Tensorisation of vectors and their efficient convolution

Abstract

In recent papers the tensorisation of vectors has been discussed. In principle, 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 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 corresponding procedure to univariate functions.