Arithmetic Circuit Lower Bounds via Max Rank

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 first main result, we prove that any homogeneous depth-3 circuit for computing the product of d matrices of dimension n x n requires Omega(n(d-1)/2(d)) size. This improves the lower bounds in [9] for d = omega(1). As our second main result, we show that there is an explicit polynomial on n variables and degree at most n/2 for which any depth-3 circuit C of product dimension at most n/10 (dimension of the space of affine forms feeding into each product gate) requires size 2(Omega(n)). This generalizes the lower bounds against diagonal circuits proved in [14]. Diagonal circuits are of product dimension 1. We prove a n(Omega(log n)) lower bound on the size of product-sparse formulas. By definition, any multilinear formula is a product-sparse formula. Thus, this result extends the known super-polynomial lower bounds on the size of multilinear formulas [11]. We prove a 2(Omega(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 [7].