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A Newton-Grassmann method for computing the best multilinear rank-(r1,r2,r3) approximation of a tensor
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.ORCID iD: 0000-0003-2281-856X
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-1542-2690
2009 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, Vol. 32, no 2, p. 248-271Article in journal (Refereed) Published
##### Abstract [en]

We derive a Newton method for computing the best rank-$(r_1,r_2,r_3)$ approximation of a given $J\times K\times L$ tensor $\mathcal{A}$. The problem is formulated as an approximation problem on a product of Grassmann manifolds. Incorporating the manifold structure into Newton's method ensures that all iterates generated by the algorithm are points on the Grassmann manifolds. We also introduce a consistent notation for matricizing a tensor, for contracted tensor products and some tensor-algebraic manipulations, which simplify the derivation of the Newton equations and enable straightforward algorithmic implementation. Experiments show a quadratic convergence rate for the Newton–Grassmann algorithm.

##### Place, publisher, year, edition, pages
2009. Vol. 32, no 2, p. 248-271
##### Keyword [en]
tensor, multilinear, rank, approximation, Grassmann manifold, Newton
Mathematics
##### Identifiers
OAI: oai:DiVA.org:liu-13193DiVA, id: diva2:18009
Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2013-10-11
##### In thesis
1. Algorithms in data mining using matrix and tensor methods
Open this publication in new window or tab >>Algorithms in data mining using matrix and tensor methods
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors.

The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is

$\min_{\rank(X) = k} \det (B - X A)(B - X A)\tp,$

where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix.

Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations.

The remaining two papers discuss computational methods for the best multilinear

rank approximation problem

$\min_{\cB} \| \cA - \cB\|$

where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product

$\cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu},$

where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.

##### Place, publisher, year, edition, pages
Matematiska institutionen, 2008. p. 29
##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1178
##### Keyword
Volume, Minimization criterion, Determinant, Rank deficient matrix, Reduced rank regression, System identification, Rank reduction, Volume minimization, General algorithm, Handwritten digit classification, Tensors, Higher order singular value decomposition, Tensor approximation, Least squares, Tucker model, Multilinear algebra, Notation, Contraction, Tensor matricization, Newton's method, Grassmann manifolds, Product manifolds, Quasi-Newton algorithms, BFGS and L-BFGS, Symmetric tensor approximation, Local/intrinsic coordinates, Global/embedded coordinates;
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:liu:diva-11597 (URN)978-91-7393-907-2 (ISBN)
##### Public defence
2008-05-27, Glashuset, B-huset, ing. 25, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
##### Supervisors
Available from: 2008-04-29 Created: 2008-04-29 Last updated: 2013-10-11

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