A Note on Nonclosed Tensor Formats

Hackbusch, Wolfgang

doi:10.1007/s10013-019-00372-4

Cited by 2 publications

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“…The following statements are described in detail and proved in [8] and [3, § § 9.5.3-9.5.6, p. 312ff]. We consider a format satisfying the following simple conditions.…”

Section: General Divergence Behaviour Of Nonclosed Formatsmentioning

confidence: 99%

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Minimal divergence for border rank-2 tensor approximation

Hackbusch

2021

Linear and Multilinear Algebra

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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 r is the minimum integer such that w is a limit of rank-r tensors. Usually, the set of rankr 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 v − w < ε and rank(v) ≤ 2, the parameters of v increase at least proportionally to ε −1/2 .

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“…The following statements are described in detail and proved in [8] and [3, § § 9.5.3-9.5.6, p. 312ff]. We consider a format satisfying the following simple conditions.…”

Section: General Divergence Behaviour Of Nonclosed Formatsmentioning

confidence: 99%

“…Proof: δ U (w, ε) ≥ δ(w, ε) is the trivial estimate. Fix some v ∈ F, w − v < ε, from the right-hand side in (8). Setû := Pv ∈ F ∩ U(w) with P from Lemma 3.3 and note that w −û = P(w − v) ≤ P w − v < ε P .…”

Section: Lemma 33mentioning

confidence: 99%