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MiS Preprint
49/2024
Minimal Divergence for Border Rank-3 Tensor Approximation
Wolfgang Hackbusch
Abstract
It is known that, in general, the set $\mathcal{R}_{n}$ of tensors of rank $\leq n$ is nonclosed. Hence, there are tensors $\mathbf{w}$ in the closure of $\mathcal{R}_{n}$ but not in $\mathcal{R}_{n}$ which are the limits of sequences $\mathbf{v}_{\nu}\in\mathcal{R}_{n}.$ Let $\mathbf{v}_{\nu} =\sum_{i=1}^{n}\mathbf{z}_{i,\nu}$ be a representation by elementary tensors $\mathbf{z}_{i,\nu}.$ It is well-known that $\delta_{\nu}:=\max\{\left\Vert \mathbf{z}_{i,\nu}\right\Vert :1\leq i\leq n\}$ is unbounded as $\nu \rightarrow\infty.$ The error $\varepsilon_{\nu}:=\left\Vert \mathbf{w} -\mathbf{v}_{\nu}\right\Vert $ tends to zero. Since $\delta_{\nu} \rightarrow\infty$ describes an instability, it is of interest how $\delta_{\nu}$ depends on $\varepsilon_{\nu}.$ In a previous paper the author proved for $n=2$ that both quantities are related by $\delta_{\nu}\geq O(\varepsilon_{\nu}^{-1/2}).$ This article concerns the case of $n=3$ and proves $\delta_{\nu}\geq O(\varepsilon_{\nu}^{-1/3}).$
