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Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits
[article]

2011
*
arXiv
*
pre-print

In particular, we show that polynomial time

arXiv:1111.0582v1
fatcat:4o7q7jv6jjaghhs2ic5uic3ucu
*hitting*-*set*generators*for*identity testing of the two seemingly different*and*well studied models -*depth*-*3**circuits*with*bounded*top fanin,*and*constant-*depth*... By exploiting the*Jacobian*, we design the first efficient*hitting*-*set*generators*for*broad generalizations of the above-mentioned models, namely: (1)*depth*-*3*(Sigma-Pi-Sigma)*circuits*with constant*transcendence*... A.2*Hitting*-*set**for*constant*transcendence**degree**depth*-*3**circuits*Lemma 3.1 (restated). ...##
###
Jacobian Hits Circuits: Hitting Sets, Lower Bounds for Depth-$D$ Occur-$k$ Formulas and Depth-3 Transcendence Degree-$k$ Circuits

2016
*
SIAM journal on computing (Print)
*

In particular, we show that polynomial (in the size of the

doi:10.1137/130910725
fatcat:hpsqcu3x7japnnhpwplplox6xy
*circuit*) time*hitting*-*set*generators*for*identity testing of the two seemingly different*and*well studied models-*depth*-*3**circuits*with*bounded*... By exploiting the*Jacobian*, we design the first efficient*hitting*-*set*generators*for*broad generalizations of the above-mentioned models, namely, (a)*depth*-*3*(ΣΠΣ)*circuits*with constant*transcendence*... This work was done when MA, RS,*and*NS visited the Max-Planck Institute*for*Informatics (Saarbrücken)*and*would like to thank the institute*for*its generous hospitality. ...##
###
Jacobian hits circuits

2012
*
Proceedings of the 44th symposium on Theory of Computing - STOC '12
*

In particular, we show that polynomial time

doi:10.1145/2213977.2214033
dblp:conf/stoc/AgrawalSSS12
fatcat:micc6dhlqrfztncc35ddkuv3qq
*hitting*-*set*generators*for*identity testing of the two seemingly different*and*well studied models -*depth*-*3**circuits*with*bounded*top fanin,*and*constant-*depth*... By exploiting the*Jacobian*, we design the first efficient*hitting*-*set*generators*for*broad generalizations of the above-mentioned models, namely: ... A.2*Hitting*-*set**for*constant*transcendence**degree**depth*-*3**circuits*Lemma 3.1 (restated). ...##
###
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)-

arXiv:1701.05328v2
fatcat:ueg2mlf5l5febo7hojpqri67dq
*degree*polylog(N)-size*circuits*is a*hitting**set**for*the class of poly(N)-*degree*poly(N)-size*circuits*. ... Just as with the natural proofs notion of Razborov*and*Rudich*for*boolean*circuit**lower**bounds*, our notion of algebraically natural*lower**bounds*captures nearly all*lower**bound*techniques known. ... We also thank the anonymous reviewers*for*their careful reading of this paper*and**for*many useful comments. ...##
###
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)-

doi:10.1145/3055399.3055496
dblp:conf/stoc/ForbesSV17
fatcat:amglevgewvdlpozsxppzhdqify
*degree*polylog(N)-size*circuits*is a*hitting**set**for*the class of poly(N)-*degree*poly(N)-size*circuits*. ... Just as with the natural proofs notion of Razborov*and*Rudich [RR97]*for*boolean*circuit**lower**bounds*, our notion of algebraically natural*lower**bounds*captures nearly all*lower**bound*techniques known. ... Acknowledgements We thank Scott Aaronson, Andy Drucker, Josh Grochow, Mrinal Kumar, Shubhangi Saraf*and*Dor Minzer*for*useful conversations regarding this work. ...##
###
Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits

2018
*
Theory of Computing
*

However, Razborov

doi:10.4086/toc.2018.v014a018
dblp:journals/toc/ForbesSV18
fatcat:xca443ndhzfxfjnclcmx6y57py
*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 ... polynomial identity testing problem, that is, to the existence of a*hitting**set**for*the class of poly(N)-*degree*poly(N)-size*circuits*which consists of coefficient vectors of polynomials of polylog(N) ... We also thank the anonymous reviewers*for*their careful reading of this paper*and**for*many useful comments. ...##
###
Progress on Polynomial Identity Testing - II
[article]

2014
*
arXiv
*
pre-print

*Bounded*fanin

*depth*-

*3*blackbox PIT Let C = i∈[

*k*] T i be a

*depth*-

*3*

*circuit*. When

*k*is constant, C is naturally called

*bounded*fanin

*depth*-

*3*. ... [ASS13] gave the first quasi-poly-time

*hitting*-

*set*

*for*

*set*-multilinear

*depth*-

*3*(

*and*extensions to constant-

*depth*, non-multilinear versions). ...

