A∣A∈Csn×n(R,S),XHAX=B} and S2={A∣A∈Csn×n(R,S),‖XHAX−B‖=min} in the Frobenius norm sense, where the superscript H denotes conjugate transpose. Then we characterize the unique matrices A˜=argminA∈S1‖A−A∗‖ and Â=argminA∈S2‖A−A∗‖ when R=RH and S=SH. By using the generalized singular value decomposition (GSVD), the necessary and sufficient conditions for the non-emptiness of S1 and the general representations of the elements in S1 and A˜ are derived, respectively. The analytical expressions of A∈S2 and  are also obtained by using the GSVD, the canonical correlation decomposition (CCD) and the projection theorem. Finally, a corresponding numerical algorithm and some illustrated examples are presented.

Joint DOD and DOA Estimation for Bistatic MIMO Radar in Unknown Correlated Noise

In this paper, joint direction-of-departure (DOD) and direction-of-arrival (DOA) estimation for bistatic multiple-input–multi-output (MIMO) radar in an unknown spatially correlated noise environment is investigated. The signal model is based on the assumption that the waveforms are transmitted by two separated subarrays having $M_1$ and $M_2$ sensors and received by two separated subarrays having $N_1$ and $N_2$ sensors, respectively. The received data are pulse-compressed using a matching matrix consisting of $M=M_1+M_2$ orthogonally transmitted waveforms. The joint covariance matrix of unknown correlated noise is analyzed. A novel algorithm is proposed by jointly estimating the DOD and DOA with transmitter and receiver subarrays in unknown noise. The joint estimation algorithm is based on the canonical correlation decomposition (CCD) and exploits the shift-invariance properties in the Kronecker product structure of each column of the various steering matrices. The estimated DOA and DOD can be automatically paired correspondingly. In addition, the formulas of stochastic Cramer–Rao bounds (CRB) for DOD and DOA estimation are derived. Simulations show that our method can effectively improve the performance of estimation in unknown correlated noise environments and is insensitive to having different noise environments in the two subarrays.

The *Congruence Class of the Solutions to a System of Matrix Equations

We present the *congruence class of the least-square and the minimum norm least-square solutions to the system of complex matrix equationAX=C, XB=Dby generalized singular value decomposition and canonical correlation decomposition.

SSHI methods for solving general linear matrix equations

PurposeThe purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X+ XA2+A3XH+XHA4=B (including Lyapunov and Sylvester matrix equations as special cases) with the unknown complex (reflexive) matrix X.Design/methodology/approachBy applying the principle of hierarchical identification and the Hermitian/skew‐Hermitian splitting of the coefficient matrix quadruplet A1; A2; A3; A4 the authors propose a shift‐splitting hierarchical identification (SSHI) method to solve the general linear matrix equation A1X+XA2+A3XH+XHA4=B. Also, the proposed algorithm is extended for finding the reflexive solution to this matrix equation.FindingsThe authors propose two iterative methods for finding the solution and reflexive solution of the general linear matrix equation, respectively. The proposed algorithms have a simple, neat and elegant structure. The convergence analysis of the methods is also discussed. Some numerical results are given which illustrate the power and effectiveness of the proposed algorithms.Originality/valueSo far, several methods have been presented and used for solving the matrix equations by using vec operator and Kronecker product, generalized inverse, generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD) of matrices. In several cases, it is difficult to find the solutions by using matrix decomposition and generalized inverse. Also vec operator and Kronecker product enlarge the size of the matrix greatly therefore the computations are very expensive in the process of finding solutions. To overcome these complications and drawbacks, by using the hierarchical identification principle and the Hermitian=skew‐Hermitian splitting of the coefficient matrix quadruplet (A1; A2; A3; A4), the authors propose SSHI methods for solving the general matrix equation.

Inverse problem for symmetric $P$-symmetric matrices with a submatrix constraint

For a fixed generalized reflection matrix $P$, i.e., $P^T=P, P^2= I$, and $P\neq \pm I$, then a matrix $A $ is called a symmetric $P$-symmetric matrix if $A=A^T$ and $(PA)^T=PA$. This paper is mainly concerned with finding the least squares symmetric $P$-symmetric solutions to the matrix inverse problem $AX=B$ with a submatrix constraint, where $X$ and $B$ are given matrices of suitable size. By applying the generalized singular value decomposition and the canonical correlation decomposition, an analytical expression of the least squares solutions is derived basing on the Projection Theorem in Hilbert inner products spaces. Moreover, in the corresponding solution set, the analytical expression of the unique minimum-norm solution is described in detail.

