R-Term Representation

The r-term representation v = Xr =1 Od j=1 v(j) ;i.e., a representation by sums of r elementary tensors, is already used in the algebraic definition (3.9) of tensors. In different fields, the r-term representation has different names: ‘canonical decomposition’ in psychometrics (cf. [48]), ‘parallel factors model’ (cf. [158]) in chemometrics.1 The word ‘representation’ is often replaced by ‘format’. The short form ‘CP’ is proposed by Comon [62] meaning ‘canonical polyadic decomposition’. Sometimes the polyadic decomposition (PD) is distinguished from the canonical polyadic decomposition (CPD) by the fact that in the latter case the used number of terms should be minimal (equal to the tensor rank; cf. Vervliet et al [296]). Here, the notation ‘r-term representation’ is used, where ‘r’ is considered as a variable from N0; which may be replaced by other variable names or numbers. Before we discuss the r-term representation in Section 7.3, we consider representations in general (Section 7.1) and the full representation (Section 7.2). The sensitivity of the r-term representation is analysed in Section 7.4. Section 7.5 discusses possible representations of the vectors v(j) 2Vj : We briefly mention the conversion from full format to r-term representation (cf. 7.6.1). We conclude with the discussion of (anti-)symmetric tensors (in Section 7.7) and other modifications of the r-term format (in Section 7.8). The discussion of arithmetical operations with tensors in r-term representation is postponed to Chapter 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 Chapter 9.