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Arithmetic circuit lower bounds via maximum-rank of partial derivative matrices
Gaurav Maheshwari,
Published in Association for Computing Machinery
2016
Volume: 8

Issue: 3
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 superpolynomial 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 (nd −1/2d) 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 most2 n for which any depth-3 circuit of product dimension at most10 n (dimension of the space of affine forms feeding into each product gate) requires size 2(n). This generalizes the lower bounds against diagonal circuits proved in Saxena [2008]. Diagonal circuits are of product dimension 1. —We prove a n(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(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]. c 2016 ACM