Singular values of nonnegative rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. Some properties concerning the singular values of a real rectangular tensor were discussed by K. C. Chang et al. [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we give some new results on the Perron-Frobenius Theorem for nonnegative rectangular tensors. We show that the weak Perron-Frobenius keeps valid and the largest singular value is really geometrically simple under some conditions. In addition, we establish the convergence of an algorithm proposed by K. C. Chang et al. for finding the largest singular value of nonnegative primitive rectangular tensors.     

Linear convergence of an algorithm for largest singular value of a nonnegative rectangular tensor

An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we establish a linear convergence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.     

两个基于不同张量乘法的四阶张量分解

提出了两个基于不同张量乘法的四阶张量分解. 首先, 在矩阵乘法的基础上, 定义第一种四阶张量乘法(F-乘), 基于F-乘提出了第一种四阶张量分解(F-TD). 其次, 基于三阶张量t-product给出了第二种四阶张量乘法(B-乘)和分解(FT-SVD). 同时, 利用两种分解方法, 分别给出两个张量逼近定理. 最后, 三个数值算例阐明提出的两种分解方法的准确性和可行性.     

Strictly nonnegative tensors and nonnegative tensor partition

We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.     

Geometric simplicity of spectral radius of nonnegative irreducible tensors

We study the real and complex geometric simplicity of nonnegative irreducible tensors. First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an evenorder nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.     

Robust tensor completion using transformed tensor singular value decomposition

In this article, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform matrix. This would be more effective for robust tensor completion. Experimental results for hyperspectral, video and face datasets have shown that the recovery performance for the robust tensor completion problem by using transformed tensor SVD is better in peak signal‐to‐noise ratio than that by using Fourier transform and other robust tensor completion methods.     

A new definition of geometric multiplicity of eigenvalues of tensors and some results based on it

We give a new definition of geometric multiplicity of eigenvalues of tensors, and based on this, we study the geometric and algebraic multiplicity of irreducible tensors’ eigenvalues. We get the result that the eigenvalues with modulus ρ(A) have the same geometric multiplicity. We also prove that twodimensional nonnegative tensors’ geometric multiplicity of eigenvalues is equal to algebraic multiplicity of eigenvalues.     

High-order copositive tensors and its applications

With the coming of the big data era, high-order high-dimensional structured tensors received much attentions of researchers" in recent years, and now they are developed into a new research branch in mathematics named multilinear algebra. As a special kind of structured tensor, the copositive tensor receives a special concern due to its wide applications in vacuum stability of a general scalar potential, polynomial optimization, tensor complementarity problem and tensor eigenvalue complementarity problem. In this review, we will give a simple survey on recent advances of high-order copositive tensors and its applications. Some potential research directions in the future are also listed in the paper.     

l k,s -Singular values and spectral radius of rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of l ^k,s -singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of l ^k,s -singular values /vectors, some properties of the related l ^k,s -spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.     

Singular values of a real rectangular tensor

Real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we systematically study properties of singular values of a real rectangular tensor, and give an algorithm to find the largest singular value of a nonnegative rectangular tensor. Numerical results show that the algorithm is efficient.     

A randomized tensor singular value decomposition based on the t‐product

The tensor SVD (t‐SVD) for third‐order tensors, previously proposed in the literature, has been applied successfully in many fields, such as computed tomography, facial recognition, and video completion. In this paper, we propose a method that extends a well‐known randomized matrix method to the t‐SVD. This method can produce a factorization with similar properties to the t‐SVD, but it is more computationally efficient on very large data sets. We present details of the algorithms and theoretical results and provide numerical results that show the promise of our approach for compressing and analyzing image‐based data sets. We also present an improved analysis of the randomized and simultaneous iteration for matrices, which may be of independent interest to the scientific community. We also use these new results to address the convergence properties of the new and randomized tensor method as well.     

Expression of curvature tensors and some applications

We get an explicit expression of curvature operators in terms of at most eight terms of sectional curvatures. Some applications of this result are also given, particularly we improve a result of Chen-Tian related to the first Chern class of admissible surfaces in pinched manifolds. We also characterize in a simple way all functionsk(x, y) which can be sectional curvatures of some curvature operatorR.Supported by CNPq, Brazil and NNSFC.     

复半正定矩阵的奇异值不等式

用Mn表示所有复矩阵组成的集合.对于A∈Mn,σ(A)=(σ1(A),…,σn(A)),其中σ1(A)≥…≥σn(A)是矩阵A的奇异值.本文给出证明:对于任意实数α,A,B∈Mn为半正定矩阵,优化不等式σ(A-|α|B) wlogσ(A+αB)成立,改进和推广了文[5]的结果.     

奇异值函数的二阶方向导数

本文用一个直接的方法给出了奇异值函数的二阶方向导数公式. 作为应用, 利用这一公式建立了谱范数的上图集合与核范数的上图集合的切锥和二阶切集的具体表达式, 这些表达式在矩阵优化的一阶和二阶最优条件的研究中起着重要作用.     

Two perturbation bounds for singular values and eigenvalues

The relative error in as an approximation to α is measured by

Positive solutions of nonlinear singular third-order two-point boundary value problem

In this paper, we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third-order two-point boundary value problem

     

An iterative algorithm for low-rank tensor completion problem with sparse noise and missing values

Robust low-rank tensor completion plays an important role in multidimensional data analysis against different degradations, such as sparse noise, and missing entries, and has a variety of applications in image processing and computer vision. In this paper, an optimization model for low-rank tensor completion problems is proposed and a block coordinate descent algorithm is developed to solve this model. It is shown that for one of the subproblems, the closed-form solution exists and for the other, a Riemannian conjugate gradient algorithm is used. In particular, when all elements are known, that is, no missing values, the block coordinate descent is simplified in the sense that both subproblems have closed-form solutions. The convergence analysis is established without requiring the latter subproblem to be solved exactly. Numerical experiments illustrate that the proposed model with the algorithm is feasible and effective.