A 2n2-log2(n)-1 lower bound for the border rank of matrix multiplication

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2018
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Landsberg, Joseph M.
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10.1093/imrn/rnx025
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International Mathematics Research Notices ; 2018 (2018), 15. - pp. 4722-4733. - Oxford University Press (OUP). - ISSN 1073-7928. - eISSN 1687-0247
Abstract
Let M⟨n⟩ ∈ Cn2⊗Cn2⊗Cn2 denote the matrix multiplication tensor for n×n matrices. We use the border substitution method [2, 3, 6] combined with Koszul flattenings [8] to prove the border rank lower bound R(M⟨n,n,n⟩)≥2n2−⌈log2(n)⌉−1⁠.
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ISO 690LANDSBERG, Joseph M., Mateusz MICHALEK, 2018. A 2n2-log2(n)-1 lower bound for the border rank of matrix multiplication. In: International Mathematics Research Notices. Oxford University Press (OUP). 2018(15), pp. 4722-4733. ISSN 1073-7928. eISSN 1687-0247. Available under: doi: 10.1093/imrn/rnx025
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@article{Landsberg2018lower-53233,
  year={2018},
  doi={10.1093/imrn/rnx025},
  title={A 2n<sup>2</sup>-log<sub>2</sub>(n)-1 lower bound for the border rank of matrix multiplication},
  number={15},
  volume={2018},
  issn={1073-7928},
  journal={International Mathematics Research Notices},
  pages={4722--4733},
  author={Landsberg, Joseph M. and Michalek, Mateusz}
}
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    <dcterms:abstract xml:lang="eng">Let M&lt;sub&gt;⟨n⟩&lt;/sub&gt; ∈ C&lt;sup&gt;n2&lt;/sup&gt;⊗C&lt;sup&gt;n2&lt;/sup&gt;⊗C&lt;sup&gt;n2&lt;/sup&gt; denote the matrix multiplication tensor for n×n matrices. We use the border substitution method [2, 3, 6] combined with Koszul flattenings [8] to prove the border rank lower bound R(M&lt;sub&gt;⟨n,n,n⟩&lt;/sub&gt;)≥2n&lt;sup&gt;2&lt;/sup&gt;−⌈log&lt;sub&gt;2&lt;/sub&gt;(n)⌉−1⁠.</dcterms:abstract>
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