2019-03-11

# Asymptotic tensor rank of graph tensors: Beyond matrix multiplication

## Publication

### Publication

*Computational Complexity , Volume 28 - Issue 1 p. 57- 111*

We present 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. For k≥ 4 , we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication (k = 3), which is approximately 0.79. We raise the question whether for some k the exponent per edge can be below 2/3, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalize to higher-order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity.

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Keywords | tensor rank, graph tensors, subrank, Dicke tensors, algebraic complexity, matrix multiplication |

Persistent URL | dx.doi.org/10.1007/s00037-018-0172-8 |

Journal | Computational Complexity |

Citation |
Christandl, M, Vrana, P, & Zuiddam, J. (2019). Asymptotic tensor rank of graph tensors: Beyond matrix multiplication.
Computational Complexity, 28(1), 57–111. doi:10.1007/s00037-018-0172-8 |