Arithmetic Circuit Lower Bounds via Maximum-Rank of Partial Derivative Matrices

Abstract

We introduce the polynomial coefficient matrix and identify the 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 × n requires Ω( n d − 1 /2 d ) size. This improves the lower bounds in Nisan and Wigderson [1995] for d = ω(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 of product dimension at most n /10 (dimension of the space of affine forms feeding into each product gate) requires size 2 Ω( n ) . This generalizes the lower bounds against diagonal circuits proved in Saxena [2008]. Diagonal circuits are of product dimension 1. —We prove a n Ω(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 [Raz 2006]. —We prove a 2 Ω( 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 [Jansen 2008].