arXiv subjects

Liqun Qi

Publications and source records attributed to Liqun Qi.

At least 37 records · Page 2Linked to original sources

arXiv2025

The SOS Rank of Biquadratic Forms

In 1973, Calderón proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ squares of quadratic forms. Very recently, by applying Hilbert's theorem on ternary quartics, we proved that a $2 \times 2$ psd biquadratic form can always be expressed as the sum of three squares of bilinear forms. This improved Calderón's result for $m=2$, and left the sos (sum-of-squares) rank problem of $m \times 2$ biquadratic forms for $m \ge 3$ to further exploration. In this paper, we show that an $3 \times 2$ psd biquadratic form can always be expressed as four squares of bilinear forms. We make a conjecture that an $m \times 2$ psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms.

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arXiv2025

Structured Symmetric Tensors

In this paper, we study structured symmetric tensors. We introduce several new classes of structured symmetric tensors: completely decomposable (CD) tensors, strictly sum of squares (SSOS) tensors and SOS$^*$ tensors. CD tensors have applications in data analysis and signal processing. Complete Hankel tensors are CD tensors. SSOS tensors are defined as SOS tensors with a positive definite Gram matrix, ensuring structural stability under perturbations. The SOS$^*$ cone is defined as the dual cone of the SOS tensor cone, with characterizations via moment matrices and polynomial nonnegativity. We study the relations among completely positive (CP) cones, CD cones, sum of squares (SOS) cones, positive semidefinite (PSD) cones and copositive (COP) cones. We identify the interiors of PSD, SOS, CP, COP and CD cones for even-order tensors. These characterizations are crucial for interior-point methods and stability analysis in polynomial and tensor optimization. We generalize the classical Schur product theorem to CD and CP tensors, including the case of strongly completely decomposable (SCD) and strongly completely positive (SCP) tensors. We identify equivalence between strictly CD (SCD) and positive definite (PD) for CD tensors. Furthermore, we give an example of a PSD but not SOS Hankel tensor. This answers an open question raised in the literature.

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arXiv2025

Completely Positive Biquadratic Tensors

In this paper, we systemically introduce completely positive biquadratic (CPB) tensors and copositive biquadratic tensors. We show that all weakly CPB tensors are sum of squares tensors, the CPB tensor cone and the copositive biquadratic tensor cone are dual cone to each other. We also show that the outer product of two completely positive matrices is a CPB tensor, and the outer product of two copositive matrices is a copositive biquadratic tensor. We then study two easily checkable subclasses of CPB tensors, namely positive biquadratic Cauchy tensors and biquadratic Pascal tensors. We show that a biquadratic Pascal tensor is both strongly CPB and positive definite.

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arXiv2025

A Dual Quaternion Control Law for Formation Control of Multiple 3-D Rigid Bodies

This paper studies the integrated position and attitude control problem for multi-agent systems of 3D rigid bodies. While the state-of-the-art method in [Olfati-Saber and Murray, 2004] established the theoretical foundation for rigid-body formation control, it requires all agents to asymptotically converge to identical positions and attitudes, limiting its applicability in scenarios where distinct desired relative configurations must be maintained. In this paper, we develop a novel dual-quaternion-based framework that generalizes this paradigm. By introducing a unit dual quaternion directed graph (UDQDG) representation, we derive a new control law through the corresponding Laplacian matrix, enabling simultaneous position and attitude coordination while naturally accommodating directed interaction topologies. Leveraging the recent advances in UDQDG spectral theory, we prove global asymptotic convergence to desired relative configurations modulo a right-multiplicative constant and establish an R-linear convergence rate determined by the second smallest eigenvalue of the UDQDG Laplacian. A projected iteration method is proposed to compute the iterative states. Finally, the proposed solution is verified by several numerical experiments.

