Tensor Approximation Research Articles

The spectral theory of tensors and its applications

Over the last few years, there has been a realization that there is a reasonably complete and consistent theory of eigenvalues and singular values for tensors of higher order. This theory, which generalizes the theory of matrix eigenvalues and singular values in various manners and extent, was proposed by Lim 1 and Qi 2 independently. We will use the term ‘spectral theory of tensors’ to refer to this body of work. This special issue contains nine papers with new results on the spectral theory of tensors and its applications. The first paper 3 by Chang et al. is a survey on some recent developments in the spectral theory of nonnegative tensors: H-eigenvalue and Z-eigenvalue problems, Perron–Frobenius theory, applications to higher order Markov chains, spectral theory of hypergraphs, and quantum entanglement. The next two papers are about the spectral theory of nonnegative tensors. In 4, Zhou et al. proposed new spectral characterizations of symmetric nonnegative tensors, while in 5, Zhang et al. extended the concept of essential nonnegativity from matrices to higher order tensors and established the convexity and log convexity of dominant eigenvalues for such a class of tensors. The next three papers concern the numerical computations of tensor eigenvalues or singular values. In 6, Friedland et al. proposed an algorithm called the alternating singular value decomposition for computing a best rank-one approximation of tensors based on computations of maximal singular values and singular vectors of matrices. In 7, Dupont and Scott studied the angular orientations of convergent iterates generated by Newton's method in multiple spatial dimensions for solving a tensor eigenproblem. In 8, Hu et al. studied tensor conic linear programming and employed it to solve the extreme Z-eigenvalue problem. In 9, Li et al. studied the perturbation bound for the Perron vector of a transition probability tensor P, and derived a new perturbation bound for the Perron vector of a transition probability matrix. Finally, the last papers are related to the applications of the spectral theory of tensors to hypergraphs. In 10, Li et al. examined some fundamental analytic properties of Z-eigenvalues of even order symmetric tensors; they showed that the maximum Z-eigenvalue function of the associated characteristic tensor provides a link between the combinatorial and analytic structures of the underlying hypergraph. In 11, Xie and Chang generalized the signless Laplacian matrices for graphs to an analogous notion for even uniform hypergraphs and deduced some of its fundamental properties. We hope that the reader will find these articles interesting and be spurred to either commence or continue working in this exciting new area.

Tensor Approximation Approach to Calculation of Singular Values and Vectors for SVD Problem

A new method of calculation of singular values and left and right singular vectors of arbitrary nonsquare matrices is proposed. The method allows one to avoid solutions of high rank systems of linear equations of singular value decomposition problems, which makes it not sensitive to ill-conditioness of the decomposed matrix. On the base of the Eckart–Young theorem, it was shown that each second order r-rank tensor can be represented as a sum of the first rank r-order “coordinate” tensors. A new system of equations for “coordinate” tensor generator vectors was obtained. An iterative method of solution of the system was elaborated. Results of the method were compared with classical methods of solutions of singular value decomposition problems.

Computing molecular correlation energies with guaranteed precision

We present an approach to compute accurate correlation energies for atoms and molecules in the framework of multiresolution analysis (MRA), using an adaptive discontinuous multiresolution spectral-element representation for the six-dimensional (two-electron) pair function. The key features of our approach that make it feasible, namely (1) low-rank tensor approximations of functions and operators and (2) analytic elimination of operator singularities via explicit correlation, were retained from the previous work [F. A. Bischoff, R. J. Harrison, and E. F. Valeev, J. Chem. Phys. 137, 104103 (2012)]. Here we generalized the working equations to handle general (non-symmetric) many-electron systems at the MP2 level. The numerical performance is shown for the beryllium atom and the water molecule where literature data for the basis set limits could be reproduced to a few tens of μE(h). The key advantages of molecular MRA-MP2 are the absence of bias and arbitrariness in the choice of the basis set, high accuracy, and low scaling with respect to the system size.

