arXiv subjects

Xinzhen Zhang

Publications and source records attributed to Xinzhen Zhang.

At least 19 recordsLinked to original sources

arXiv2024

Efficient Low Rank Matrix Recovery With Flexible Group Sparse Regularization

In this paper, we present a novel approach to the low rank matrix recovery (LRMR) problem by casting it as a group sparsity problem. Specifically, we propose a flexible group sparse regularizer (FLGSR) that can group any number of matrix columns as a unit, whereas existing methods group each column as a unit. We prove the equivalence between the matrix rank and the FLGSR under some mild conditions, and show that the LRMR problem with either of them has the same global minimizers. We also establish the equivalence between the relaxed and the penalty formulations of the LRMR problem with FLGSR. We then propose an inexact restarted augmented Lagrangian method, which solves each subproblem by an extrapolated linearized alternating minimization method. We analyze the convergence of our method. Remarkably, our method linearizes each group of the variable separately and uses the information of the previous groups to solve the current group within the same iteration step. This strategy enables our algorithm to achieve fast convergence and high performance, which are further improved by the restart technique. Finally, we conduct numerical experiments on both grayscale images and high altitude aerial images to confirm the superiority of the proposed FLGSR and algorithm.

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arXiv2022

Optimal Sizing and Pricing of Renewable Power to Ammonia Systems Considering the Limited Flexibility of Ammonia Synthesis

Converting renewable energy into ammonia has been recognized as a promising way to realize ``green hydrogen substitution" in the chemical industry. However, renewable power to ammonia (RePtA) requires an essential investment in facilities to provide a buffer against the strong volatility of renewable energy and the limited flexibility of ammonia synthesis, which involves the three main stakeholders, namely, power, hydrogen, and ammonia. Therefore, the sizing and pricing of RePtA play a core role in balancing the interest demands of investors. This paper proposes an optimal sizing and pricing method for RePtA system planning. First, power to ammonia (P2A) is modeled as a flexible load, especially considering the limited flexibility of ammonia synthesis, which has been verified using real dynamic regulation data. Second, the multi-investor economic (MIE) model is established considering both external and internal trading modes. Then, a two-stage decomposed sizing and pricing method is proposed to solve the problem caused by the strong coupling of planning, operation, and trading, and information gap decision theory (IGDT) method is utilized to handle the uncertainty of renewable generation. Finally, real data from a real-life system in Inner Mongolia are utilized to verify the proposed approach. The results show that the system proposed has a yield of 8.15%.

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arXiv2022

Deterministic Dynamic Stackelberg Games: Time-Consistent Open-Loop Solution

In this paper, the known deterministic linear-quadratic Stackelberg game is revisited, whose open-loop Stackelberg solution actually possesses the nature of time inconsistency. To handle this time inconsistency, {a two-tier game framework is introduced, where the upper-tier game works according to Stackelberg's scenario with a leader and a follower, and two lower-tier intertemporal games give the follower's and leader's equilibrium response mappings that mimic the notion of time-consistent open-loop equilibrium control in existing literature. The resulting open-loop equilibrium solution of the two-tier game} is shown to be weakly time-consistent in the sense that the adopted policies will no longer be denied in the future only if past policies are consistent with the equilibrium policies. On the existence and uniqueness of such a solution, necessary and sufficient conditions are obtained, which are characterized via the solutions of several Riccati-like equations.

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arXiv2022

Tensor factorization based method for low rank matrix completion and its application on tensor completion

Low rank matrix and tensor completion problems are to recover the incomplete two and higher order data by using their low rank structures. The essential problem in the matrix and tensor completion problems is how to improve the efficiency. To this end, we first establish the relationship between matrix rank and tensor tubal rank, and then reformulate matrix completion problem as a tensor completion problem. For the reformulated tensor completion problem, we adopt a two-stage strategy based on tensor factorization algorithm. In this way, a matrix completion problem of big size can be solved via some matrix computations of smaller sizes. For a third order tensor completion problem, to fully exploit the low rank structures, we introduce the double tubal rank which combines the tubal rank and the rank of the mode-3 unfolding matrix. For the mode-3 unfolding matrix rank, we follow the idea of matrix completion. Based on this, we establish a novel model and modify the tensor factorization based algorithm for third order tensor completion. Extensive numerical experiments demonstrate that the proposed methods outperform state-of-the-art methods in terms of both accuracy and running time.

