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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
48/2010

Tensorisation of Vectors and their Efficient Convolution

Wolfgang Hackbusch

Abstract

In recent papers the tensorisation of vectors has been discussed. Inprinciple, this is the isomorphic representation of an $\mathbb{R}^{n}$~vector as a tensor. Black-box tensor approximation methods can be used to reduce the data size of the tensor representation. In particular, if the vector corresponds to a grid function, the resulting data size can become much smaller than $n,$ e.g., $O(\log n)\ll n$. In this article we discuss vector operations, in particular, the convolution of two vectors which are given via a sparse tensor representation. We want to obtain the result again in the tensor representation. Furthermore, the cost of the convolution algorithm should be related to the operands' data sizes.

While $\mathbb{R}^{n}$~vectors can be considered as grid values of function, we also apply the same procedure to univariate functions.

Received:
Sep 8, 2010
Published:
Sep 8, 2010
MSC Codes:
15A69, 15A99, 44A35, 65F99, 65T99
Keywords:
tensorisation, tensor representation, hierarchical tensor representation, convolution, matrix-vector multiplication

Related publications

inJournal
2011 Repository Open Access
Wolfgang Hackbusch

Tensorisation of vectors and their efficient convolution

In: Numerische Mathematik, 119 (2011) 3, pp. 465-488