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Super-polynomial lower bounds for depth-4 homogeneous arithmetic formulas

2014
*
Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14
*

dq 2 Prior to this work, Nisan [17] showed an exponential

doi:10.1145/2591796.2591823
dblp:conf/stoc/KayalLSS14
fatcat:sjeo4teybjde7dar4qflhjzygq
*lower**bound**for*noncommutative*arithmetic**formulas*3 In fact, a very recent work of [14] shows a*super*-*polynomial*separation between general*formulas*... We show that any*depth*-*4**homogeneous**arithmetic**formula*computing the Iterated Matrix Multiplication*polynomial*IMM n,d -the p1, 1q-th entry of the product of d generic nˆn matrices -has size n Ωplog nq ... d log N q size*lower**bound**for**depth*-*4**homogeneous**formulas*1 , computing a degree-d, N -variate*polynomial*(in a*polynomial*family), implies a*super*-*polynomial**lower**bound**for*general circuits. ...##
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Tensor-Rank and Lower Bounds for Arithmetic Formulas

2013
*
Journal of the ACM
*

This shows that strong enough

doi:10.1145/2535928
fatcat:eozbyhiosrbsvohbpiih4tk3eq
*lower**bounds**for*the size of*arithmetic**formulas*of*depth*3 imply*super*-*polynomial**lower**bounds**for*the size of general*arithmetic**formulas*. ... This refutes a conjecture of Nisan and Wigderson [NW95] and shows that*super*-*polynomial**lower**bounds**for**homogeneous**formulas**for**polynomials*of small degree imply*super*-*polynomial**lower**bounds**for*general ... Thus,*super*-*polynomial**lower**bounds**for**homogeneous**formulas**for**polynomials*of degree up to O(log n) imply*super*-*polynomial**lower**bounds**for*general*arithmetic**formulas*. ...##
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Tensor-rank and lower bounds for arithmetic formulas

2010
*
Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10
*

This shows that strong enough

doi:10.1145/1806689.1806780
dblp:conf/stoc/Raz10
fatcat:w7gp4d3obzbz7iblbub73676se
*lower**bounds**for*the size of*arithmetic**formulas*of*depth*3 imply*super*-*polynomial**lower**bounds**for*the size of general*arithmetic**formulas*. ... This refutes a conjecture of Nisan and Wigderson [NW95] and shows that*super*-*polynomial**lower**bounds**for**homogeneous**formulas**for**polynomials*of small degree imply*super*-*polynomial**lower**bounds**for*general ... Thus,*super*-*polynomial**lower**bounds**for**homogeneous**formulas**for**polynomials*of degree up to O(log n) imply*super*-*polynomial**lower**bounds**for*general*arithmetic**formulas*. ...##
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An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

2017
*
SIAM journal on computing (Print)
*

We show here a 2 Ω( √ d·log N ) size

doi:10.1137/151002423
fatcat:r7hl2rzlbjgf3j43rxyyqvr2uq
*lower**bound**for**homogeneous**depth*four*arithmetic**formulas*. ... Our work builds on the recent*lower**bound*results [Kay12, GKKS13a, KSS14, FLMS14, KS14] and yields an improved quantitative*bound*as compared to the quasi-*polynomial**lower**bound*of [KLSS14] and the N Ω ... Acknowledgements NK would like to thank Avi Wigderson*for*many helpful discussions including pointing out the use of random restrictions to reduce a general*homogeneous*ΣΠΣΠ circuit into one with low support ...##
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An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

2014
*
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
*

We show here a 2 Ω( √ d·log N ) size

doi:10.1109/focs.2014.15
dblp:conf/focs/KayalLSS14
fatcat:r6yt25req5fs5fr4z5ylckpi3y
*lower**bound**for**homogeneous**depth*four*arithmetic**formulas*. ... Our work builds on the recent*lower**bound*results [Kay12, GKKS13a, KSS14, FLMS14, KS14] and yields an improved quantitative*bound*as compared to the quasi-*polynomial**lower**bound*of [KLSS14] and the N Ω ... Acknowledgements NK would like to thank Avi Wigderson*for*many helpful discussions including pointing out the use of random restrictions to reduce a general*homogeneous*ΣΠΣΠ circuit into one with low support ...##
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Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

2015
*
Computational Complexity Conference
*

log(n)-

doi:10.4230/lipics.ccc.2015.412
dblp:conf/coco/LiTW15
fatcat:hhbf52sujbehvgumh62uolm4y4
*depth*circuits denoted NC 1 (equivalently, of*polynomial*-size*formulas*[32]), considered to be a strong computational model*for*which no (explicit)*super*-*polynomial**lower**bounds*are currently known ... Fourth, that proving*super*-*polynomial**lower**bounds*on Frege proofs seems to a certain extent out of reach of current techniques. ... We thank Joshua Grochow*for*helpful comments. ...##
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Recent Progress on Arithmetic Circuit Lower Bounds

2014
*
Bulletin of the European Association for Theoretical Computer Science
*

We also look at some results on

dblp:journals/eatcs/Saptharishi14
fatcat:yg7aifxgzrfohaota3azux7drm
*depth*reduction and some approaches aimed towards proving superpolynomial*lower**bounds**for**homogeneous**formulas*. ... Several of these results are centered*homogeneous**depth*four circuits, and come tantalizingly close to separating the algebraic analogue of P from the algebraic analogue of NP. ... Any*super*-*polynomial**lower**bound**for*the class of O(log d)*depth*circuits automatically yields a*super*-*polynomial**lower**bound**for*general circuits. ...##
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Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin

2016
*
Computational Complexity
*

Meanwhile, Nisan and Wigderson [18] had posed the problem of proving superpolynomial

doi:10.1007/s00037-016-0132-0
fatcat:s6mnfmxlizhzve5aq4xwit2aum
*lower**bounds**for**homogeneous**depth*five*arithmetic*circuits. ... We resolve this problem by proving a N Ω( d τ )*lower**bound**for*(nonhomogeneous)*depth*three*arithmetic*circuits with bottom fanin at most τ computing an explicit N -variate*polynomial*of degree d over ... The authors would like to thank Amit Chakrabarti, Mrinal Kumar, Satya Lokam and Ramprasad Saptharishi*for*helpful discussions. ...##
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Hardness vs Randomness for Bounded Depth Arithmetic Circuits

2018
*
Computational Complexity Conference
*

[SICOMP, 2009], where they showed that

doi:10.4230/lipics.ccc.2018.13
dblp:conf/coco/ChouKS18
fatcat:agv7nwore5ggngdwc4vix7du5m
*super*-*polynomial**lower**bounds**for**depth*∆ circuits*for*any explicit family of*polynomials*(of potentially high degree) implies sub-exponential time deterministic ... In this paper, we study the question of hardness-randomness tradeoffs*for**bounded**depth**arithmetic*circuits. ... Acknowledgements We are thankful to Rafael Oliveira and Guy Moshkovitz*for*helpful discussions. ...##
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Arithmetic Circuit Lower Bounds via MaxRank
[article]

2013
*
arXiv
*
pre-print

We use our techniques to prove

arXiv:1302.3308v1
fatcat:f5sbvgb7fzdxlcvf6b672rvlli
*super*-*polynomial**lower**bounds*against several classes of non-multilinear*arithmetic*circuits. ... Thus, our result extends the known*super*-*polynomial**lower**bounds*on the size of multilinear*formulas*by Raz(2006). ... Acknowledgements The authors thank the anonymous referees*for*suggesting a simplified view of the proof*for*Lemma 4.2. ...##
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Arithmetic Circuit Lower Bounds via MaxRank
[chapter]

2013
*
Lecture Notes in Computer Science
*

Thus, our result extends the known

doi:10.1007/978-3-642-39206-1_56
fatcat:mj37mdyox5gofo4gckfvwo6apq
*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. ... We use our techniques to prove*super*-*polynomial**lower**bounds*against several classes of non-multilinear*arithmetic*circuits. ... Acknowledgements The authors thank the anonymous referees*for*suggesting a simplified view of the proof*for*Lemma 4.2. ...##
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The Limits of Depth Reduction for Arithmetic Formulas: It's all about the top fan-in
[article]

2013
*
arXiv
*
pre-print

*lower*

*bounds*

*for*regular

*arithmetic*

*formulas*via an improved

*depth*reduction

*for*these

*formulas*. ... We do it via studying the class of

*depth*

*4*

*homogeneous*

*arithmetic*circuits. We show: (1) the first superpolynomial

*lower*

*bounds*

*for*the class of

*homogeneous*

*depth*

*4*circuits with top fan-in o( n). ... We would also like to thank Swastik Kopparty and Avi Wigderson

*for*many helpful comments on an earlier version of this paper. ...

##
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Some Closure Results for Polynomial Factorization and Applications
[article]

2018
*
arXiv
*
pre-print

This is incomparable to a beautiful result of Dvir et al., where they showed that

arXiv:1803.05933v1
fatcat:z3nw4d3f4jcjhowfwappzrfeai
*super*-*polynomial**lower**bounds**for*constant*depth**arithmetic*circuits*for*any explicit family of*polynomials*(of potentially ...*for**polynomial*identity testing (PIT)*for**bounded**depth**arithmetic*circuits. ... Acknowledgment We thank Rafael Oliveira*for*making us aware of the question about the complexity of factors of*polynomials*in VNP, and Guy Moshkovitz*for*helpful discussions. ...##
###
Barriers for Rank Methods in Arithmetic Complexity
[article]

2017
*
arXiv
*
pre-print

Despite many successes and rapid progress, however, challenges like proving

arXiv:1710.09502v1
fatcat:q77slcdltfa7hh5awrcko2wmbq
*super*-*polynomial**lower**bounds*on circuit or*formula*size*for*explicit*polynomials*, or*super*-linear*lower**bounds*on explicit 3 ... (In particular, they cannot prove*super*-linear, indeed even >8n tensor rank*lower**bounds**for*any 3-dimensional tensors.) 2. ... The works [GKKS14, KLSS14, FLMS15, KS14, KS15] use rank methods to prove matching*lower**bounds*of exp(Õ( √ n))*for**homogeneous**depth*-*4**formulas*. ...##
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A super-polynomial lower bound for regular arithmetic formulas

2014
*
Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14
*

We consider

doi:10.1145/2591796.2591847
dblp:conf/stoc/KayalSS14
fatcat:azd3igvcyzhtpo5psasa6z3d6a
*arithmetic**formulas*consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. ... We refer to such*formulas*as ΣΠ [b] ΣΠ [a]*formulas*. We show that there exists an n 2 -variate*polynomial*of degree n in VNP such that any ΣΠ [O( ... and the choice of a candidate hard*polynomial*. ...
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