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

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Published:25 May 2016Publication History
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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(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 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 2Ω(n). This generalizes the lower bounds against diagonal circuits proved in Saxena [2008]. Diagonal circuits are of product dimension 1.

—We prove a nΩ(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 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 [Jansen 2008].

References

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 8, Issue 3
        May 2016
        105 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/2936308
        Issue’s Table of Contents

        Copyright © 2016 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 25 May 2016
        • Revised: 1 December 2015
        • Accepted: 1 December 2015
        • Received: 1 March 2015
        Published in toct Volume 8, Issue 3

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