Bivariate least squares approximation with linear constraints

In this article linear least squares problems with linear equality constraints are considered, where the data points lie on the vertices of a rectangular grid. A fast and efficient computational method for the case when the linear equality constraints can be formulated in a tensor product form is presented. Using the solution of several univariate approximation problems the solution of the bivariate approximation problem can be derived easily. AMS subject classification (2000)  65D05, 65D07, 65D10, 65F05, 65F20     

Cauchy constraints and particle content of fourth-order gravity in n dimensions

The full class of purely metrical gravitational theories in n⩾3 dimensions which follows from a Lagrangian composed of linear and quadratic curvature terms is analyzed. The type of the field equations in a suitable gauge is discussed. The principal symbol and the particle content of the linearized field equations are investigated. The space+time decomposition and the ADM formalism are used to derive the constraints and evolution equations for the variational derivative tensor.     

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.     

求解具有张量积结构线性系统的共轭梯度法

本文考虑具有张量积结构线性系统的数值解法．该线性系统常常来源于高维立方体上线性偏微分方程的有限差分离散化．利用张量一矩阵乘法,给出了基于张量格式的求解这类线性系统的共轭梯度法．与求解标准线性系统的共轭梯度法比较,新的算法能够节约大量的计算量及存储空间．     

屈服条件由应力张量的二次齐次函数与一次齐次函数之和来表达的极限分析定理

本文建立了由应力张量σ_ij的二次齐次函数与一次齐次函数的和来表达其屈服条件的刚理想塑性体的极限分析变分原理,它可用于岩土力学的极限分析问题,并把屈服条件为应力张量σ_ij 的二次齐次函数或一次齐次函数来表达的情况作为其特例.     

Jacobi fields for a nonholonomic distribution

The variational problem with nonholonomic constraints was considered in detail by Bliss. A distribution is a special case of constraints. Horizontal geodesics on a manifold with flat metric and constant tensor of nonholonomity are considered. It is proved that, in the classical adjoint problem, conjugate points appear, which does not involve any loss of optimality. The second variation of the length (or energy) functional of admissible (horizontal) geodesics for a distribution on a smooth manifold is expressed in terms of the distribution curvature tensor.     

张量分析和多项式优化的若干进展

张量分析 (也称多重数值线性代数) 主要包括张量分解和张量特征值的理论和算法,多项式优化主要包括目标和约束均为多项式的一类优化问题的理论和算法. 主要介绍这两个研究领域中若干新的研究结果. 对张量分析部分,主要介绍非负张量H-特征值谱半径的一些性质及求解方法,还介绍非负张量最大 (小) Z-特征值的优化表示及其解法;对多项式优化部分,主要介绍带单位球约束或离散二分单位取值、目标函数为齐次多项式的优化问题及其推广形式的多项式优化问题和半定松弛解法. 最后对所介绍领域的发展趋势做了预测和展望.     

Factorization strategies for third-order tensors

Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array. The specifics of the approximate representation (such as how many terms to use in the sum, orthogonality constraints, etc.) depend on the application.In this paper, we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem. Reminiscent of matrix factorizations, we present a new factorization of a tensor as a product of tensors. To derive the new factorization, we define a closed multiplication operation between tensors. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications.Specifically, this new multiplication allows us to introduce concepts such as tensor transpose, inverse, and identity, which lead to the notion of an orthogonal tensor. The multiplication also gives rise to a linear operator, and the null space of the resulting operator is identified. We extend the concept of outer products of vectors to outer products of matrices. All derivations are presented for third-order tensors. However, they can be easily extended to the order-p(p>3) case. We conclude with an application in image deblurring.     

Linear invariants of a Cartesian tensor

The number of linear invariants under SO(3) as well as SO(2)of a Cartesian tensor of an arbitrary rank is studied. A linearform is defined in terms of elements of a tensor. It is establishedthat the number of linear invariants of a tensor of rank n underSO(3) equals the dimension of the space of isotropic tensorsof rank n. Formulas for the number of invariants in the twocases are also derived. For the elasticity tensor, our analysisconfirms the results of Norris.     

Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (?) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (?) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.     

