Arithmetic Circuit Lower Bounds via MaxRank

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 in [9] when d = ω(1). • There is an explicit polynomial on n variables and degree at most n2 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 2Ω(n). This generalizes the lower bounds against diagonal circuits proved in [14]. 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 [11]. • 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 [7]. Part of this work was done while the first two authors were undergraduate students at IIT Madras Department of Computer Science, Rutgers University, Piscataway, NJ 08855, USA. Email : [email protected] Department of Computer Science and Engineering, Indian Institute of Technology Madras, Chennai 600036. Email:[email protected] Department of Computer Science and Engineering, Indian Institute of Technology Madras, Chennai 600036. Email :[email protected]