Tensorisation

Tensorisation has been introduced by Oseledets [153] (applied to matrices instead of vectors). The tensorised version of a

$$\mathbb{K}^{n}$$

vector can easily be truncated in a black-box fashion. Under suitable conditions, the data size reduces drastically. Operations applied to these tensors instead of the original vectors have a cost related to the (much smaller) tensor data size.

Section 14.1

describes the main principle, the hierarchical format

$$\mathcal{H}^{{\rm tens}}_{\rho}$$

corresponding to the TT format, operations with tensorised vectors, and the generalisation to matrices. The reason, why the data size can be reduced so efficiently is analysed in

Sect. 14.2

. Tensorisation mimics classical analytical approximations method which exploit the smoothness of a function to obtain an approximation with much less degrees of freedom.

Section 14.3

presents in detail the (exact) convolution of vectors performed by means of their tensorisations. It is shown that the cost corresponds to the data size of the tensors.

Section 14.4

is devoted to the tensorised counterpart of the fast Fourier transform (FFT). During the algorithm one has to insert truncation steps, since otherwise a maximal representation rank arises. While the original tensorisation technique applies to discrete data,

Sect. 14.5

generalises the approach to functions.