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arXiv2021

A Nash-Type Fictitious Game Framework to Time-Inconsistent Stochastic Control Problems

In this paper, a Nash-type fictitious game framework is introduced to handle a time-inconsistent linear-quadratic optimal control. The Nash-type game in this framework is called fictitious as it is between the decision maker (called real player) and an auxiliary control variable (called fictitious player) with the real player and fictitious player looking for time-consistent policy and precomitted optimal policy, respectively. Namely, the fictitious-game framework is actually an auxiliary-variable-based mechanism where the fictitious player is our particular design. Noting that the real player's cost functional is revised in accordance with that of fictitious player, the equilibrium policy of real player is called an open-loop self-coordination control of original linear-quadratic problem. As a generalization, a time-inconsistent nonzero-sum stochastic linear-quadratic dynamic game is investigated, where one player is to look for precommitted optimal policy and the other player is to search time-consistent policy. Necessary and sufficient conditions are presented to ensure the existence of open-loop equilibrium of the nonzero-sum game, which resort to a set of Riccati-like equations and linear equations. By applying the developed theory of nonzero-sum game, open-loop self-coordination control of the linear-quadratic optimal control is fully characterized, and multi-period mean-variance portfolio selection is also investigated. Finally, numerical simulations are presented, which show the efficiency of the proposed fictitious-game framework.

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arXiv2021

A smoothing proximal gradient algorithm for matrix rank minimization problem

In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both problems have the same minimzers and the same optimal value. In particular, we introduce a class of lifted stationary point of the relaxed problem and show that any local minimizer of the relaxed problem must be a lifted stationary point. In addition, we derive lower bound property for the nonzero singular values of the lifted stationary point and hence also of the local minimizers of the relaxed problem. Then the smoothing proximal gradient (SPG) algorithm is proposed to find a lifted stationary point of the continuous relaxation model. Moreover, it is shown that the whole sequence generated by SPG algorithm converges to a lifted stationary point. At last, numerical examples show the efficiency of the SPG algorithm.

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arXiv2021

T-Quadratic Forms and Spectral Analysis of T-Symmetric Tensors

An $n \times n \times p$ tensor is called a T-square tensor. It arises from many applications, such as the image feature extraction problem and the multi-view clustering problem. We may symmetrize a T-square tensor to a T-symmetric tensor. For each T-square tensor, we define a T-quadratic form, whose variable is an $n \times p$ matrix, and whose value is a $p$-dimensional vector. We define eigentuples and eigenmatrices for T-square tensors. We show that a T-symmetric tensor has unique largest and smallest eigentuples, and a T-quadratic form is positive semi-definite (definite) if and only if its smallest eigentuple is nonnegative (positive). The relation between the eigen-decomposition of T-symmetric tensors, and the TSVD of general third order tensors are also studied.

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arXiv2020

Multi-Tubal Rank of Third Order Tensor and Related Low Rank Tensor Completion Problem

Recently, a tensor factorization based method for a low tubal rank tensor completion problem of a third order tensor was proposed, which performed better than some existing methods. Tubal rank is only defined on one mode of third order tensor without low rank structure in the other two modes. That is, low rank structures on the other two modes are missing. Motivated by this, we first introduce multi-tubal rank, and then establish a relationship between multi-tubal rank and Tucker rank. Based on the multi-tubal rank, we propose a novel low rank tensor completion model. For this model, a tensor factorization based method is applied and the corresponding convergence anlysis is established. In addition, spatio-temporal characteristics are intrinsic features in video and internet traffic tensor data. To get better performance, we make full use of such features and improve the established tensor completion model. Then we apply tensor factorization based method for the improved model. Finally, numerical results are reported on the completion of image, video and internet traffic data to show the efficiency of our proposed methods. From the reported numerical results, we can assert that our methods outperform the existing methods.

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arXiv2020

Quaternion Matrix Optimization and The Underlying Calculus

Optimization models involving quaternion matrices are widely used in color image process and other engineering areas. These models optimize real functions of quaternion matrix variables. In particular, $\ell_0$-norms and rank functions of quaternion matrices are discrete. Yet calculus with derivatives, subdifferentials and generalized subdifferentials of such real functions is needed to handle such models. In this paper, we introduce first and second order derivatives and establish their calculation rules for such real functions. Our approach is consistent with the subgradient concept for norms of quaternion matrix variables, recently introduced in the literature. We develop the concepts of generalized subdifferentials of proper functions of quaternion matrices, and use them to analyze the optimality conditions of a sparse low rank color image denoising model. We introduce R-product for two quaternion matrix vectors, as a key tool for our calculus. We show that the real representation set of low-rank quaternion matrices is closed and semi-algebraic. We also establish first order and second order optimality conditions for constrained optimization problems of real functions in quaternion matrix variables.

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arXiv2020

A Low Rank Quaternion Decomposition Algorithm and Its Application in Color Image Inpainting

In this paper, we propose a lower rank quaternion decomposition algorithm and apply it to color image inpainting. We introduce a concise form for the gradient of a real function in quaternion matrix variables. The optimality conditions of our quaternion least squares problem have a simple expression with this form. The convergence and convergence rate of our algorithm are established with this tool.

