Tensor Subspace Approximation

The exact representation of $${\rm v} \in {\rm V} = {\bigotimes}_{j=1}^{d} V_{j}$$ by a tensor subspace representation (8.6b) may be too expensive because of the high dimensions of the involved subspaces or even impossible since v is a topological tensor admitting no finite representation. In such cases we must be satisfied with an approximation u ≈ v which is easier to handle. We require that $$\mathbf{u} \in {\mathcal{T}}_{\mathbf{r}}$$ , i.e., there are bases $$\{{b}_{1}^{(j)},\ldots,{b}_{{r}_{j}}^{(j)}\} \subset {V }_{j}$$ such that 1 $$\mathbf{u} ={ \sum \nolimits }_{{i}_{1}=1}^{{r}_{1} }\cdots {\sum \nolimits }_{{i}_{d}=1}^{{r}_{d} }\mathbf{a}[{i}_{1}\cdots {i}_{d}]{\bigotimes}_{j=1}^{d}{b}_{{ i}_{j}}^{(j)}.$$ The basic task of this chapter is the following problem: 2 $$\text{ Given }\mathbf{v} \in \mathbf{V}\text{, find a suitable approximation }\mathbf{u} \in {\mathcal{T}}_{\mathbf{r}} \subset \mathbf{V,}$$ where $$\mathbf{r} = \left ({r}_{1},\ldots,{r}_{d}\right ) \in {\mathbb{N}}^{d}.$$ Finding $$\mathbf{u} \in {\mathcal{T}}_{\mathbf{r}}$$ means finding coefficients $$\mathbf{a}[{i}_{1}\cdots {i}_{d}]$$ as well as basis vectors b i (j) ∈ V j in (10.1). Problem (10.2) is formulated rather vaguely. If an accuracy ε > 0 is prescribed, r ∈ ℕ d as well as $$\mathbf{u} \in {\mathcal{T}}_{\mathbf{r}}$$ are to be determined. The strict minimisation of $$\left \Vert \mathbf{v} -\mathbf{u}\right \Vert$$ is often replaced by an appropriate approximation $$\mathbf{u}$$ requiring low computational cost. Instead of ε > 0, we may prescribe the rank vector r ∈ ℕ d in (10.2).Optimal approximations (so-called ‘best approximations’) will be studied in Sect. 10.2. While best approximations require an iterative computation, quasi-optimal approximations can be determined explicitly using the HOSVD basis introduced in Sect. 8.3. The latter approach is explained in Sect. 10.1.