Border rank is not multiplicative under the tensor product

Abstract

It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper bounds were obtained with help of border rank. It was left open whether border rank itself can be strictly submultiplicative. We answer this question in the affirmative. In order to do so, we construct lines in projective space along which the border rank drops multiple times and use this result in conjunction with a previous construction for a tensor rank drop. Our results also imply strict submultiplicativity for cactus rank and border cactus rank.

arXiv comment25 pages, 1 figure - Revised version

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• Partial Degeneration of Tensors
• The Tensor as an Informational Resource
• Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
• Border Waring Rank via Asymptotic Rank
• Collineation varieties of tensors
• Nuclear Norm Under Tensor Kronecker Products