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arXiv2025

Postive Semidefinite and Sum of Squares Biquadratic Polynomials

Hilbert proved in 1888 that a positive semi-definite (PSD) homogeneous quartic polynomial of three variables always can be expressed as the sum of squares (SOS) of three quadratic polynomials, and a psd homogeneous quartic polynomial of four variables may not be sos. Only after 87 years, in 1975, Choi gave the explicit expression of such a psd-not-sos (PNS) homogeneous quartic polynomial of four variables. An $m \times n$ biquadratic polynomial is a homogeneous quartic polynomial of $m+n$ variables. In this paper, we show that an $m \times n$ biquadratic polynomial can be expressed as a tripartite homogeneous quartic polynomial of $m+n-1$ variables. Therefore, {by Hilbert's theorem}, a $2 \times 2$ PSD biquadratic polynomial can be expressed as the sum of squares of three quadratic polynomials. This improves the result of Calderón in 1973, who proved that a $2 \times 2$ biquadratic polynomial can be expressed as the sum of squares of nine quadratic polynomials. Furthermore, we present a necessary and sufficient condition for an $m \times n$ psd biquadratic polynomial to be sos, and show that if such a polynomial is sos, then its sos rank is at most $mn$. Then we give a constructive proof of the sos form of a $2 \times 2$ psd biquadratic polynomial in three cases.

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arXiv2025

Quasi-Irreducibility of Nonnegative Biquadratic Tensors

While the adjacency tensor of a bipartite 2-graph is a nonnegative biquadratic tensor, it is inherently reducible. To address this limitation, we introduce the concept of quasi-irreducibility in this paper. The adjacency tensor of a bipartite 2-graph is quasi-irreducible if that bipartite 2-graph is not bi-separable. This new concept reveals important spectral properties: although all M$^+$-eigenvalues are M$^{++}$-eigenvalues for irreducible nonnegative biquadratic tensors, the M$^+$-eigenvalues of a quasi-irreducible nonnegative biquadratic tensor can be either M$^0$-eigenvalues or M$^{++}$-eigenvalues. Furthermore, we establish a max-min theorem for the M-spectral radius of a nonnegative biquadratic tensor.

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arXiv2025

Nonnegative Biquadratic Tensors

An M-eigenvalue of a nonnegative biquadratic tensor is referred to as an M$^+$-eigenvalue if it has a pair of nonnegative M-eigenvectors. If furthermore that pair of M-eigenvectors is positive, then that M$^+$-eigenvalue is called an M$^{++}$-eigenvalue. A nonnegative biquadratic tensor has at least one M$^+$ eigenvalue, and the largest M$^+$-eigenvalue is both the largest M-eigenvalue and the M-spectral radius. For irreducible nonnegative biquadratic tensors, all the M$^+$-eigenvalues are M$^{++}$-eigenvalues. Although the M$^+$-eigenvalues of irreducible nonnegative biquadratic tensors are not unique in general, we establish a sufficient condition to ensure their uniqueness. For an irreducible nonnegative biquadratic tensor, the largest M$^+$-eigenvalue has a max-min characterization, while the smallest M$^+$-eigenvalue has a min-max characterization. A Collatz algorithm for computing the largest M$^+$-eigenvalues is proposed. Numerical results are reported.

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arXiv2025

Biquadratic Tensors: Eigenvalues and Structured Tensors

The covariance tensors in statistics{, elasticity tensor in solid mechanics, Riemann curvature tensor in relativity theory are all biquadratic tensors that are weakly symmetric, but not symmetric in general. Motivated by this, in this paper, we consider nonsymmetric biquadratic tensors, and study possible conditions and algorithms for identifying positive semi-definiteness and definiteness of such biquadratic tensors. We extend M-eigenvalues to nonsymmetric biquadratic tensors, prove that a general biquadratic tensor has at least one M-eigenvalue, and show that a general biquadratic tensor is positive semi-definite if and only if all of its M-eigenvalues are nonnegative, and a general biquadratic tensor is positive definite if and only if all of its M-eigenvalues are positive. We present a Gershgorin-type theorem for biquadratic tensors, and show that (strictly) diagonally dominated biquadratic tensors are positive semi-definite (definite). We introduce Z-biquadratic tensors, M-biquadratic tensors, strong M-biquadratic tensors, B$_0$-biquadratic tensors and B-biquadratic tensors. We show that M-biquadratic tensors and symmetric B$_0$-biquadratic tensors are positive semi-definite, and that strong M-biquadratic tensors and symmetric B-biquadratic tensors are positive definite. A Riemannian LBFGS method for computing the smallest M-eigenvalue of a general biquadratic tensor is presented. Numerical results are reported.

