### Abstract

We show that any explicit example for a tensor A : [ n]^{r}\to F with tensor-rank /ge n^{r·(1-o(1))}, where r = r(n) ≤ log n/ log log n is super-constant, implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over 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 homogeneous polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a homogeneous formula of size O (( r^{d+r+1} ) · s )for f . In particular, for any r ≤ O(log n), if there exists a polynomial size formula for f then there exists a polynomial size homogeneous formula for f . This refutes a conjecture of Nisan and Wigderson [1996] and shows that superpolynomial lower bounds for homogeneous 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 ≤ O(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 language | English (US) |
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Article number | 40 |

Journal | Journal of the ACM |

Volume | 60 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2013 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence

### Keywords

- Arithmetic circuits
- Homogenous circuits
- Lower bounds
- Multilinear circuits
- Tensor rank