Tensor-rank and lower bounds for arithmetic formulas

• Authors: Ran Raz
• Venue: Proceedings of the forty-second ACM symposium on Theory of computing
• Year: 2010

We show that any explicit example for a tensor A:[n]r - 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 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.