Tensor Subspace Representation

Abstract

This chapter is devoted to the tensor subspace representation which is also called the Tucker representation. We use the term ‘tensor subspace’ for the tensor product U := a Nd j=1Uj of subspaces Uj Vj . Obviously, U is a subspace of V:= a Nd j=1Vj . For d = 2, r-term and tensor subspace representations are identical. Therefore, both approaches can be viewed as extensions of the concept of rank-r matrices to the tensor case d 3: The resulting set Tr introduced in Section 8.1 will be characterised by a vector-valued rank r = (r1; : : : ; rd). Since by definition, tensors v 2 Tr are closely related to subspaces, their descriptions by means of frames or bases is of interest (see Section 8.2). Differently from the r-term format, algebraic tools like the singular-value decomposition can be applied and lead to a higher order singular-value decomposition (HOSVD), which is a quite important feature of the tensor subspace representation (cf. Section 8.3). Moreover, HOSVD yields a connection to the minimal subspaces from Chapter 6. In Section 8.5 we compare the formats discussed so far and describe conversions between the formats. In a natural way, a hybrid format appears using the r-term format for the coefficient tensor of the tensor subspace representation (cf. 8.2.6). Section 8.6 deals with the problem of joining two representation systems, as it is needed when we add two tensors involving different tensor subspaces.