Higher-order principal component pursuit via tensor approximation and convex optimization

Recovering the low-rank structure of data matrix from sparse errors arises in the principal component pursuit （PCP）. This paper exploits the higher-order generalization of matrix recovery, named higher-order principal component pursuit （HOPCP）, since it is critical in multi-way data analysis. Unlike the convexification （nuclear norm） for matrix rank function, the tensorial nuclear norm is stil an open problem. While existing preliminary works on the tensor completion field provide a viable way to indicate the low complexity estimate of tensor, therefore, the paper focuses on the low multi-linear rank tensor and adopt its convex relaxation to formulate the convex optimization model of HOPCP. The paper further propose two algorithms for HOPCP based on alternative minimization scheme： the augmented Lagrangian alternating direction method （ALADM） and its truncated higher-order singular value decomposition （ALADM-THOSVD） version. The former can obtain a high accuracy solution while the latter is more efficient to handle the computationally intractable problems. Experimental results on both synthetic data and real magnetic resonance imaging data show the applicability of our algorithms in high-dimensional tensor data processing.

Preservation of Linear Constraints in Approximation of Tensors

For an arbitrary tensor(multi-index array) with linear constraints at each direction,it is proved that the factors of any minimal canonical tensor approximation to this tensor satisfy the same linear constraints for the corresponding directions.

Non-Convex Optimization of Resource Allocation in Fog Computing Using Successive Approximation

Fog computing can deliver low delay and advanced IT services to end users with substantially reduced energy consumption.Nevertheless,with soaring demands for resource service and the limited capability of fog nodes,how to allocate and manage fog computing resources properly and stably has become the bottleneck.Therefore,the paper investigates the utility optimization-based resource allocation problem between fog nodes and end users in fog computing.The authors first introduce four types of utility functions due to the diverse tasks executed by end users and build the resource allocation model aiming at utility maximization.Then,for only the elastic tasks,the convex optimization method is applied to obtain the optimal results;for the elastic and inelastic tasks,with the assistance of Jensen’s inequality,the primal non-convex model is approximated to a sequence of equivalent convex optimization problems using successive approximation method.Moreover,a two-layer algorithm is proposed that globally converges to an optimal solution of the original problem.Finally,numerical simulation results demonstrate its superior performance and effectiveness.Comparing with other works,the authors emphasize the analysis for non-convex optimization problems and the diversity of tasks in fog computing resource allocation.

Relativistic symmetries with the trigonometric Pschl-Teller potential plus Coulomb-like tensor interaction

The Dirac equation is solved to obtain its approximate bound states for a spin-1/2 particle in the presence of trigonometric Poeschl-Teller （tPT） potential including a Coulomb-like tensor interaction with arbitrary spin-orbit quantum number κ using an approximation scheme to substitute the centrifugal terms κ（κ± i 1）r^-2. In view of spin and pseudo-spin （p-spin） symmetries, the relativistic energy eigenvalues and the corresponding two-component wave functions of a particle moving in the field of attractive and repulsive tPT potentials are obtained using the asymptotic iteration method （AIM）. We present numerical results in the absence and presence of tensor coupling A and for various values of spin and p-spin constants and quantum numbers n and κ. The non-relativistic limit is also obtained.

Construction and Application of 3-Point Tensor Product Scheme

In this paper, we propose and analyze a tensor product subdivision scheme which is the extension of three point scheme for curve modeling. The usefulness of the scheme is illustrated by considering different examples along with its application in surface modeling.

Double Transformed Tubal Nuclear Norm Minimization for Tensor Completion

Non-convex methods play a critical role in low-rank tensor completion for their approximation to tensor rank is tighter than that of convex methods.But they usually cost much more time for calculating singular values of large tensors.In this paper,we propose a double transformed tubal nuclear norm(DTTNN)to replace the rank norm penalty in low rank tensor completion(LRTC)tasks.DTTNN turns the original non-convex penalty of a large tensor into two convex penalties of much smaller tensors,and it is shown to be an equivalent transformation.Therefore,DTTNN could take advantage of non-convex envelopes while saving time.Experimental results on color image and video inpainting tasks verify the effectiveness of DTTNN compared with state-of-the-art methods.

