arXiv subjects

Liqun Qi

Publications and source records attributed to Liqun Qi.

At least 55 records · Page 3Linked to original sources

arXiv2024

Eigenvalues of Dual Hermitian Matrices with Application in Formation Control

We propose a supplement matrix method for computing eigenvalues of a dual Hermitian matrix, and discuss its application in multi-agent formation control. Suppose we have a ring, which can be the real field, the complex field, or the quaternion ring. We study dual number symmetric matrices, dual complex Hermitian matrices and dual quaternion Hermitian matrices in a unified frame of dual Hermitian matrices. An $n \times n$ dual Hermitian matrix has $n$ dual number eigenvalues. We define determinant, characteristic polynomial and supplement matrices for a dual Hermitian matrix. Supplement matrices are Hermitian matrices in the original ring. The standard parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of the standard part Hermitian matrix in the original ring, while the dual parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of those supplement matrices. Hence, by applying any practical method for computing eigenvalues of Hermitian matrices in the original ring, we have a practical method for computing eigenvalues of a dual Hermitian matrix. We call this method the supplement matrix method. In multi-agent formation control, a desired relative configuration scheme may be given. People need to know if this scheme is reasonable such that a feasible solution of configurations of these multi-agents exists. By exploring the eigenvalue problem of dual Hermitian matrices, and its link with the unit gain graph theory, we open a cross-disciplinary approach to solve the relative configuration problem. Numerical experiments are reported.

math.NA↗

arXiv2023

Boundedness from below conditions for a general scalar potential of two real scalars fields and the Higgs boson

The most general scalar potential of two real scalar fields and a Higgs boson is a quartic homogeneous polynomial about 3 variables, which defines a 4th order 3 dimensional symmetric tensor. Hence, the boundedness from below of such a scalar potential involves the positive (semi-)definiteness of the corresponding tensor. So, we mainly discuss analytical expressions of positive (semi-)definiteness for such a special 4th order 3-dimension symmetric tensor in this paper. Firstly, an analytically necessary and sufficient condition is given to test the positive (semi-)definiteness of a 4th order 2 dimensional symmetric tensor. Furthermore, by means of such a result, the necessary and sufficient conditions of the boundedness from below are obtained for a general scalar potential of two real scalar fields and the Higgs boson.

math-ph↗

arXiv2023

Low-rank Tensor Train Decomposition Using TensorSketch

Tensor train decomposition is one of the most powerful approaches for processing high-dimensional data. For low-rank tensor train decomposition of large tensors, the alternating least squares (ALS) algorithm is widely used by updating each core tensor alternatively. However, it may suffer from the curse of dimensionality due to the large scale of subproblems. In this paper, a novel randomized proximal ALS algorithm is proposed for low-rank tensor train decomposition by using TensorSketch, which allows for efficient implementation via fast Fourier transform. The theoretical lower bounds of sketch size are estimated for approximating the optimal value of subproblems. Numerical experiments on synthetic and real-world data also demonstrate the effectiveness and efficiency of the proposed algorithm.

math.NA↗

arXiv2023

A Randomized Block Krylov Method for Tensor Train Approximation

Tensor train decomposition is a powerful tool for dealing with high-dimensional, large-scale tensor data, which is not suffering from the curse of dimensionality. To accelerate the calculation of the auxiliary unfolding matrix, some randomized algorithms have been proposed; however, they are not suitable for noisy data. The randomized block Krylov method is capable of dealing with heavy-tailed noisy data in the low-rank approximation of matrices. In this paper, we present a randomized algorithm for low-rank tensor train approximation of large-scale tensors based on randomized block Krylov subspace iteration and provide theoretical guarantees. Numerical experiments on synthetic and real-world tensor data demonstrate the effectiveness of the proposed algorithm.

