Abstract
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.
| Originalsprog | Engelsk |
|---|---|
| Tidsskrift | Computational Complexity |
| Vol/bind | 28 |
| Udgave nummer | 1 |
| Sider (fra-til) | 57-111 |
| Antal sider | 55 |
| ISSN | 1016-3328 |
| DOI | |
| Status | Udgivet - 2019 |