An equi-directional generalization of adaptive cross approximation for higher-order tensors

A scheme for the equi-directional approximation of third-order tensors by sums of products of vectors is presented and analyzed. The new method is a generalization of the adaptive cross approximation (ACA) method initially designed for the low-rank approximation of matrices using only few of its original entries. Moreover, a recursive formula is presented which generalizes the three-way ACA to tensors of arbitrary order.

A literature survey of low‐rank tensor approximation techniques

AbstractDuring the last years, low‐rank tensor approximation has been established as a new tool in scientific computing to address large‐scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function‐related tensors. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Direct strain tensor approximation for full‐field strain measurement methods

SUMMARYFull‐field strain measurement techniques are based on computing the spatial derivatives of numerical or functional approximations of the underlying displacement fields extracted from digital imaging methods. These methods implicitly assume that the medium satisfies the strain compatibility conditions, which are only true in the case of a continuum body that remains continuum throughout its deformation history. In the present work, we introduce a method that can be used to calculate the strain components directly from typical digital imaging data, without the need of the continuum hypothesis and the need for displacement field differentiation. Thus, it enables the measurement of strain fields from imaged surfaces that may or may not contain discontinuities. Numerical comparisons are performed on the basis synthetic data produced from an analytical solution for an elastically orthotropic open‐hole domain in tension. For performance comparison purposes, the mean absolute error distributions are calculated for the cases of both the traditional meshless random grid method, and the direct strain method introduced herein. It is established that the more refined representation of strain provided by our present approach is more accurate everywhere in the domain, but most importantly, near its boundaries. Published 2013. This article is a US Government work and is in the public domain in the USA.

Reflective mode of deformed-helix ferroelectric liquid crystal cells for sensing applications

We present a detailed theoretical and experimental study of the reflectance response of a deformed-helix ferroelectric (DHF) liquid crystal (LC) cell to an applied voltage under cross-polarisers. Using a model based on the effective dielectric tensor approximation, we derive simple analytical formulas to design a LC cell with maximum modulation depth and optimal linearity of the electro-optical response intensity versus the electric field. Our experimental results show that the cell works at frequencies up to 10 kHz and exhibits excellent linearity, with a total harmonic distortion as low as −70 dB. These findings suggest that DHF-LCs can be exploited to develop simple and accurate optical sensors.

TAMRESH – Tensor Approximation Multiresolution Hierarchy for Interactive Volume Visualization

AbstractInteractive visual analysis of large and complex volume datasets is an ongoing and challenging problem. We tackle this challenge in the context of state‐of‐the‐art out‐of‐core multiresolution volume rendering by introducing a novel hierarchical tensor approximation (TA) volume visualization approach. The TA framework allows us (a) to use a rank‐truncated basis for compact volume representation, (b) to visualize features at multiple scales, and (c) to visualize the data at multiple resolutions. In this paper, we exploit the special properties of the TA factor matrix bases and define a novel multiscale and multiresolution volume rendering hierarchy. Different from previous approaches, to represent one volume dataset we use but one set of global bases (TA factor matrices) to reconstruct at all resolution levels and feature scales. In particular, we propose a coupling of multiscalable feature visualization and multiresolution DVR through the properties of global TA bases. We demonstrate our novel TA multiresolution hierarchy based volume representation and visualization on a number of μCT volume datasets.

Enrichment of linear hexahedral finite elements using rotations of a virtual space fiber

SUMMARYThe present paper deals with the enrichment of 3D low‐order finite elements. The used concept is based on the idea that a 3D virtual fiber, after a spatial rotation, introduces an enhancement of the strain field tensor approximation. A consistent stiffness matrix is obtained, allowing a better approximation of the actual solution compared with that resulting from low‐order finite elements. Implemented for two eight‐node hexahedral elements, the performance of the space fiber rotation concept is assessed by running some classical beam, plate, and shell benchmarks, and the obtained results are compared especially with those given by linear eight‐node and quadratic 20‐node hexahedral elements. In particular, it is shown that the developed elements accuracy is significantly superior to that of the classical eight‐node hexahedral element and close to that of the classical 20‐node hexahedral element. Copyright © 2013 John Wiley & Sons, Ltd.

