A Depth-Five Lower Bound For Iterated Matrix Multiplication
CCC '15: Proceedings of the 30th Conference on Computational Complexity(2015)
摘要
We prove that certain instances of the iterated matrix multiplication (IMM) family of polynomials with N variables and degree n require N-Omega(root n) gates when expressed as a homogeneous depth-five Sigma Pi Sigma Pi Sigma arithmetic circuit with the bottom fan-in bounded by N1/2-epsilon. By a depth-reduction result of Tavenas, this size lower bound is optimal and can be achieved by the weaker class of homogeneous depth-four Sigma Pi Sigma Pi circuits.Our result extends a recent result of Kumar and Saraf, who gave the same N-Omega(root n) lower bound for homogeneous depth-four Sigma Pi Sigma Pi circuits computing IMM. It is analogous to a recent result of Kayal and Saha, who gave the same lower bound for homogeneous Sigma Pi Sigma Pi Sigma circuits (over characteristic zero) with bottom fan-in at most N1-epsilon, for the harder problem of computing certain polynomials defined by Nisan-Wigderson designs.
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关键词
arithmetic circuits,iterated matrix multiplication,depth five circuits,lower bound
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