The tensor Golub–Kahan–Tikhonov method applied to the solution of ill-posed problems with a t-product structure

This paper discusses an application of partial tensor Golub–Kahan bidiagonalization to the solution of large-scale linear discrete ill-posed problems based on the t-product formalism for third-order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658). The solution methods presented first reduce a given (large-scale) problem to a problem of small size by application of a few steps of tensor Golub–Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third-order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third-order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill-posed problems defined by a matrix operator to linear discrete ill-posed problems defined by a third-order tensor operator. An interlacing property of singular tubes for third-order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t-product approach, in comparison with solution methods that are based on matricization of the tensor equation.     

A tensor-based dictionary learning approach to tomographic image reconstruction

We consider tomographic reconstruction using priors in the form of a dictionary learned from training images. The reconstruction has two stages: first we construct a tensor dictionary prior from our training data, and then we pose the reconstruction problem in terms of recovering the expansion coefficients in that dictionary. Our approach differs from past approaches in that (a) we use a third-order tensor representation for our images and (b) we recast the reconstruction problem using the tensor formulation. The dictionary learning problem is presented as a non-negative tensor factorization problem with sparsity constraints. The reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary. Numerical results show that our tensor formulation leads to very sparse representations of both the training images and the reconstructions due to the ability of representing repeated features compactly in the dictionary.     

A note on linear preservers of certain ranks of tensor products of matrices

In this note, we give a simple proof as well as an extension of a very recent result of B. Zheng, J. Xu and A. Fosner concerning linear maps between vector spaces of complex square matrices that preserve the rank of tensor products of matrices by using a structure theorem of R. Westwick on linear maps between tensor product spaces that preserve non-zero decomposable elements.     

On algorithms for and computing with the tensor ring decomposition

Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high‐dimensional data, achieving linear scaling with the input dimension instead of exponential scaling. In this paper, we investigate even lower storage‐cost representations in the tensor ring format, which is an extension of the tensor train format with variable end‐ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage‐cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph‐based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.     

Reductions of the object and the affine torsion tensor of a centroprojective connection on a distribution of planes

The projective group is represented as a bundle of centroprojective frames. This bundle is endowed with a centroprojective connection and becomes the space of this centroprojective connection. Structure equations of this space are found, which include the affine torsion tensor and the centroprojective curvature tensor containing the affine curvature subtensor. A distribution of planes in projective space and its associated principal bundle (which has two simplest and two simple (in the sense of [1]) quotient principal bundles) are considered. On the associated bundle, a group connection is defined. The object of the centroprojective connection is reduced to the object of the group connection. The object of the group connection contains the objects of the flat and normal linear connections, the centroprojective subconnection, and the affine-group connection as subobjects. The torsion object of the affine-group connection is determined. It is proved that it forms a tensor, which contains the torsion tensor of the normal linear connection as a subtensor. It is shown that the affine torsion tensor of the centroprojective connection reduces to the torsion tensor of the affine-group connection.     

Small‐strain heterogeneous elastoplasticity revisited

The elastoplastic quasi‐static evolution of a multiphase material—a material with a pointwise varying yield surface and elasticity tensor, together with interfaces between the phases—is revisited in the context of conservative globally minimizing movements. Existence is shown, and classical evolutions are recovered under natural constraints on the plastic dissipation potential. Special attention is paid to the interfaces where the correct dissipation has to be enforced on the interfaces. Further, the evolution is shown to be a limit of that obtained for a model with linear isotropic hardening as the hardening becomes vanishingly small. The duality between plastic strains and admissible stresses is also revisited for Lipschitz boundaries, and its role in deriving a classical evolution is circumscribed. © 2012 Wiley Periodicals, Inc.     

Constraints on singularities under Ricci curvature bounds

We announce results giving constraints on the singularities of spaces which are Gromov-Hausdorf'f limits of sequences of Riemannian manifolds whose Ricci curvature and volume are bounded from below and whose curvature tensor is bounded in an integral sense.     

On the elementary divisors of the Sylvester and Lyapunov maps

The paper addresses the problem of computing the elementary divisors of the tensor product of linear transformations using the analysis of the tensor products of polynomial models, as developed in Fuhrmann and Helmke [5]. We use this to study the elementary divisors of the Lyapunov and complementary Lyapunov maps.     

Linear Precision for Toric Surface Patches

We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification includes, as a proper subset, the classification of toric surface patches from geometric modeling which have linear precision. Besides the well-known tensor product patches and Bézier triangles, we identify a family of toric patches with trapezoidal shape, each of which has linear precision. Furthermore, Bézier triangles and tensor product patches are special cases of trapezoidal patches.