E-characteristic Polynomials of Real Rectangular Tensor

By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.

APPROXIMATING THE STATIONARY BELLMAN EQUATION BY HIERARCHICAL TENSOR PRODUCTS

We treat infinite horizon optimal control problems by solving the associated stationary Bellman equation numerically to compute the value function and an optimal feedback law.The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations(PDE),which means that the Bellman equation suffers from the curse of dimensionality.Its non linearity is handled by the Policy Iteration algorithm,where the problem is reduced to a sequence of linear equations,which remain the computational bottleneck due to their high dimensions.We reformulate the linearized Bellman equations via the Koopman operator into an operator equation,that is solved using a minimal residual method.Using the Koopman operator we identify a preconditioner for operator equation,which deems essential in our numerical tests.To overcome computational infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats,in particular tensor trains(TT tensors)and multi-polynomials,together with high-dimensional quadrature,e.g.Monte-Carlo.By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidence is given.

基于并行小波算法的多模态数据近似匹配模型构建

针对近似匹配过程易受数据冗余性、异构成分等的影响,提出了基于并行小波算法的多模态数据近似匹配模型。该模型首先采用并行小波算法剔除多模态数据中的噪声,避免噪声对匹配过程产生影响;其次采用张量分解的聚类算法将不同相似度的数据划分到不同类簇中,以消除不同类簇的数据差异度;最后将预处理后的数据输入到基于空间方向近似性的数据匹配模型中,通过计算参考数据与待匹配数据之间的空间方向近似度、编辑距离完成多模态数据的近似匹配。实验结果表明,所提方法的匹配查准率高、查全率高、匹配时间短。

Best rank one approximation of real symmetric tensors can be chosen symmetric

We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.

LIMITATION OF AVERAGE ESHELBY TENSOR AND ITS APPLICATION IN ANALYSIS OF ELLIPSE APPROXIMATION

By the aid of irreducible decomposition, the average Eshelby tensor can be expressed by two complex coefficients in 2D Eshelby problem. This paper proved the limitation of complex coefficients based on the span of elastic strain energy density. More discussions yielded the constraints on the sampling of module and phase difference of complex coefficients. Using this information, we obtained that the maximum relative error is 65.78% after an ellipse approximation. These results, as a supplement to our previous paper, further implied that Eshelby＇s solution for an ellipsoidal inclusion could not be applied to non-ellipsoidal inclusions without taking care.

Fast nonnegative tensor ring decomposition based on the modulus method and low-rank approximation

Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on frequent reshaping and permutation operations in the optimization process and use a shrinking step size or projection techniques to ensure core tensor nonnegativity, which leads to a slow convergence rate, especially for large-scale problems. In this paper, we first propose an NTR algorithm based on the modulus method(NTR-MM), which constrains core tensor nonnegativity by modulus transformation. Second, a low-rank approximation(LRA) is introduced to NTR-MM(named LRA-NTR-MM), which not only reduces the computational complexity of NTR-MM significantly but also suppresses the noise. The simulation results demonstrate that the proposed LRA-NTR-MM algorithm achieves higher computational efficiency than the state-of-the-art algorithms while preserving the effectiveness of feature extraction.

The Low-Rank Approximation of Fourth-Order Partial-Symmetric and Conjugate Partial-Symmetric Tensor

We present an orthogonal matrix outer product decomposition for the fourth-order conjugate partial-symmetric(CPS)tensor and show that the greedy successive rank-one approximation(SROA)algorithm can recover this decomposition exactly.Based on this matrix decomposition,the CP rank of CPS tensor can be bounded by the matrix rank,which can be applied to low-rank tensor completion.Additionally,we give the rank-one equivalence property for the CPS tensor based on the SVD of matrix,which can be applied to the rank-one approximation for CPS tensors.

