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Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas
[article]

2014
*
arXiv
*
pre-print

In this paper we give

arXiv:1411.7492v1
fatcat:tktkstrumbbhpf6lvymrq6mtda
*subexponential**size**hitting**sets**for**bounded**depth**multilinear*arithmetic*formulas*. ...*For**depth*-3*multilinear**formulas*, of*size*(n^δ), we give a*hitting**set*of*size*(Õ(n^2/3 + 2δ/3)). ... Acknowledgments The authors would like to thank Zeev Dvir and Avi Wigderson*for*helpful discussions during the course of this work. ...##
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Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

2016
*
Computational Complexity
*

In this paper we give

doi:10.1007/s00037-016-0131-1
fatcat:m55ugwfj6jbm5gqqirfxszw5le
*subexponential**size**hitting**sets**for**bounded**depth**multilinear*arithmetic*formulas*. ...*For**depth*-3*multilinear**formulas*, of*size*exp(n δ ), we give a*hitting**set*of*size*exp Õ n 2/3+2δ/3 . ... The authors would like to thank Zeev Dvir and Avi Wigderson*for*helpful discussions during the course of this work. ...##
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Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in

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

Using ideas from previous works on identity testing of sums of read-once

doi:10.1145/1806689.1806779
dblp:conf/stoc/KarninMSV10
fatcat:sklwzqypifbuhmw6snvzaanpka
*formulas*and of*depth*-3*multilinear*circuits, we are able to exploit this structure and obtain an identity testing algorithm*for*... More precisely, he shows that the black-box derandomization of PIT implies that an explicit*multilinear*polynomial has no*subexponential**size*arithmetic circuit. ... The best algorithms today are*for*sums of read-once*formulas*[SV09] and*for**set*-*multilinear**depth*3*formulas*(non black-box) [RS05] . ...##
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Deterministic Identity Testing of Depth-4 Multilinear Circuits with Bounded Top Fan-in

2013
*
SIAM journal on computing (Print)
*

Using ideas from previous works on identity testing of sums of read-once

doi:10.1137/110824516
fatcat:7kwyeftqrbbcho5vpw4hu6syze
*formulas*and of*depth*-3*multilinear*circuits, we are able to exploit this structure and obtain an identity testing algorithm*for*... More precisely, he shows that the black-box derandomization of PIT implies that an explicit*multilinear*polynomial has no*subexponential**size*arithmetic circuit. ... The best algorithms today are*for*sums of read-once*formulas*[SV09] and*for**set*-*multilinear**depth*3*formulas*(non black-box) [RS05] . ...##
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Proving Lower Bounds Via Pseudo-random Generators
[chapter]

2005
*
Lecture Notes in Computer Science
*

In this paper, we formalize two stepwise approaches, based on pseudo-random generators,

doi:10.1007/11590156_6
fatcat:utnfe6pyvfczxkyaqpy6gchrdu
*for*proving P = NP and its arithmetic analog: Permanent requires superpolynomial*sized*arithmetic circuits. ... This implies that f B,d cannot be a*hitting**set*generator*for*all d (because SAC 1 circuits can be transformed to*subexponential**sized*constant*depth*circuits as observed earlier). ... We conjecture that above lemma holds*for*all*depths*: Conjecture. Function f A,d is a*hitting**set*generator against*depth*d,*size*n arithmetic circuits*for*every d > 0. ...##
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Deterministic polynomial identity tests for multilinear bounded-read formulae

2015
*
Computational Complexity
*

Our results encompass recent deterministic identity tests

doi:10.1007/s00037-015-0097-4
fatcat:5z7zqjuifjetnltve7qeugx5se
*for*sums of a constant number of read-once*formulae*, and*for**multilinear**depth*-four*formulae*. ... Before our work no*subexponential*-time deterministic algorithm was known*for*this class of*formulae*. ... Acknowledgements The authors would like to thank Amir Shpilka*for*bringing them in touch with each other, and the anonymous reviewers*for*their comments. ...##
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Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds
[article]

2018
*
arXiv
*
pre-print

That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-

arXiv:1701.05328v2
fatcat:ueg2mlf5l5febo7hojpqri67dq
*size*circuits is a*hitting**set**for*the class of poly(N)-degree poly(N)-*size*circuits. ... Nevertheless, our succinct*hitting**sets*have relevance to the GCT program as they imply lower*bounds**for*the complexity of the defining equations of polynomials computed by small circuits. ... We also thank the anonymous reviewers*for*their careful reading of this paper and*for*many useful comments. ...##
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Succinct hitting sets and barriers to proving algebraic circuits lower bounds

2017
*
Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017
*

That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-

doi:10.1145/3055399.3055496
dblp:conf/stoc/ForbesSV17
fatcat:amglevgewvdlpozsxppzhdqify
*size*circuits is a*hitting**set**for*the class of poly(N)-degree poly(N)-*size*circuits. ... Nevertheless, our succinct*hitting**sets*have relevance to the GCT program as they imply lower*bounds**for*the complexity of the defining equations of polynomials computed by small circuits. ... Acknowledgements We thank Scott Aaronson, Andy Drucker, Josh Grochow, Mrinal Kumar, Shubhangi Saraf and Dor Minzer*for*useful conversations regarding this work. ...##
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Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs

2018
*
ACM Transactions on Computation Theory
*

*hitting*of [22] was quasipolynomial

*sized*

*for*

*bounded*individual degree, but the subsequent

*hitting*

*set*of [2] is quasipolynomial

*sized*

*for*any d = poly(n)). ... -4 circuits, that maps

*subexponential*

*size*to

*subexponential*

*size*[3] . ... Shpilka and Volkovich [61] constructed quasipolynomial-

*size*

*hitting*

*set*

*for*read-once

*formulas*, and Anderson, van Melkebeek and Volkovich [8] extended this result to

*multilinear*read-k

*formulas*and ...