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arXiv2020

Positivity Conditions for Cubic, Quartic and Quintic Polynomials

We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the points where the cubic polynomial attains zero. We then reformulate a necessary and sufficient condition for a quartic polynomial to be nonnegative for all positive reals. From this, we derive a necessary and sufficient condition for a quartic polynomial to be nonnegative and positive for all reals. Our condition explicitly exhibits the scope and role of some coefficients, and has strong geometrical meaning. In the interior of the nonnegativity region for all reals, there is an appendix curve. The discriminant is zero at the appendix, and positive in the other part of the interior of the nonnegativity region. By using the Sturm sequences, we present a necessary and sufficient condition for a quintic polynomial to be positive and nonnegative for all positive reals. We show that for polynomials of a fixed even degree higher than or equal to four, if they have no real roots, then their discriminants take the same sign, which depends upon that degree only, except on an appendix set of dimension lower by two, where the discriminants attain zero.

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arXiv2020

Copositivity of Three-Dimensional Symmetric Tensors

In this paper, we seek analytically checkable necessary and sufficient condition for copositivity of a three-dimensional symmetric tensor. We first show that for a general third order three-dimensional symmetric tensor, this means to solve a quartic equation and some quadratic equations. All of them can be solved analytically. Thus, we present an analytical way to check copositivity of a third order three dimensional symmetric tensor. Then, we consider a model of vacuum stability for $\mathbb{Z}_3$ scalar dark matter. This is a special fourth order three-dimensional symmetric tensor. We show that an analytically expressed necessary and sufficient condition for this model bounded from below can be given, by using a result given by Ulrich and Watson in 1994.

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arXiv2020

A Tensor Rank Theory and Maximum Full Rank Subtensors

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

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arXiv2020

Triple Decomposition and Tensor Recovery of Third Order Tensors

In this paper, we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order low rank tensors in a balanced way. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. For a third order tensor, its CP decomposition can be regarded as a special case of its triple decomposition. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank, and is strictly less than the middle value of the Tucker rank for an essential class of examples. These indicate that practical data can be approximated by low rank triple decomposition as long as it can be approximated by low rank CP or Tucker decomposition. This theoretical discovery is confirmed numerically. Numerical tests show that third order tensor data from practical applications such as internet traffic and video image are of low triple ranks. A tensor recovery method based on low rank triple decomposition is proposed. Its convergence and convergence rate are established. Numerical experiments confirm the efficiency of this method.

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arXiv2019

Tensor Norm, Cubic Power and Gelfand Limit

We establish two inequalities for the nuclear norm and the spectral norm of tensor products. The first inequality indicates that the nuclear norm of the square matrix is a matrix norm. We extend the concept of matrix norm to tensor norm. We show that the $1$-norm, the Frobenius norm and the nuclear norm of tensors are tensor norms, but the infinity norm and the spectral norm of tensors are not tensor norms. We introduce the cubic power for a general third order tensor, and show that a Gelfand formula holds for a general third order tensor. In that formula, for any norm, a common spectral radius-like limit exists for that third order tensor. We call such a limit the Gelfand limit. The Gelfand limit is zero if the third order tensor is nilpotent, and is one or zero if the third order tensor is idempotent. The Gelfand limit is not greater than any tensor norm of that third order tensor, and the cubic power of that third order tensor tends to zero as the power increases to infinity if and only if the Gelfand limit is less than one. The cubic power and the Gelfand limit can be extended to any higher odd order tensors.

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arXiv2019

Biquadratic Tensors, Biquadratic Decomposition and Norms of Biquadratic Tensors

Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.

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arXiv2019

Calculating Entanglement Eigenvalues for Non-Symmetric Quantum Pure States Based on the Jacobian Semidefinite Programming Relaxation Method

The geometric measure of entanglement is a widely used entanglement measure for quantum pure states. The key problem of computation of the geometric measure is to calculate the entanglement eigenvalue, which is equivalent to computing the largest unitary eigenvalue of a corresponding complex tensor. In this paper, we propose a Jacobian semidefinite programming relaxation method to calculate the largest unitary eigenvalue of a complex tensor. For this, we first introduce the Jacobian semidefinite programming relaxation method for a polynomial optimization with equality constraint, and then convert the problem of computing the largest unitary eigenvalue to a real equality constrained polynomial optimization problem, which can be solved by the Jacobian semidefinite programming relaxation method. Numerical examples are presented to show the availability of this approach.

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arXiv2017

A Complete Semidefinite Algorithm for Detecting Copositive Matrices and Tensors

A real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. The problem of detecting their copositivity is NP-hard. This paper proposes a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor. If it is copositive, the algorithm can get a certificate for the copositivity. If it is not, the algorithm can get a point that refutes the copositivity. We show that the detection can be done by solving a finite number of semidefinite relaxations, for all matrices and tensors.

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