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MiS Preprint

Tensorisation of Vectors and their Efficient Convolution

Wolfgang Hackbusch


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.

MSC Codes:
15A69, 15A99, 44A35, 65F99, 65T99
tensorisation, tensor representation, hierarchical tensor representation, convolution, matrix-vector multiplication

Related publications

2011 Repository Open Access
Wolfgang Hackbusch

Tensorisation of vectors and their efficient convolution

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