math.NA↗

arXiv2023

A Genuine Extension of The Moore-Penrose Inverse to Dual Matrices

The Moore-Penrose inverse is a genuine extension of the matrix inverse. Given a complex matrix, there uniquely exists another complex matrix satisfying the four Moore-Penrose conditions, and if the original matrix is nonsingular, it is exactly the inverse of that matrix. In the last one and half decade, in the study of approximate synthesis in kinematic, two generalizations of the Moore-Penrose inverse appeared for dual real matrices,including Moore-Penrose dual generalized inverse and dual Moore-Penrose generalized inverse (DMPGI). DMPGI satisfies the four Moore-Penrose conditions, but does not exist for uncountably many dual real matrices. %Another generalization, called Moore-Penrose dual generalized inverse (MPDGI), is different from DMPGI even if DMPGI exists. In this paper, based upon the singular value decomposition of dual matrices, we extend the first Moore-Penrose condition to dual matrices and introduce a genuine extension of the Moore-Penrose inverse to dual matrices, referred to as GMPI. Given a dual complex matrix, its GMPI is the unique dual complex matrix which satisfy the first extended and the other three Moore-Penrose conditions. If the original matrix is a complex matrix, its GMPI is exactly the Moore-Penrose inverse of that matrix. And if the original matrix is a dual real matrix and its DMPGI exists, its GMPI coincides with its DMPGI.

math.RA↗

arXiv2023

Dual Number Matrices with Primitive and Irreducible Nonnegative Standard Parts

In this paper, we extend the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts. One motivation of our research is to consider probabilities as well as perturbation, or error bounds, or variances, in the Markov chain process. We show that such a dual number matrix always has a positive dual number eigenvalue with a positive dual number eigenvector. The standard part of this positive dual number eigenvalue is larger than or equal to the modulus of the standard part of any other eigenvalue of this dual number matrix. We present an explicit formula to compute the dual part of this positive dual number eigenvalue. The Collatz minimax theorem also holds here.The results are nontrivial as even a positive dual number matrix may have no eigenvalue at all. An algorithm based upon the Collatz minimax theorem is constructed. The convergence of the algorithm is studied. We show the upper bounds on the distance of stationary states between the dual Markov chain and the perturbed Markov chain. Numerical results on both synthetic examples and dual Markov chain including some real world examples are reported.

math.RA↗

arXiv2023

Eigenvalues and Jordan Forms of Dual Complex Matrices

Dual complex matrices have found applications in brain science. There are two different definitions of the dual complex number multiplication. One is noncommutative. Another is commutative. In this paper, we use the commutative definition. This definition is used in the research related with brain science. Under this definition, eigenvalues of dual complex matrices are defined. However, there are cases of dual complex matrices which have no eigenvalues or have infinitely many eigenvalues. We show that an $n \times n$ dual complex matrix is diagonalizable if and only if it has exactly $n$ eigenvalues with $n$ appreciably linearly independent eigenvectors. Hermitian dual complex matrices are diagonalizable. We present the Jordan form of a dual complex matrix with a diagonalizable standard part, and the Jordan form of a dual complex matrix with a Jordan block standard part. Based on these, we give a description of the eigenvalues of a general square dual complex matrix.

math.RA↗

arXiv2023

Variable T-Product and Zero-Padding Tensor Completion with Applications

The T-product method based upon Discrete Fourier Transformation (DFT) has found wide applications in engineering, in particular, in image processing. In this paper, we propose variable T-product, and apply the Zero-Padding Discrete Fourier Transformation (ZDFT), to convert third order tensor problems to the variable Fourier domain. An additional positive integer parameter is introduced, which is greater than the tubal dimension of the third order tensor. When the additional parameter is equal to the tubal dimension, the ZDFT reduces to DFT. Then, we propose a new tensor completion method based on ZDFT and TV regularization, called VTCTF-TV. Extensive numerical experiment results on visual data demonstrate that the superior performance of the proposed method. In addition, the zero-padding is near the size of the original tubal dimension, the resulting tensor completion can be performed much better.

