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

Neeraj Kayal, Nutan Limaye, Chandan Saha, Srikanth Srinivasan
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 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.  ... 
doi:10.1145/2591796.2591823 dblp:conf/stoc/KayalLSS14 fatcat:sjeo4teybjde7dar4qflhjzygq

Tensor-Rank and Lower Bounds for Arithmetic Formulas

Ran Raz
2013 Journal of the ACM  
This shows that strong enough 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.  ... 
doi:10.1145/2535928 fatcat:eozbyhiosrbsvohbpiih4tk3eq

Tensor-rank and lower bounds for arithmetic formulas

Ran Raz
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
This shows that strong enough 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.  ... 
doi:10.1145/1806689.1806780 dblp:conf/stoc/Raz10 fatcat:w7gp4d3obzbz7iblbub73676se

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

Neeraj Kayal, Nutan Limaye, Chandan Saha, Srikanth Srinivasan
2017 SIAM journal on computing (Print)  
We show here a 2 Ω( √ d·log N ) size 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  ... 
doi:10.1137/151002423 fatcat:r7hl2rzlbjgf3j43rxyyqvr2uq

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

Neeraj Kayal, Nutan Limaye, Chandan Saha, Srikanth Srinivasan
2014 2014 IEEE 55th Annual Symposium on Foundations of Computer Science  
We show here a 2 Ω( √ d·log N ) size 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  ... 
doi:10.1109/focs.2014.15 dblp:conf/focs/KayalLSS14 fatcat:r6yt25req5fs5fr4z5ylckpi3y

Non-Commutative Formulas and Frege Lower Bounds: a New Characterization of Propositional Proofs

Fu Li, Iddo Tzameret, Zhengyu Wang, Marc Herbstritt
2015 Computational Complexity Conference  
log(n)-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.  ... 
doi:10.4230/lipics.ccc.2015.412 dblp:conf/coco/LiTW15 fatcat:hhbf52sujbehvgumh62uolm4y4

Recent Progress on Arithmetic Circuit Lower Bounds

Ramprasad Saptharishi
2014 Bulletin of the European Association for Theoretical Computer Science  
We also look at some results on 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.  ... 
dblp:journals/eatcs/Saptharishi14 fatcat:yg7aifxgzrfohaota3azux7drm

Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin

Neeraj Kayal, Chandan Saha
2016 Computational Complexity  
Meanwhile, Nisan and Wigderson [18] had posed the problem of proving superpolynomial 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.  ... 
doi:10.1007/s00037-016-0132-0 fatcat:s6mnfmxlizhzve5aq4xwit2aum

Hardness vs Randomness for Bounded Depth Arithmetic Circuits

Chi-Ning Chou, Mrinal Kumar, Noam Solomon, Marc Herbstritt
2018 Computational Complexity Conference  
[SICOMP, 2009], where they showed that 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.  ... 
doi:10.4230/lipics.ccc.2018.13 dblp:conf/coco/ChouKS18 fatcat:agv7nwore5ggngdwc4vix7du5m

Arithmetic Circuit Lower Bounds via MaxRank [article]

Mrinal Kumar, Gaurav Maheshwari, Jayalal Sarma M.N
2013 arXiv   pre-print
We use our techniques to prove 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.  ... 
arXiv:1302.3308v1 fatcat:f5sbvgb7fzdxlcvf6b672rvlli

Arithmetic Circuit Lower Bounds via MaxRank [chapter]

Mrinal Kumar, Gaurav Maheshwari, Jayalal Sarma M.N.
2013 Lecture Notes in Computer Science  
Thus, our 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.  ...  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.  ... 
doi:10.1007/978-3-642-39206-1_56 fatcat:mj37mdyox5gofo4gckfvwo6apq

The Limits of Depth Reduction for Arithmetic Formulas: It's all about the top fan-in [article]

Mrinal Kumar, Shubhangi Saraf
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.  ... 
arXiv:1311.6716v1 fatcat:crbdfxqvtfcthfrhccgdekjzvm

Some Closure Results for Polynomial Factorization and Applications [article]

Chi-Ning Chou, Mrinal Kumar, Noam Solomon
2018 arXiv   pre-print
This is incomparable to a beautiful result of Dvir et al., where they showed that 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.  ... 
arXiv:1803.05933v1 fatcat:z3nw4d3f4jcjhowfwappzrfeai

Barriers for Rank Methods in Arithmetic Complexity [article]

Klim Efremenko and Ankit Garg and Rafael Oliveira and Avi Wigderson
2017 arXiv   pre-print
Despite many successes and rapid progress, however, challenges like proving 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.  ... 
arXiv:1710.09502v1 fatcat:q77slcdltfa7hh5awrcko2wmbq

A super-polynomial lower bound for regular arithmetic formulas

Neeraj Kayal, Chandan Saha, Ramprasad Saptharishi
2014 Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14  
We consider 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.  ... 
doi:10.1145/2591796.2591847 dblp:conf/stoc/KayalSS14 fatcat:azd3igvcyzhtpo5psasa6z3d6a
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