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arXiv2025

Even Order Pascal Tensors are Positive Definite

In this paper, we show that even order Pascal tensors are positive definite, and odd order Pascal tensors are strongly completely positive. The significance of these is that our induction proof method also holds for some other families of completely positives tensors, whose construction satisfies certain rules, such an inherence property holds. We show that for all tensors in such a family, even order tensors would be positive definite, and odd order tensors would be strongly completely positive, as long as the matrices in this family are positive definite. In particular, we show that even order generalized Pascal tensors would be positive definite, and odd order generalized Pascal tensors would be strongly completely positive, as long as generalized Pascal matrices are positive definite. We also investigate even order positive definiteness and odd order strongly completely positivity for fractional Hadamard power tensors. Furthermore, we study determinants of Pascal tensors. We prove that the determinant of the $m$th order two dimensional symmetric Pascal tensor is equal to the $m$th power of the factorial of $m-1$.

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arXiv2025

Strongly Positive Semi-Definite Tensors and Strongly SOS Tensors

We introduce {odd-order} strongly PSD (positive semi-definite) tensors which map real vectors to nonnegative vectors. We then introduce odd-order strongly SOS (sum-of-squares) tensors. A strongly SOS tensor maps real vectors to nonnegative vectors whose components are all SOS polynomials. Strongly SOS tensors are strongly PSD tensors. Odd order completely positive tensors are strongly SOS tensors. We also introduce strict Hankel tensors, which are also strongly SOS tensors. Odd order Hilbert tensors are strict Hankel tensors. However, the Laplacian tensor of a uniform hypergraph may not be strongly PSD. This motivates us to study wider PSD-like tensors. A cubic tensor is said to be a generalized PSD tensor if its corresponding symmetrization tensor has no negative H-eigenvalue. In the odd order case, this extension contains a peculiar tensor class, whose members have no H-eigenvalues at all. We call such tensors barren tensors, and the other generalized PSD symmetric tensors genuinely PSD symmetric tensors. In the odd order case, genuinely PSD tensors embrace various useful structured tensors, such as strongly PSD tensors, symmetric M-tensors and Laplacian tensors of uniform hypergraphs. However, there are exceptions. It is known an even order symmetric B-tensor is a PSD tensor. We give an example of an odd order symmetric B-tensor, which is a barren tensor.

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arXiv2024

Triangular Decomposition of Third Order Hermitian Tensors

We define lower triangular tensors, and show that all diagonal entries of such a tensor are eigenvalues of that tensor. We then define lower triangular sub-symmetric tensors, and show that the number of independent entries of a lower triangular sub-symmetric tensor is the same as that of a symmetric tensor of the same order and dimension. We further introduce third order Hermitian tensors, third order positive semi-definite Hermitian tensors, and third order positive semi-definite symmetric tensors. Third order completely positive tensors are positive semi-definite symmetric tensors. Then we show that a third order positive semi-definite Hermitian tensor is triangularly decomposable. This generalizes the classical result of Cholesky decomposition in matrix analysis.

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arXiv2024

A metric function for dual quaternion matrices and related least-squares problems

Solving dual quaternion equations is an important issue in many fields such as scientific computing and engineering applications. In this paper, we first introduce a new metric function for dual quaternion matrices. Then, we reformulate dual quaternion overdetermined equations as a least squares problem, which is further converted into a bi-level optimization problem. Numerically, we propose two implementable proximal point algorithms for finding approximate solutions of dual quaternion overdetermined equations. The relevant convergence theorems %and computational complexity estimates have also been established. Preliminary simulation results on synthetic and color image datasets demonstrate the effectiveness of the proposed algorithms.