Least squares (P,Q)-orthogonal symmetric solutions of the matrix equation and its optimal approximation

In this paper, the relationship between the (P,Q)-orthogonal symmetric and symmetric matrices is derived. By applying the generalized singular value decomposition, the general expression of the least square (P,Q)-orthogonal symmetric solutions for the matrix equation AT XB = C is provided. Based on the projection theorem in inner space, and by using the canonical correlation decomposition, an analytical expression of the optimal approximate solution in the least squares (P, Q)-orthogonal symmetric solution set of the matrix equation AT XB = C to a given matrix is obtained. An explicit expression of the least square (P, Q)-orthogonal symmetric solution for the matrix equation AT XB = C with the minimum-norm is also derived. An algorithm for finding the optimal approximation solution is described and some numerical results are given.

Inverse eigenproblem for [formula omitted]-symmetric matrices and their approximation

Let R ∈ C n × n be a nontrivial involution, i.e., R = R − 1 ≠ ± I n . We say that G ∈ C n × n is R -symmetric if R G R = G . The set of all n × n R -symmetric matrices is denoted by GSC n × n . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors { x i } i = 1 m in C n and a set of complex numbers { λ i } i = 1 m , find a matrix A ∈ GSC n × n such that { x i } i = 1 m and { λ i } i = 1 m are, respectively, the eigenvalues and eigenvectors of A . We then consider the following approximation problem: Given an n × n matrix A ̃ , find A ˆ ∈ S E such that ‖ A ̃ − A ˆ ‖ = min A ∈ S E ‖ A ̃ − A ‖ , where S E is the solution set of IEP and ‖ ⋅ ‖ is the Frobenius norm. We provide an explicit formula for the best approximation solution A ˆ by means of the canonical correlation decomposition.

Fixed rank solutions of the matrix equation with statistical applications

By applying the canonical correlation decomposition of matrix pairs, the general fixed rank least square solutions of matrix equation Xβ=Y are derived. As statistical applications, an algorithm for computing the least square estimator of the multivariate reduced rank regression model Y=Xβ+ϵ, r(β)=t is given.

An optimal approximation problem for a matrix equation

Chang and Wang discussed the consistent conditions for the symmetric solutions of the linear matrix equation (A T XA, B T XB)=(C, D) and obtained the general expressions for its symmetric solution and symmetric least-squares solution, but they could not give the minimum Frobenius norm symmetric solution for this equation or the related least-squares problem. In this paper, the explicit analytical expression of the symmetric optimal approximation solution (or the minimum Frobenius norm symmetric solution as a special case) for the least-squares problem of this linear matrix equation is obtained by using the projection theorem in the Hilbert space, the quotient singular value decomposition and the canonical correlation decomposition in the matrix theory for efficient tools; therefore, this result gives an answer to the open problem in Chang and Wang's paper in 1993.

Generalized Correlation Decomposition-Based Blind Channel Estimation in DS-CDMA Systems With Unknown Wide-Sense Stationary Noise

A new blind subspace-based channel estimation technique is proposed for direct-sequence code-division multiple access (DS-CDMA) systems operating in the presence of unknown wide-sense stationary noise. Unlike most of the blind techniques developed for unknown correlated noise environments so far, the proposed algorithm is applicable to an arbitrary symbol constellation and does not require any auxiliary antennas at the receiver or any prior knowledge of the interfering user spreading codes. Our approach exploits the centro-Hermitian property of the unknown noise covariance matrix and makes use of the generalized correlation decomposition (GCD) to obtain an accurate estimate of the noise subspace and, consequently, of the user-of- interest channel vector. We also obtain optimal values of the GCD weighting matrices which maximally preserve the orthogonality of the estimated noise subspace to the actual signal subspace in the high signal-to-noise ratio (SNR) regime. It is shown that such an optimal choice of the weighting matrices transforms GCD to an extended form of the conventional canonical correlation decomposition (CCD). Simulation results further demonstrate that if such an extended CCD-based approach is used to estimate the user-of-interest channel vector, then the estimation performance can be substantially improved as compared to earlier SVD-based blind channel estimation techniques.

