Minimal Divergence for Border Rank-2 Tensor Approximation

Wolfgang Hackbusch

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Submission date: 26. Oct. 2020
Pages: 14
Bibtex
MSC-Numbers: 14N07, 15A69, 46A32
Keywords and phrases: tensor approximation, nonclosed tensor formats, border rank
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Abstract:
A tensor v is the sum of at least rank(v) elementary tensors. In addition, a ‘border rank’ is defined: rank(w) = r holds if w is a limit of rank-r tensors. Usually, the set of rank-r tensors is not closed, i.e., tensors with r =rank(w) < rank(w) may exist. It is easy to see that in such a case the representation of rank-r tensors v contains diverging elementary tensors as v approaches 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 rank(w) = 2 < rank(w) the divergence strength is ≳ 𝜀^−1∕2, i.e., if < 𝜀 and rank(v) ≤ 2, the parameters of v increase at least proportionally to 𝜀^−1∕2.