Abstract
Various tensor formats exist which allow a data-sparse representation of tensors. Some of these formats are not closed. The consequences are (i) possible non-existence of best approximations and (ii) divergence of the representing parameters when a tensor within the format tends to a border tensor outside. The paper tries to describe the nature of this divergence. A particular question is whether the divergence is uniform for all border tensors.
Highlights
Given vector spaces Vj we denote the corresponding tensor space by ⊗d V = Vj . j =1Since usually the dimension ∏d j =1 dim(Vj ) ofV is rather huge, the numerical treatment of tensor needs special data-sparse representation techniques
We study the order of divergence in a neighbourhood of a border tensor and whether this quantity behaves continuously
A tensor representation is given by a map ρ from a parameter set into the tensor space
Summary
Given (finite-dimensional) vector spaces Vj we denote the corresponding tensor space by. The oldest one is the r-term format: given a representation rank r we form all tensors which can be written as a sum of r elementary tensors, where r ∈ N0 := N ∪ {0}. For matrices it is well known that a convergent sequence {Mk} with rank(Mk) ≤ r has a limit M with rank(M) ≤ r, i.e., the set of matrices of rank ≤ r is closed This is not true for tensors of order d ≥ 3. Part (c) states that, in general, the r-term representation is not closed This leads to the notation of the border rank. We study the order of divergence in a neighbourhood of a border tensor and whether this quantity behaves continuously
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