The geometry of algorithms using hierarchical tensors

In this paper, the differential geometry of the novel hierarchical Tucker format for tensors is derived. The set HT,k of tensors with fixed tree T and hierarchical rank k is shown to be a smooth quotient manifold, namely the set of orbits of a Lie group action corresponding to the non-unique basis representation of these hierarchical tensors. Explicit characterizations of the quotient manifold, its tangent space and the tangent space of HT,k are derived, suitable for high-dimensional problems. The usefulness of a complete geometric description is demonstrated by two typical applications. First, new convergence results for the nonlinear Gauss–Seidel method on HT,k are given. Notably and in contrast to earlier works on this subject, the task of minimizing the Rayleigh quotient is also addressed. Second, evolution equations for dynamic tensor approximation are formulated in terms of an explicit projection operator onto the tangent space of HT,k. In addition, a numerical comparison is made between this dynamical approach and the standard one based on truncated singular value decompositions.

Structured multi-way arrays and their applications

Based on the structure of the rank-1 matrix and the different unfolding ways of the tensor, we present two types of structured tensors which contain the rank-1 tensors as special cases. We study some properties of the ranks and the best rank-r approximations of the structured tensors. By using the upper-semicontinuity of the matrix rank, we show that for the structured tensors, there always exist the best rank-r approximations. This can help one to better understand the sequential unfolding singular value decomposition (SVD) method for tensors proposed by J. Salmi et al. [IEEE Trans Signal Process, 2009, 57(12): 4719–4733] and offer a generalized way of low rank approximations of tensors. Moreover, we apply the structured tensors to estimate the upper and lower bounds of the best rank-1 approximations of the 3rd-order and 4th-order tensors, and to distinguish the well written and non-well written digits.

On best rank one approximation of tensors

SUMMARYToday, compact and reduced data representations using low rank data approximation are common to represent high‐dimensional data sets in many application areas as for example, genomics, multimedia, quantum chemistry, social networks, or visualization. To produce such low rank data representations, the input data is typically approximated by the so‐called alternating least squares (ALS) algorithms. However, not all of these ALS algorithms are guaranteed to converge. To address this issue, we suggest a new algorithm for the computation of a best rank one approximation of tensors, called alternating singular value decomposition. This method is based on the computation of maximal singular values and the corresponding singular vectors of matrices. We also introduce a modification for this method and the ALS method, which ensures that alternating iterations will always converge to a semi‐maximal point (a critical point in several vector variables is semi‐maximal if it is maximal with respect to each vector variable, while other vector variables are kept fixed). We present several numerical examples that illustrate the computational performance of the new method in comparison to the ALS method. Copyright © 2013 John Wiley & Sons, Ltd.

Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors

The problem discussed in this paper is the symmetric best low multilinear rank approximation of third-order symmetric tensors. We propose an algorithm based on Jacobi rotations, for which symmetry is preserved at each iteration. Two numerical examples are provided indicating the need for such algorithms. An important part of the paper consists of proving that our algorithm converges to stationary points of the objective function. This can be considered an advantage of the proposed algorithm over existing symmetry-preserving algorithms in the literature.

Symmetric Tensor Approximation Hierarchies for the Completely Positive Cone

In this paper we construct two approximation hierarchies for the completely positive cone based on symmetric tensors. We show that one hierarchy corresponds to dual cones of a known polyhedral approximation hierarchy for the copositive cone, and the other hierarchy corresponds to dual cones of a known semidefinite approximation hierarchy for the copositive cone. As an application, we consider a class of bounds on the stability number of a graph obtained from the polyhedral approximation hierarchy, and we construct a primal optimal solution with its tensor lifting for each of such linear programs.

