ABSTRACT

A tensor $\mathbf{v}$ is the sum of at least $\mathrm{rank}(\mathbf{v})$ elementary tensors. In addition, a ‘border rank’ is defined: $\underset{\_}{\mathrm{rank}}(\mathbf{w})=r$ holds if r is the minimum integer such that $\mathbf{w}$ is a limit of rank-r tensors. Usually, the set of rank-r tensors is not closed, i.e. tensors with $r=\underset{\_}{\mathrm{rank}}(\mathbf{w})<\mathrm{rank}(\mathbf{w})$ 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 $\underset{\_}{\mathrm{rank}}(\mathbf{w})=2<\mathrm{rank}(\mathbf{w})$ the divergence strength is $\gtrsim {\epsilon }^{-1/2}$, i.e. if $\Vert \mathbf{v}-\mathbf{w}\Vert <\epsilon$ and $\mathrm{rank}(\mathbf{v})\le 2,$ the parameters of $\mathbf{v}$ increase at least proportionally to ${\epsilon }^{-1/2}.$

摘要

张量$\mathbf{v}$是至少$秩(\mathbf{v})$初等张量。此外，定义了“边界等级”：$\underset{\_}{秩}(\mathbf{w})=r$如果r是满足以下条件的最小整数，则成立$\mathbf{w}$是秩- r张量的极限。通常，rank- r张量的集合不是封闭的，即具有$r=\underset{\_}{秩}(\mathbf{w})<秩(\mathbf{w})$可能存在。很容易看出，在这种情况下，rank -r张量的表示$\mathbf{v}$包含发散的基本张量作为$\mathbf{v}$方法$\mathbf{w}\mathbf{.}$在第一部分中，我们回顾了在一般非封闭张量格式（限于有限维）的情况下散度的均匀强度的结果。第二部分讨论无限维张量空间的r项格式。结果表明，一般情况与有限维模型空间的行为非常相似。第三部分包含主要结果：证明了在$\underset{\_}{秩}(\mathbf{w})=\mathrm{2个}<秩(\mathbf{w})$发散强度是$\gtrsim {\epsilon }^{-\mathrm{1个}/\mathrm{2个}}$, 即如果$\Vert \mathbf{v}-\mathbf{w}\Vert <\epsilon$和$秩(\mathbf{v})\le \mathrm{2个},$的参数$\mathbf{v}$至少成比例地增加到${\epsilon }^{-\mathrm{1个}/\mathrm{2个}}.$