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MiS Preprint
98/2020
Minimal Divergence for Border Rank-2 Tensor Approximation
Wolfgang Hackbusch
Abstract
A tensor $\mathbf{v}$ is the sum of at least $\limfunc{rank}(\mathbf{v})$ elementary tensors. In addition, a 'border rank' is defined: \underline{$ \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{$\limfunc{rank}$}$(\mathbf{w})<\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 $\mathbf{w.}$ 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{$\limfunc{rank}$}$(\mathbf{w})=2<\limfunc{rank}(\mathbf{w})$ the divergence strength is $\gtrsim \varepsilon ^{-1/2}$, i.e., if $\left\Vert \mathbf{v}-\mathbf{w}\right\Vert <\varepsilon $ and $\limfunc{rank}(\mathbf{v})\leq 2,$ the parameters of $\mathbf{v}$ increase at least proportionally to $\varepsilon ^{-1/2}.$
