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Arithmetic circuit lower bounds via MaxRank
Gaurav Maheshwari,
Published in
2013
Volume: 7965 LNCS
   
Issue: PART 1
Pages: 661 - 672
Abstract
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 Ω(nd-1/2d) size. This improves the lower bounds in [9] 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 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(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 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 [7]. © 2013 Springer-Verlag.
About the journal
JournalLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
ISSN03029743
Open AccessYes
Concepts (13)
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    Arithmetic circuit
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    Branching programs
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    Complexity measures
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    DEPTH-3 CIRCUITS
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    MAXIMUM RANK
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    Multivariate polynomial
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    Polynomial coefficients
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    VARIABLE SUBSTITUTION
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    AUTOMATA THEORY
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    Integrating circuits
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    Logic circuits
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    Polynomials
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    Matrix algebra