##
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Quasi-polynomial hitting-set for set-depth-Δ formulas

2013
*
Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13
*

In this work, we give a

doi:10.1145/2488608.2488649
dblp:conf/stoc/AgrawalSS13
fatcat:5ajt6l7etneghgq2kqxyssukmu
*hitting*-*set*generator*for**set*-*depth*-∆*formulas*(over any field) with running time polynomial in exp((∆ 2 log s) ∆−1 ), where s is the*size**bound*on the input*set*-*depth*-∆*formula*... with*bounded*top fanin [ASSS12, SS11], •*depth*-4 (*bounded**depth*) constant-occur*formulas*[ASSS12], and a quasi-polynomial time*hitting*-*set*generator*for*•*multilinear*constant-read*formulas*[AvMV11], ... Acknowledgments This work was initiated when MA and NS visited Max Planck Institute*for*Informatics, and would like to thank the institute*for*its generous hospitality. ...##
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Quasi-polynomial Hitting-set for Set-depth-Delta Formulas
[article]

2012
*
arXiv
*
pre-print

In this work, we give a

arXiv:1209.2333v1
fatcat:52f2hvrvfbgh5g35c5yic7jsei
*hitting*-*set*generator*for**set*-*depth*-Delta*formulas*(over any field) with running time polynomial in exp((Delta^2 log s)^Delta - 1), where s is the*size**bound*on the input*set*-*depth*-Delta ... Previously, the very special case of Delta=3 (also known as*set*-*multilinear**depth*-3 circuits) had no known sub-exponential time*hitting*-*set*generator. ... Acknowledgments This work was initiated when MA and NS visited Max Planck Institute*for*Informatics, and would like to thank the institute*for*its generous hospitality. ...##
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Hardness-Randomness Tradeoffs for Algebraic Computation

2019
*
Bulletin of the European Association for Theoretical Computer Science
*

The interplay between the question of proving lower

dblp:journals/eatcs/0001S19
fatcat:kwvkfr7ghrd47l4ytdord7jaii
*bounds*and that of derandomization, in various*settings*, is one of the central themes in complexity theory. ... We also talk about an application of this machinery to the phenomenon of bootstrapping*for*polynomial identity testing and mention some open problems. ... Acknowledgements We are thankful to Chi-Ning Chou, Zeyu Guo, Swastik Kopparty, Noam Solomon, Anamay Tengse and Ben Lee Volk*for*insightful discussions on many of the themes addressed in this survey. ...##
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Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits

2018
*
Theory of Computing
*

polynomial identity testing problem, that is, to the existence of a

doi:10.4086/toc.2018.v014a018
dblp:journals/toc/ForbesSV18
fatcat:xca443ndhzfxfjnclcmx6y57py
*hitting**set**for*the class of poly(N)-degree poly(N)-*size*circuits which consists of coefficient vectors of polynomials of polylog(N) ... However, Razborov and Rudich [86] showed that (possibly after a small modification) most circuit lower*bounds*(such as those*for*constant-*depth*circuits ([9, 41, 106, 53, 85, 98])) yield large and constructive ... We also thank the anonymous reviewers*for*their careful reading of this paper and*for*many useful comments. ...##
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Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits

2015
*
SIAM journal on computing (Print)
*

The 1-distance strictly subsumes the

doi:10.1137/140975103
fatcat:bcs2sbyu2nchrlzzympgl7eowu
*set*-*multilinear*model, while n-distance captures general*multilinear**depth*-3. We design a*hitting*-*set*in time (nk) O(∆ log n)*for*∆-distance. ... Its restriction to*multilinearity*has known exponential lower*bounds*but no nontrivial blackbox identity tests. In this paper, we take a step towards designing such*hitting*-*sets*. ... On the other hand, there are exponential lower*bounds**for**depth*-3*multilinear*circuits [RY09] . ...##
###
Hitting-sets for ROABP and Sum of Set-Multilinear circuits
[article]

2014
*
arXiv
*
pre-print

We give the first

arXiv:1406.7535v1
fatcat:qlqaxcqvkzasnpd7rdd62azbba
*subexponential*whitebox PIT*for*the sum of constantly many*set*-*multilinear**depth*-3 circuits. To achieve this, we define notions of distance and base*sets*. ... Distance,*for*a*multilinear**depth*-3 circuit, measures how far are the partitions from a mere refinement. We design a*hitting*-*set*in time n^O(d n)*for*d-distance. ... On the other hand, there are exponential lower*bounds**for**depth*-3*multilinear*circuits [RY09] . ...
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