Tensor-Rank and Lower Bounds for Arithmetic Formulas

Abstract

We show that any explicit example for a tensor A : [ n ] r → F with tensor-rank ≥ n rċ(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 (( d + r +1 r) ċ 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 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 ċ 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.