Tensor Subspace Representation

We use the term ‘

tensor subspace

’ for the tensor product

$$\mathbf{U}:=a{\bigotimes}^{d}_{j=1}\,\,\rm U_{j}$$

of subspaces

$$\rm U_{j}{\subset}\rm V_{j}$$

. Obviously, U is a subspace of

$$\mathbf{V}:=a{\bigotimes}^{d}_{j=1}\,\,\rm V_{j}$$

, but not any subspace of V is a tensor subspace. For

d

= 2,

r

-term and tensor subspace representations (also called

Tucker representation

) 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

$$\mathcal{T}_{{\rm r}}$$

introduced in

Sect. 8.1

will be characterised by a vector-valued rank r = (

r

1

, …,

r

d

). Since by definition, tensors

$${\rm v} \in \mathcal{T}_{{\rm r}}$$

are closely related to subspaces, their descriptions by means of frames or bases is of interest (see

Sect. 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.

Sect. 8.3

). Moreover, HOSVD yields a connection to the minimal subspaces from Chap. 6. In

Sect. 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. §8.2.4).

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.