We introduce the polynomial coefficient matrix and identify maximum rank of
this matrix under variable substitution as a complexity measure for
multivariate polynomials. We use our techniques to prove super-polynomial lower
bounds against several classes of non-multilinear arithmetic circuits. In
particular, we obtain the following results :
As our main result, we prove that any homogeneous depth-3 circuit for
computing the product of d matrices of dimension n×n requires
Ω(nd−1/2d) size. This improves the lower bounds by Nisan and
Wigderson(1995) when d=ω(1).
There is an explicit polynomial on n variables and degree at most
2n for which any depth-3 circuit C of product dimension at most
10n (dimension of the space of affine forms feeding into each
product gate) requires size 2Ω(n). This generalizes the lower bounds
against diagonal circuits proved by Saxena(2007). Diagonal circuits are of
product dimension 1.
We prove a nΩ(logn) lower bound on the size of product-sparse
formulas. By definition, any multilinear formula is a product-sparse formula.
Thus, our result extends the known super-polynomial lower bounds on the size of
multilinear formulas by Raz(2006).
We prove a 2Ω(n) lower bound on the size of partitioned arithmetic
branching programs. This result extends the known exponential lower bound on
the size of ordered arithmetic branching programs given by Jansen(2008).Comment: 22 page