A Three-Way Jordan Canonical Form as Limit of Low-Rank Tensor Approximations

A best rank-$R$ approximation of an order-3 tensor or three-way array may not exist due to the fact that the set of three-way arrays with rank at most $R$ is not closed. In this case, we are trying to compute the approximation results in diverging rank-1 terms. We show that this phenomenon can be seen as a three-way generalization of approximate diagonalization of a nondiagonalizable (real) matrix. Moreover, we show that, analogous to the matrix case, the limit point of the approximating rank-$R$ sequence satisfies a three-way generalization of the real Jordan canonical form. Recently, it was shown how to obtain the limit point and its three-way Jordan form for $R\le\min(I,J,K)$ and groups of two or three diverging rank-1 terms, where $I\times J\times K$ is the size of the array. We extend this to groups of four diverging rank-1 terms and show that $R>\min(I,J,K)$ is possible as long as no groups of more than $\min(I,J,K)$ diverging rank-1 terms occur. We demonstrate our results by means of numerical experiments.

Convergence Rates of Multilevel and Sparse Tensor Approximations for a Random Elliptic PDE

Partial differential equations with random coefficients can be cast as parametric problems with a potentially infinite-dimensional parameter domain. For a class of elliptic equations, we derive convergence rates of approximations based on a tensorized polynomial basis on the parameter domain and a sequence of spatial discretizations. We prove that a sparse tensor product construction achieves essentially the same convergence rate as a multilevel approximation in which the spatial discretization level is chosen separately for each coefficient of the solution with respect to the polynomial basis. In some cases, the same rate is attained also by a single level approximation, using just one spatial discretization for all coefficients. We suggest an adaptive algorithm that reaches the optimal convergence rate with respect to the total number of degrees of freedom in a multilevel setting, without prior knowledge of this rate. Numerical computations confirm theoretical results.

A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties

We here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables. It is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales. An efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale, involving a deterministic operator, and local problems at a micro scale for which we have the possibility to use fine approximation spaces. Global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions. Convergence properties of these tensor based methods, which are closely related to spectral decompositions, benefit from the separation of scales. Different types of uncertainties are considered at the micro level. They may be associated with some variability in the operator or source terms, or even with some geometrical variability. In the latter case, specific reformulations of local problems using fictitious domain methods are introduced.

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.

Tensor-based methods for numerical homogenization from high-resolution images

We present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images. We first introduce specific numerical treatments for the translation of image-based homogenization problems into a tensor framework. It includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images. We then introduce some variants of proper generalized decomposition (PGD) methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems. A new definition of PGD is introduced which allows the progressive construction of a Tucker decomposition of the solution. This tensor format is well adapted to the present application and improves convergence properties of tensor decompositions. Finally, we use a dual-based error estimator on quantities of interest which was recently introduced in the context of PGD. We exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material. We also provide a complete goal-oriented adaptive strategy for the progressive construction of tensor decompositions (of primal and dual solutions) yielding to predictions of homogenized quantities with a prescribed accuracy.

Adaptive cross approximation of tensors arising in the discretization of boundary integral operator shape derivatives

The shape derivative of a dense N × N BEM matrix is a sparse three-way tensor with O ( N 2 ) non-zero entries, to which standard BEM acceleration techniques such as the adaptive cross approximation (ACA) and FMM cannot be directly applied. The tensor can be used to compute shape sensitivities, or via adjoint equations, the gradient of an objective function. Although for many PDEs, calculation of the tensor can be avoided by expressing the shape derivative of the solution as the solution of a related PDE, this approach is not always easily amenable to BEM. Therefore, the computation of shape derivatives via the sparse three-way tensor is a valuable alternative, provided that efficient acceleration techniques exist. We propose a new algorithm for the approximation of BEM shape derivative tensors based on ACA that achieves the same complexity and error bounds as ACA for the BEM matrix itself. Numerical examples show that despite the much larger amount of data involved, the tensor approximation is only moderately slower than the matrix approximation. We also demonstrate the method on a shape optimization problem from the literature.