Tensorisation of Vectors and their Efficient Convolution

Wolfgang Hackbusch

Contact the author: Please use for correspondence this email.
Submission date: 08. Sep. 2010
Pages: 19
published in: Numerische Mathematik, 119 (2011) 3, p. 465-488 
DOI number (of the published article): 10.1007/s00211-011-0393-0
Bibtex
MSC-Numbers: 15A69, 15A99, 44A35, 65F99, 65T99
Keywords and phrases: tensorisation, tensor representation, hierarchical tensor representation, convolution, matrix-vector multiplication
Download full preprint: PDF (271 kB), PS ziped (274 kB)

Abstract:
In recent papers the tensorisation of vectors has been discussed. In principle, this is the isomorphic representation of an  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., . 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  vectors can be considered as grid values of function, we also apply the same procedure to univariate functions.