MiS Preprint Repository

A tensor $\mathbf{v}$ is the sum of at least $\text{limfunc}rank(\mathbf{v})$ elementary tensors. In addition, a 'border rank' is defined: \underline{$\text{limfunc}rank$}$(\mathbf{w})=r$ holds if $\mathbf{w}$ is a limit of rank-$r$ tensors. Usually, the set of rank-$r$ tensors is not closed, i.e., tensors with $r=$\underline{$\text{limfunc}rank$}$(\mathbf{w})<\text{limfunc}rank(\mathbf{w})$ may exist. It is easy to see that in such a case the representation of rank-$r$ tensors $\mathbf{v}$ contains diverging elementary tensors as $\mathbf{v}$ approaches $\mathbf{w}\mathbf{.}$ In a first part we recall results about the uniform strength of the divergence in the case of general nonclosed tensor formats (restricted to finite dimensions). The second part discusses the $r$-term format for infinite-dimensional tensor spaces. It is shown that the general situation is very similar to the behaviour of finite-dimensional model spaces. The third part contains the main result: it is proved that in the case of \underline{$\text{limfunc}rank$}$(\mathbf{w})=2<\text{limfunc}rank(\mathbf{w})$ the divergence strength is $\gtrsim {\epsilon }^{-1/2}$, i.e., if $\Vert \mathbf{v}-\mathbf{w}\Vert <\epsilon$ and $\text{limfunc}rank(\mathbf{v})\le 2,$ the parameters of $\mathbf{v}$ increase at least proportionally to ${\epsilon }^{-1/2}.$