Tensor-rank and lower bounds for arithmetic formulas

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

23 Scopus citations

Abstract

We show that any explicit example for a tensor A:[n]r - → double-struck F with tensor-rank ≥ nr·(1- o(1)), (where r ≤ log n / log log n), implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over double-struck F. This shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply super-polynomial lower bounds for the size of general arithmetic formulas. One component of our proof is a new approach for homogenization and multilinearization of arithmetic formulas, that gives the following results: We show that for any n-variate homogenous polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a homogenous formula of size O ( d+r+1/r·s) for f. In particular, for any r ≤ log n / log log n, r ≤ log n, if there exists a polynomial size formula for f then there exists a polynomial size homogenous formula for f. This refutes a conjecture of Nisan and Wigderson [10] and shows that super-polynomial lower bounds for homogenous formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas. We show that for any n-variate set-multilinear polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a set-multilinear formula of size O ( (d+2)r·s ) for f. In particular, for any r ≤ log n / log log n, if there exists a polynomial size formula for f then there exists a polynomial size set-multilinear formula for f. This shows that super-polynomial lower bounds for set-multilinear formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas.

Original languageEnglish (US)
Title of host publicationSTOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing
Pages659-666
Number of pages8
DOIs
StatePublished - Jul 23 2010
Externally publishedYes
Event42nd ACM Symposium on Theory of Computing, STOC 2010 - Cambridge, MA, United States
Duration: Jun 5 2010Jun 8 2010

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Other

Other42nd ACM Symposium on Theory of Computing, STOC 2010
CountryUnited States
CityCambridge, MA
Period6/5/106/8/10

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • arithmetic circuits
  • homogenous circuits
  • lower bounds
  • multilinear circuits
  • tensor rank

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  • Cite this

    Raz, R. (2010). Tensor-rank and lower bounds for arithmetic formulas. In STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing (pp. 659-666). (Proceedings of the Annual ACM Symposium on Theory of Computing). https://doi.org/10.1145/1806689.1806780