Tensor rank: Some lower and upper bounds

Boris Alexeev, Michael A. Forbes, Jacob Tsimerman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The results of Strassen [1] and Raz [2] show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we construct field-independent explicit 0/1 tensors T: [n]d → F with rank at least 2n[d/2] + n - Θ(d lg n). This improves the lower-order terms in known lower bounds for any odd d ≥ 3. We also explore a generalization of permutation matrices, which we denote permutation tensors. We show, by applying known counting lower bounds, that there exist order-3 permutation tensors with super-linear rank as well as orderd permutation tensors with high rank. We also explore a natural class of permutation tensors, which we call group tensors. For any group G, we define the group tensor TGd: Gd → F, by T Gd(g1,⋯,gd) = 1 iff g 1 ··· gd = 1G. We give two upper bounds for the rank of these tensors. The first uses representation theory and works over "large" fields F, showing (among other things) that rankF(TGd) ≤ |G|d/2. In the case that d = 3, we are able to show that rankF(TG d) ≤ O|G|ω/2) ≤ O|G|1.19), where ω is the exponent of matrix multiplication. The next upper bound uses interpolation and only works for abelian G, showing that over any field F that rankF(TGd) ≤ O(|G|1+lg d lgd-1 |G|). In either case, this shows that many permutation tensors have far from maximal rank, which is very different from the matrix case and thus eliminates many natural candidates for high tensor rank. We also explore monotone tensor rank. We give explicit 0/1 tensors T: [n]d → F that have tensor rank at most dn but have monotone tensor rank exactly nd-1. This is a nearly optimal separation.

Original languageEnglish (US)
Title of host publicationProceedings - 26th Annual IEEE Conference on Computational Complexity, CCC 2011
Pages283-291
Number of pages9
DOIs
StatePublished - Aug 29 2011
Externally publishedYes
Event26th Annual IEEE Conference on Computational Complexity, CCC 2011 - San Jose, CA, United States
Duration: Jun 8 2011Jun 10 2011

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Other

Other26th Annual IEEE Conference on Computational Complexity, CCC 2011
CountryUnited States
CitySan Jose, CA
Period6/8/116/10/11

Keywords

  • Algebraic complexity
  • Tensor rank

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Computational Mathematics

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