基于半张量理论的电力系统稳定域边界逼近 (一)理论基础

电力系统稳定域边界的逼近一直是应用能量函数方法分析电力系统暂态稳定的难点。基于半张量理论,文中给出了逼近电力系统稳定域边界的矩阵方程。通过矩阵运算,可以获得电力系统稳定域边界的高阶逼近表达式。所提出的算法没有对系统进行任何形式的非线性变换,不需要求解系统所有的特征根与特征向量,也无需进行时域积分;算法充分保留了系统的非线性结构,形式简洁。由于该算法完全基于矩阵运算,因此非常适于计算机实现。

纤维增强复合材料的力学性能预测的数值模拟

纤维增强复合材料的力学性能和热物理性能依赖于纤维的取向状态。在注射成型过程中纤维最终的取向状态依赖于充填过程的速度场 ,因此最终的产品性质依赖于成型的详细过程。研究发现 ,注塑成型制品的结构呈层状分布 ,层数依赖于模具几何和成型条件 ,不过大多数的结构在成型表面为沿流动方向取向 ,而在中心层为横向排列 ,有时在制件表面还有一层薄的介于二者之间排列的取向层。本文主要给出两个简单模型中纤维取向预测的理论和数值方法 ,这两个模型分别为 :中心浇口圆盘和边浇口长条。

极分解中转动张量的计算及近似分析

该文利用Ｃａｙｌｅｙ－Ｈａｍｉｌｔｏｎ定理给出了转动张量Ｒ的显式表示，其中只包含变形梯度Ｆ的较低次幂及Ｕ和Ｆ的主不变量，由此获得了转动角的计算公式和转动轴的显式表达。最后通过近似分析，给出了在小变形中等转动情况下转动张量和伸长张量的近似表示。

基于张量积结构的多维小波网络(英文)

针对多维函数逼近的‘维数灾’问题 ,依据小波框架理论提出了一种张量积结构小波网络 ,其主要特点是在网络输出层将各维输入的小波重构相乘 ,从而得到自动覆盖函数输入空间的多维小波框架 ,最后通过权系数的在线或离线学习实现多维函数的小波逼近 .理论分析和仿真结果证实了该结构设计方法应用于多维函数逼近时的有效性 .

基于稳定域边界二阶逼近的暂态电压稳定分析

电力系统中暂态电压失稳事故时有发生从而引起电力界广泛关注。目前的暂态电压稳定分析方法尚存在不足之处。本文基于非线性动力学的半张量积方法研究电力系统暂态电压稳定问题,以期在电力系统发生大扰动后快速判断故障后系统的电压稳定性,从而有利于及时采取控制措施避免系统崩溃。论文首先建立了暂态电压稳定分析的系统综合模型,主要包括对暂态电压失稳至关重要的发电机、励磁、感应电动机和输电网四个子系统。在此基础上,提出了基于稳定域边界二阶逼近的暂态电压稳定判据;在一个简单系统进行的实例分析表明所提出的暂态电压失稳判据具有速度快、准确度高及易于计算机实现的特点,具有较高的工程实用性,可望有效判断故障后系统的暂态电压稳定性。

张量积Bézier曲面的S幂基降多阶逼近

文章给出了张量积Bézier曲面一次降多阶的算法。给定张量积Bézier曲面,采用了分向降阶算法,对u向、v向Bézier曲线分别一次降多阶。这里曲线降阶,利用基转换矩阵将Bézier曲线的Bernstein基函数表示成S幂基函数,通过截断曲线中的高次项,可以得到相应的降多阶逼近曲线,所得的降多阶逼近曲面自动保角点高阶插值;最后给出了数值实例。

强扰动理论及其发展

从矩阵形式的麦克斯韦方程出发,对强扰动理论有了重要的发展,使它不仅能处理强扰动随机电介质的问题,而且可以处理强扰动电、磁混合媒质电磁散射的计算。作为应用新理论的例子,文中详细地讨论了颗粒媒质等效介电常数和等效导磁率的计算。结果表明,当粒子线度比波长小得多时,媒质粒子的介电常数和导磁率之间互不影响。