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arXiv2024

Unit Dual Quaternion Directed Graphs, Formation Control and General Weighted Directed Graphs

We study the multi-agent formation control problem in a directed graph. The relative configurations are expressed by unit dual quaternions (UDQs). We call such a weighted directed graph a unit dual quaternion directed graph (UDQDG). We show that a desired relative configuration scheme is reasonable or balanced in a UDQDG if and only if the dual quaternion Laplacian is similar to the unweighted Laplacian of the underlying directed graph. A direct method and a unit gain graph method are proposed to solve the balance problem of general unit weighted directed graphs. We then study the balance problem of general non-unit weighted directed graphs. Numerical experiments for UDQDG are reported.

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arXiv2024

An Efficient Two-Sided Sketching Method for Large-Scale Tensor Decomposition Based on Transformed Domains

Large tensors are frequently encountered in various fields such as computer vision, scientific simulations, sensor networks, and data mining. However, these tensors are often too large for convenient processing, transfer, or storage. Fortunately, they typically exhibit a low-rank structure that can be leveraged through tensor decomposition. However, performing large-scale tensor decomposition can be time-consuming. Sketching is a useful technique to reduce the dimensionality of the data. In this paper, we propose a novel two-sided sketching method based on the $\star_{L}$-product decomposition and transformed domains like the discrete cosine transformation. A rigorous theoretical analysis is also conducted to assess the approximation error of the proposed method. Specifically, we improve our method with power iteration to achieve more precise approximate solutions. Extensive numerical experiments and comparisons on low-rank approximation of synthetic large tensors and real-world data like color images and grayscale videos illustrate the efficiency and effectiveness of the proposed approach in terms of both CPU time and approximation accuracy.

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arXiv2024

Non-convex Pose Graph Optimization in SLAM via Proximal Linearized Riemannian ADMM

Pose graph optimization (PGO) is a well-known technique for solving the pose-based simultaneous localization and mapping (SLAM) problem. In this paper, we represent the rotation and translation by a unit quaternion and a three-dimensional vector, and propose a new PGO model based on the von Mises-Fisher distribution. The constraints derived from the unit quaternions are spherical manifolds, and the projection onto the constraints can be calculated by normalization. Then a proximal linearized Riemannian alternating direction method of multipliers (PieADMM) is developed to solve the proposed model, which not only has low memory requirements, but also can update the poses in parallel. Furthermore, we establish the iteration complexity of $O(1/ε^{2})$ of PieADMM for finding an $ε$-stationary solution of our model. The efficiency of our proposed algorithm is demonstrated by numerical experiments on two synthetic and four 3D SLAM benchmark datasets.

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arXiv2024

Moore Determinant of Dual Quaternion Hermitian Matrices

In this paper, we extend the Chen and Moore determinants of quaternion Hermitian} matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion Hermitian {matrices is invariant under addition, switching, multiplication, and unitary operations at the both hand sides. We then show the Chen and Moore determinants of dual quaternion Hermitian matrices are equal to each other, and they are also equal to the products of eigenvalues. The characteristic polynomial of a dual quaternion Hermitian matrix is also studied.

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arXiv2024

Spectral Properties of Dual Unit Gain Graphs

In this paper, we study dual quaternion and dual complex unit gain graphs and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if and only if the dual gain graph is balanced. By using the dual cosine functions, we establish the closed form of eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.

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arXiv2024

Quasi-determinant and right eigenvalues of dual quaternion matrices

Dual quaternion/complex matrices have important applications in brain science and multi-agent formation control. In this paper, we first study some basic properties of determinants of dual complex matrices, including Sturm theorem and Bloomfield-Watson inequality for dual complex matrices. Then, we show that every eigenvalue of a dual complex matrix must be the root of the characteristic polynomial of this matrix. With the help of the determinants of dual complex matrices, we introduce the concept of quasi-determinants of dual quaternion matrices, and show that every right eigenvalue of a dual quaternion matrix must be the root of the quasi-characteristic polynomial of this matrix, as well as the quasi-determinant of a dual quaternion Hermitian matrix is equivalent to the product of the square of the magnitudes of all eigenvalues. Our results are helpful for the further study of dual quaternion matrix theory, and their applications.

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