Tensor-Rank and Lower Bounds for Arithmetic Formulas

We show that any explicit example for a tensor A : [n](r)-> F with tensor-rank >= n(r center dot(1-o(1))), where r = r( n) <= log n/log logn 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(((d+r+1)(r))center dot s) for f. In particular, for any r <= O( logn), 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 super-polynomial 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)center dot s) for f. In particular, for any r <= O(logn/loglogn), 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.