Hierarchical Kronecker tensor-product approximations

Computational Methods in Applied Mathematics, 2006

The structured tensor-product approximation of multidimensional nonlocal operators by a two-level rank-(r1, . . . , rd) decomposition of related higher-order tensors is proposed and analysed. In this approach, the construction of the desired approximant to a target tensor is a reminiscence of the Tucker-type model, where the canonical components are represented in a fixed (uniform) basis, while the core tensor is given in the canonical format. As an alternative, the multilevel nested canonical decomposition is presented. The complexity analysis of the corresponding multilinear algebra shows an almost linear cost in the one-dimensional problem size. The existence of a low Kronecker rank two-level representation is proven for a class of function-related tensors.

International Journal of Computer Mathematics, 2013

The algebra of the Kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. The vectorisation operations and the commutation transformations that are common in multivariate statistical analysis alter the positional relationship of the matrix elements. These elements correspond to numbers that are liable to be stored in contiguous memory cells of a computer, which should remain undisturbed. It is suggested that, in the absence of an adequate index notation that enables the manipulations to be performed without disturbing the data, even the most clear-headed of computer programmers is liable to perform wholly unnecessary and time-wasting operations that shift data between memory cells.

Numerical Linear Algebra with Applications, 2007

In this paper, we consider the approximation of dense block Toeplitz-plus-Hankel matrices by sums of Kronecker products. We present an algorithm for efficiently computing the matrix approximation that requires the factorization of matrices of much smaller dimension than that of the original. The main results are described for block Toeplitz matrices with Toeplitz-plus-Hankel blocks (BTTHB), but the algorithms can be readily adjusted for other related structures that arise in image processing applications, such as block Toeplitz with Toeplitz blocks (BTTB) and block Toeplitz-plus-Hankel with Toeplitz-plus-Hankel blocks (BTHTHB). Our work extends the techniques in [11, 15], which consider similar matrices, but with the added restriction that the matrices have a banded/block-banded structure. We illustrate the effectiveness of our algorithm by using the output of the algorithm to construct preconditioners for systems from two different applications: diffuse optical tomography and atmospheric image deblurring.

Journal of the American Mathematical Society, 1998

We study maximizing vectors of Hankel operators with matrix-valued symbols. This study leads to a solution of the so-called recovery problem for unitary-valued functions and to a new approach to Wiener–Hopf factorizations for functions in a function space X X satisfying natural conditions. Finally, we improve earlier results of Peller and Young on hereditary properties of the operator of superoptimal approximation by analytic matrix functions.

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory, 2015

In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition, which allows to compute best rank k-approximations. For very big matrices a low rank approximation using SVD is not computationally feasible. In this case different approximations are available. It seems that variants of the CURdecomposition are most suitable. For d-mode tensors T ∈ ⊗ d i=1 R n i , with d > 2, many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. The most appropriate approximation seems to be best (r1,. .. , r d)-approximation, which maximizes the 2 norm of the projection of T on ⊗ d i=1 Ui, where Ui is an ri-dimensional subspace R n i. One of the most common methods is the alternating maximization method (AMM). It is obtained by maximizing on one subspace Ui, while keeping all other fixed, and alternating the procedure repeatedly for i = 1,. .. , d. Usually, AMM will converge to a local best approximation. This approximation is a fixed point of a corresponding map on Grassmannians. We suggest a Newton method for finding the corresponding fixed point. We also discuss variants of CUR-approximation method for tensors. The first part of the paper is a survey on low rank approximation of tensors. The second new part of this paper is a new Newton method for best (r1,. .. , r d)-approximation. We compare numerically different approximation methods.

2010

Tensor decompositions permit to estimate in a deterministic way the parameters in a multi-linear model. Applications have been already pointed out in antenna array processing and digital communications, among others, and are extremely attractive provided some diversity at the receiver is available. As opposed to the widely used ALS algorithm, non-iterative algorithms are proposed in this paper to compute the required tensor decomposition into a sum of rank-1 terms, when some factor matrices enjoy some structure, such as block-Hankel, triangular, band, etc.

Journal of Complexity

We are interested in approximation of a multivariate function f (x 1 ,. .. , x d) by linear combinations of products u 1 (x 1) • • • u d (x d) of univariate functions u i (x i), i = 1,. .. , d. In the case d = 2 it is a classical problem of bilinear approximation. In the case of approximation in the L 2 space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel f (x 1 , x 2). There are known results on the rate of decay of errors of best bilinear approximation in L p under different smoothness assumptions on f. The problem of multilinear approximation (nonlinear tensor product approximation) in the case d ≥ 3 is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in L p under mixed smoothness assumption on f .

Lecture Notes in Control and Information Sciences, 2006

In this paper, we will revisit the problem of generalized low rank approximation of matrices (GLRM) recently proposed in . The GLRM problem has been investigated by many researchers in the last two years due to its broad engineering applications. However, it is known that there may be no closed-form solution for this problem and the convergence for the proposed iterative algorithm Algorithm GLRAM is actually still open. Motivated by these observations, we will derive a necessary condition for the GLRAM problem via manifold theory in this paper. Further, the relationship between this derived necessary condition and the Algorithm GLRAM is investigated, which will provide some insights for the possible complete analysis of the convergence problem in the future. Finally, some illustrative examples are presented to show different cases for convergence and divergence with different initial conditions.

Mathematics and Computers in Simulation, 2008

In this paper we propose a numerical method for approximating the product of a matrix function with multiple vectors by Krylov subspace methods combined with a QR decomposition of these vectors. This problem arises in the implementation of exponential integrators for semilinear parabolic problems. We will derive reliable stopping criteria and we suggest variants using up-and downdating techniques. Moreover, we show how Ritz vectors can be included in order to speed up the computation even further. By a number of numerical examples, we will illustrate that the proposed method will reduce the total number of Krylov steps significantly compared to a standard implementation if the vectors correspond to the evaluation of a smooth function at certain quadrature points.