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Krylov-type methods for tensor computations I
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-1542-2690
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.ORCID iD: 0000-0003-2281-856X
2013 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, no 2, 891-918 p.Article in journal (Refereed) Published
Abstract [en]

Several Krylov-type procedures are introduced that generalize matrix Krylov methods for tensor computations. They are denoted minimal Krylov recursion, maximal Krylov recursion, and contracted tensor product Krylov recursion. It is proved that, for a given tensor A with multilinear rank-(p; q; r), the minimal Krylov recursion extracts the correct subspaces associated to the tensor in p+q+r number of tensor-vector-vector multiplications. An optimized minimal Krylov procedure is described that, for a given multilinear rank of an approximation, produces a better approximation than the standard minimal recursion. We further generalize the matrix Krylov decomposition to a tensor Krylov decomposition. The tensor Krylov methods are intended for the computation of low multilinear rank approximations of large and sparse tensors, but they are also useful for certain dense and structured tensors for computing their higher order singular value decompositions or obtaining starting points for the best low-rank computations of tensors. A set of numerical experiments, using real-world and synthetic data sets, illustrate some of the properties of the tensor Krylov methods.

Place, publisher, year, edition, pages
Elsevier, 2013. Vol. 438, no 2, 891-918 p.
Keyword [en]
Tensor, Krylov-type method, tensor approximation, Tucker model, multilinear algebra, multilinear rank, sparse tensor, information science
National Category
Computational Mathematics
URN: urn:nbn:se:liu:diva-73727DOI: 10.1016/j.laa.2011.12.007ISI: 000313226900017OAI: diva2:476342

Special Issue on Tensors and Multilinear Algebra

Available from: 2012-01-12 Created: 2012-01-12 Last updated: 2013-10-11Bibliographically approved

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Savas, BerkantEldén, Lars
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