math.OC↗

arXiv2023

Minimax principle for right eigenvalues of dual quaternion matrices and their generalized inverses

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency for a set of dual quaternion vectors, and study some related basic properties for the set of dual quaternion vectors and dual quaternion matrices. We present a minimax principle for right eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an important inequality for singular values of dual quaternion matrices. We finally introduce the concept of generalized inverse of dual quaternion matrices, and present the necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix.

math.NA↗

arXiv2023

A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix

In this paper, we first study the projections onto the set of unit dual quaternions, and the set of dual quaternion vectors with unit norms. Then we propose a power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. For a strict dominant eigenvalue, we show the sequence generated by the power method converges to the dominant eigenvalue and its corresponding eigenvector linearly. For a general dominant eigenvalue, we show the standard part of the sequence generated by the power method converges to the standard part of the dominant eigenvalue and its corresponding eigenvector linearly. Based upon these, we reformulate the simultaneous localization and mapping (SLAM) problem as a rank-one dual quaternion completion problem. A two-block coordinate descent method is proposed to solve this problem. One block has a closed-form solution and the other block is the best rank-one approximation problem of a dual quaternion Hermitian matrix, which can be computed by the power method. Numerical experiments are presented to show the efficiency of our proposed power method.

math.OC↗

arXiv2023

von Neumann type trace inequality for dual quaternion matrices

Dual quaternion matrices have important applications in multi-agent formation control. In this paper, we first address the concept of spectral norm of dual quaternion matrices. Then, we introduce a von Neumann type trace inequality and a Hoffman-Wielandt type inequality for general dual quaternion matrices, where the latter characterizes a simultaneous perturbation bound on all singular values of a dual quaternion matrix. In particular, we also present two variants of the above two inequalities expressed by eigenvalues of dual quaternion Hermitian matrices. Our results are helpful for the further study of dual quaternion matrix theory, algorithmic design, and applications.

math.RA↗

arXiv2023

Augmented Quaternion and Augmented Unit Quaternion Optimization

In this paper, we introduce and explore augmented quaternions and augmented unit quaternions, and present an augmented unit quaternion optimization model. An augmented quaternion consist of a quaternion and a translation vector. The multiplication rule of augmented quaternion is defined. An augmented unit quaternion consists of a unit quaternion and a translation vector. The augmented unit quaternions form a Lie group. By means of augmented unit quaternions, we study the error model and kinematics. Then we formulate two classical problems in robot research, i.e., the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as augmented unit quaternion optimization problems, which are actually real smooth spherical equality constrained optimization problems. Comparing with the corresponding unit dual quaternion optimization model, the augmented unit quaternion optimization model has less variables and removes the orthogonality constraints.

cs.RO↗

arXiv2023

Practical Sketching Algorithms for Low-Rank Tucker Approximation of Large Tensors

Low-rank approximation of tensors has been widely used in high-dimensional data analysis. It usually involves singular value decomposition (SVD) of large-scale matrices with high computational complexity. Sketching is an effective data compression and dimensionality reduction technique applied to the low-rank approximation of large matrices. This paper presents two practical randomized algorithms for low-rank Tucker approximation of large tensors based on sketching and power scheme, with a rigorous error-bound analysis. Numerical experiments on synthetic and real-world tensor data demonstrate the competitive performance of the proposed algorithms.

math.NA↗

arXiv2022

Motion, Unit Dual Quaternion and Motion Optimization

We introduce motions as real six-dimensional vectors. A motion means a rotation and a translation. We define a motion operator which maps unit dual quaternions to motions, and a UDQ operator which maps motions to unit dual quaternions. By these operators, we present the formulation of motion optimization, which is actually a real unconstrained optimization formulation. Then we formulate two classical problems in robot research, i.e., the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as motion optimization problems. This opens a new way to solve these problems via real unconstrained optimization.

