New Exponential Size Lower Bounds against Depth Four Circuits of Bounded Individual Degree
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Kayal, Saha and Tavenas [Theory of Computing, 2018] showed that for all large enough integers n and d such that d≥ω(logn), any syntactic depth four circuit of bounded individual degree δ = o(d) that computes the Iterated Matrix Multiplication polynomial (IMM_n,d) must have size n^Ω(√(d/δ)). Unfortunately, this bound deteriorates as the value of δ increases. Further, the bound is superpolynomial only when δ is o(d). It is natural to ask if the dependence on δ in the bound could be weakened. Towards this, in an earlier result [STACS, 2020], we showed that for all large enough integers n and d such that d = Θ(log^2n), any syntactic depth four circuit of bounded individual degree δ≤ n^0.2 that computes IMM_n,d must have size n^Ω(logn). In this paper, we make further progress by proving that for all large enough integers n and d, and absolute constants a and b such that ω(log^2n)≤ d≤ n^a, any syntactic depth four circuit of bounded individual degree δ≤ n^b that computes IMM_n,d must have size n^Ω(√(d)). Our bound is obtained by carefully adapting the proof of Kumar and Saraf [SIAM J. Computing, 2017] to the complexity measure introduced in our earlier work [STACS, 2020].
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