Ran Raz

Strassen V.

English


            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.
          </jats:p

Crossref

Journal of the ACM

Tensor-Rank and Lower Bounds for Arithmetic Formulas

http://www.wisdom.weizmann.ac.il/~ranraz/publications/Ptensor.pdf

Tensor-Rank and Lower Bounds for Arithmetic Formulas

Abstract

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