Blind Identification of FIR Channels in the Presence of Unknown Noise

AbstractBlind channel identification techniques based on second-order statistics (SOS) of the received data have been a topic of active research in recent years. Among the most popular is the subspace method (SS) proposed by Moulines et al. (1995). It has good performance when the channel output is corrupted by white noise. However, when the channel noise is correlated and unknown as is often encountered in practice, the performance of the SS method degrades severely. In this paper, we address the problem of estimating FIR channels in the presence of arbitrarily correlated noise whose covariance matrix is unknown. We propose several algorithms according to the different available system resources: (1) when only one receiving antenna is available, by upsampling the output, we develop the maximum a posteriori (MAP) algorithm for which a simple criterion is obtained and an efficient implementation algorithm is developed; (2) when two receiving antennae are available, by upsampling both the outputs and utilizing canonical correlation decomposition (CCD) to obtain the subspaces, we present two algorithms (CCD-SS and CCD-ML) to blindly estimate the channels. Our algorithms perform well in unknown noise environment and outperform existing methods proposed for similar scenarios.

Least-Squares Solution with the Minimum-Norm for the Matrix Equation A T XB+B T X T A = D and Its Applications

An efficient method based on the projection theorem, the generalized singular value decomposition and the canonical correlation decomposition is presented to find the least-squares solution with the minimum-norm for the matrix equation A T XB+B T X T A = D. Analytical solution to the matrix equation is also derived. Furthermore, we apply this result to determine the least-squares symmetric and sub-antisymmetric solution of the matrix equation C T XC = D with minimum-norm. Finally, some numerical results are reported to support the theories established in this paper.

Inverse problems for nonsymmetric matrices with a submatrix constraint

In this paper the following problems are considered: Problem I(a): Given matrices X ∈ R n × p with full column rank, B ∈ R p × p and A 0 ∈ R r × r , find a matrix A ∈ R n × n such that X T AX = B , A ( [ 1 , r ] ) = A 0 , where A ( [ 1 , r ] ) is the r × r leading principal submatrix of the matrix A. Problem I(b): Given matrices X ∈ R n × p , B ∈ R p × p and A 0 ∈ R r × r , find a matrix A ∈ R n × n such that ‖ X T AX - B ‖ = min s.t. A ( [ 1 , r ] ) = A 0 . Problem II: Given a matrix A ∼ ∈ R n × n with A ∼ ( [ 1 , r ] ) = A 0 , find A ^ ∈ S E such that ‖ A ∼ - A ^ ‖ = inf A ∈ S E ‖ A ∼ - A ‖ , where S E is the solution set of Problem I(a). By applying the generalized singular value decomposition (GSVD) and the canonical correlation decomposition (CCD) of a matrix pair, the solvability conditions for Problem I(a) and the general solution of Problem I are derived. The expression of the solution of Problem II is presented. A numerical algorithm for solving the problems is provided.

Inverse problems for symmetric matrices with a submatrix constraint

This paper is concerned with the following problems: Problem I(a). Given a full column rank matrix X ∈ R n × p and symmetric matrices B ∈ R p × p and A 0 ∈ R r × r , find an n × n symmetric matrix A such that X T A X = B , A ( [ 1 , r ] ) = A 0 , where A ( [ 1 , r ] ) is the r × r leading principal submatrix of the matrix A. Problem I(b). Given a matrix X ∈ R n × p and symmetric matrices B ∈ R p × p , A 0 ∈ R r × r , find an n × n symmetric matrix A such that ‖ X T A X − B ‖ = min , s.t. A ( [ 1 , r ] ) = A 0 . Problem II. Given an n × n symmetric matrix A ˜ , find A ˆ ∈ S E such that ‖ A ˜ − A ˆ ‖ = inf A ∈ S E ‖ A ˜ − A ‖ , where S E is the solution set of Problem I(a). By applying the generalized singular value decomposition (GSVD) and the canonical correlation decomposition (CCD) of a matrix pair, the solvability conditions for Problem I(a) and the general forms of the solution of Problem I are presented. The expression of the solution of Problem II is derived. A numerical algorithm for solving Problem II is provided.

The matrix nearness problem for symmetric matrices associated with the matrix equation [ ATXA, BTXB] = [ C, D

A direct method, based on the projection theorem in inner products spaces, the generalized singular value decomposition and the canonical correlation decomposition, is presented for finding the optimal approximate solution X ^ in the set S E to a given matrix X ∼ , where S E denotes the least-squares symmetric solution set of the matrix equation [ A T XA, B T XB] = [ C, D]. The analytical expression of the optimal approximate solution X ^ is obtained, and an algorithm for finding this solution is also suggested.

Least-squares solution with the minimum-norm for the matrix equation ( AXB, GXH) = ( C, D)

Based on the projection theorem in Hilbert space, by making use of the generalized singular value decomposition and the canonical correlation decomposition, an analytical expression of the least-squares solution for the matrix equation ( AXB, GXH) = ( C, D) with the minimum-norm is derived. An algorithm for finding the minimum-norm solution is described and some numerical results have been given.