math.OC↗

arXiv2022

Dual Quaternion Matrices in Multi-Agent Formation Control

Three kinds of dual quaternion matrices associated with the mutual visibility graph, namely the relative configuration adjacency matrix, the logarithm adjacency matrix and the relative twist adjacency matrix, play important roles in multi-agent formation control. In this paper, we study their properties and applications. We show that the relative configuration adjacency matrix and the logarithm adjacency matrix are all Hermitian matrices, and thus have very nice spectral properties. We introduce dual quaternion Laplacian matrices, and prove a Gershgorin-type theorem for square dual quaternion Hermitian matrices, for studying properties of dual quaternion Laplacian matrices. The role of the dual quaternion Laplacian matrices in formation control is discussed.

math.OC↗

arXiv2022

A regularization-patching dual quaternion optimization method for solving the hand-eye calibration problem

The hand-eye calibration problem is an important application problem in robot research. Based on the 2-norm of dual quaternion vectors, we propose a new dual quaternion optimization method for the hand-eye calibration problem. The dual quaternion optimization problem is decomposed to two quaternion optimization subproblems. The first quaternion optimization subproblem governs the rotation of the robot hand. It can be solved efficiently by the eigenvalue decomposition or singular value decomposition. If the optimal value of the first quaternion optimization subproblem is zero, then the system is rotationwise noiseless, i.e., there exists a ``perfect'' robot hand motion which meets all the testing poses rotationwise exactly. In this case, we apply the regularization technique for solving the second subproblem to minimize the distance of the translation. Otherwise we apply the patching technique to solve the second quaternion optimization subproblem. Then solving the second quaternion optimization subproblem turns out to be solving a quadratically constrained quadratic program. In this way, we give a complete description for the solution set of hand-eye calibration problems. This is new in the hand-eye calibration literature. The numerical results are also presented to show the efficiency of the proposed method.

cs.RO↗

arXiv2022

"Sparse + Low-Rank'' Tensor Completion Approach for Recovering Images and Videos

Recovering color images and videos from highly undersampled data is a fundamental and challenging task in face recognition and computer vision. By the multi-dimensional nature of color images and videos, in this paper, we propose a novel tensor completion approach, which is able to efficiently explore the sparsity of tensor data under the discrete cosine transform (DCT). Specifically, we introduce two ``sparse + low-rank'' tensor completion models as well as two implementable algorithms for finding their solutions. The first one is a DCT-based sparse plus weighted nuclear norm induced low-rank minimization model. The second one is a DCT-based sparse plus $p$-shrinking mapping induced low-rank optimization model. Moreover, we accordingly propose two implementable augmented Lagrangian-based algorithms for solving the underlying optimization models. A series of numerical experiments including color image inpainting and video data recovery demonstrate that our proposed approach performs better than many existing state-of-the-art tensor completion methods, especially for the case when the ratio of missing data is high.

cs.CV↗

arXiv2022

Standard Dual Quaternion Optimization and Its Applications in Hand-Eye Calibration and SLAM

Several common dual quaternion functions, such as the power function, the magnitude function, the $2$-norm function and the $k$th largest eigenvalue of a dual quaternion Hermitian matrix, are standard dual quaternion functions, i.e., the standard parts of their function values depend upon only the standard parts of their dual quaternion variables. Furthermore, the sum, product, minimum, maximum and composite functions of two standard dual functions, the logarithm and the exponential of standard unit dual quaternion functions, are still standard dual quaternion functions. On the other hand, the dual quaternion optimization problem, where objective and constraint function values are dual numbers but variables are dual quaternions, naturally arises from applications. We show that to solve an equality constrained dual quaternion optimization problem, we only need to solve two quaternion optimization problems. If the involved dual quaternion functions are all standard, the optimization problem is called a standard dual quaternion optimization problem, and some better results hold. Then, we show that the dual quaternion optimization problems arising from the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem are equality constrained standard dual quaternion optimization problems.

math.NA↗