Improved Lower Bound for Multi-R-IC Depth Four Circuits as a Function of the Number of Input Variables

Hegde, Sumant

doi:10.16943/ptinsa/2017/49224

Cited by 2 publications

(2 citation statements)

References 13 publications

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“…If we can show superpolynomial size lower bounds against multi-r-ic depth four circuits for r = n c for any constant c, then we indeed have superpolynomial circuit size lower bounds against depth four circuits. We believe that by building on the work of [20,11], Theorem 1 is a step towards that direction.…”

Section: Motivation For This Workmentioning

confidence: 99%

On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

ChillaraSuryajith

2021

ACM Trans. Comput. Theory

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In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

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Section: Motivation For This Workmentioning

confidence: 99%

On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

ChillaraSuryajith

2021

ACM Trans. Comput. Theory

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“…Towards this, in the same paper, they showed a lower bound of 2 Ω( √ N) for a N-variate polynomial that is not multilinear. Hegde and Saha [HS17] proved a bound of 2…”

Section: Introductionmentioning

confidence: 99%

New Exponential Size Lower Bounds against Depth Four Circuits of Bounded Individual Degree

Chillara

2020

Preprint

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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(log n), 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 2 n), any syntactic depth four circuit of bounded individual degree δ n 0.2 that computes IMM n,d must have size n Ω (log n) .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 2 n) 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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Improved Lower Bound for Multi-R-IC Depth Four Circuits as a Function of the Number of Input Variables

Cited by 2 publications

References 13 publications

On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

New Exponential Size Lower Bounds against Depth Four Circuits of Bounded Individual Degree

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