r-Term Representation

The

r

-term representation

$$\rm{v}={\sum}^{r}_{v=1}\,\,{\bigotimes}^{d}_{j=1}\,\,{\vartheta}_{v}^{(j)}$$

, i.e., a representation by sums of

r

elementary tensors, is already used in the algebraic definition (3.11) of tensors. In different fields, the

r

-term representation has different names: ‘canonical decomposition’ in psychometrics (cf. [30]), ‘parallel factors model’ (cf. [96]) in chemometrics.1 The word ‘representation’ is often replaced by ‘format’. The short form ‘CP’ is proposed by Comon [38] meaning ‘canonical polyadic decomposition’. Here, the notation ‘

r

-term representation’ is used with ‘

r

’ considered as a variable from

$$\mathbb{N}_0$$

, which may be replaced by other variable names or numbers.

Before we discuss the

r

-term representation in

Sect. 7.3

, we consider representations in general (

Sect. 7.1

) and the full representation (

Sect. 7.2

). The sensitivity of the

r

-term representation is analysed in

Sect. 7.4

.

Section 7.5

discusses possible representations of the vectors

$${\vartheta}_{v}^{(j)} \in V_j$$

. We briefly mention the conversion from full format to

r

-term representation (cf. §7.6.1) and modifications (cf.

Sect. 7.7

).

The discussion of arithmetical operations with tensors in

r

-term representation is postponed to Chap. 13. In this chapter we restrict our considerations to the exact representation in the

r

-term format. Approximations, which are of greater interest in practice, will be discussed in Chap. 9.