# Tensor surgery and tensor rank

```@article{Christandl2018TensorSA,
title={Tensor surgery and tensor rank},
author={Matthias Christandl and Jeroen Zuiddam},
journal={computational complexity},
year={2018},
volume={28},
pages={27-56}
}```
• Published 13 June 2016
• Mathematics, Computer Science
• computational complexity
We introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices and edges. We show that tensor surgery is capable of preserving the low rank structure of an initial tensor decomposition and thus allows to prove nontrivial upper bounds on tensor rank, border rank and…
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• Computer Science, Mathematics
computational complexity
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An upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices is presented, showing that the exponent per edge is at most 0.77, outperforming the best known upper bound for matrix multiplication.
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• Mathematics, Computer Science
SODA
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