##
###
Progress on Polynomial Identity Testing-II
[chapter]

2014
*
Perspectives in Computational Complexity
*

*Bounded*fanin

*depth*-

*3*blackbox PIT Let C = i∈[

*k*] T i be a

*depth*-

*3*

*circuit*. When

*k*is constant, C is naturally called

*bounded*fanin

*depth*-

*3*. ... Lemma

*3*. 4 ( 34

*Jacobian*criterion). Let f ⊂ F[x] be a finite

*set*of polynomials of

*degree*at most

*d*,

*and*trdeg(f ) ≤ r. ...

##
###
Complexity in Ideals of Polynomials: Questions on Algebraic Complexity of Circuits and Proofs

2020
*
Bulletin of the European Association for Theoretical Computer Science
*

*For*a few examples of speci c ideals of interest coming from

*circuit*

*lower*

*bounds*or proof complexity, some

*lower*

*bounds*on polynomials in these ideals are known using succinct

*hitting*

*sets*(Forbes-Shpilka-Volk ... Such ideals

*and*their cosets arise naturally in algebraic

*circuit*

*lower*

*bounds*, geometric complexity theory,

*and*algebraic proof complexity. ... Green,

*and*R. Santhanam

*for*the invitation

*and*support. Thanks to Peter Bürgisser

*and*Mrinal Kumar

*for*providing useful references on factoring polynomials. ...

##
###
Hitting Sets for Orbits of Circuit Classes and Polynomial Families

2021

This implies a quasi-polynomial time

doi:10.4230/lipics.approx/random.2021.50
fatcat:sphejkalhna7jmg3p5gsi6ggia
*hitting**set**for*the orbits of the family {IMM_{*3*,*d*}}_{*d*∈ ℕ}, which is complete*for*arithmetic*formulas*.*3*) Polynomials computable by constant-*depth*, constant-*occur*... This implies quasi-polynomial time*hitting**sets**for*the orbits of multilinear*depth*-4*circuits*with constant top fan-in,*and*also polynomial-time*hitting**sets**for*the orbits of the power symmetric*and*...*for*both*depth*-*3**circuits*with constant top fan-in*and*constant-*depth*, constant-*occur**formulas*. ...##
###
Circuit Simulation of All-Spin Logic

2016

Spintronic devices emerge as prime candidates

doi:10.25781/kaust-u0bzd
fatcat:2s42z7vfurdrrdorb4bg5jiaa4
*for*Beyond CMOS era by utilizing the electron spin as an extra*degree*of freedom to decrease the power consumption*and*overcome the velocity limit connected ... By using the well-known spin physics*and*the recently proposed four-component spin*circuit*formalism, ASL can be thoroughly studied*and*simulated. ... Hossein Fariborzi,*for*whom I would like to express my deepest appreciation*and*thanks*for*his advice, patience, continuous support,*and*insightful comments. ...##
###
Certifying Unstability of Switched Systems Using Sum of Squares Programming

2020
*
SIAM Journal of Control and Optimization
*

Therefore, the differential reachability

doi:10.1137/18m1173460
fatcat:ytlzbwk7vbampbuyo6snenz33m
*and*observability Gramians can be computed trajectory-wise by using impulse*and*initial output responses of*d*Σ along each φ t−t 0 (x 0 , u). ... An example of a classical mechanical system having the structure of (*3*) is a flexible-joint robot [*3*]. ... Calculate the*Jacobian*matrix of [Eζ f T η] at these points*and*stack these*Jacobians*to make a tensor J.*3*. ...##
###
On the rank of the fibers of rational elliptic surfaces

2012
*
Algebra & Number Theory
*

If a surface X of

doi:10.2140/ant.2012.6.1289
fatcat:ufqa4stz3raihf7a3mpa35ies4
*degree**d*= ω 2 X is a*k*-minimal model of a rational elliptic surface Ᏹ, then Ᏹ is isomorphic to the blow up of X in*d*points which form a Galois invariant*set*. ...*For*us*k*is a number field*and**K*=*k*(B) is the function field of a projective curve B. We now state a typical result of Kummer theory*for*elliptic curves. ... We are grateful to the AMS*for**setting*up this program,*and**for*providing a stimulating research environment. ...##
###
Aerodynamic Race Car
[article]

2021
*
Zenodo
*

the Dallara

doi:10.5281/zenodo.5725229
fatcat:eqrjuetv7zae5kzondzuoevfdq
*Formula**3*car 306. ... The following image corresponds to the same ideas applied to an "ideal" truck: INCREASE IN THE USEFUL POWER IN*FORMULA*DALLARA*3*-306 Another example of drag reduction*for*greater top speed, we did with ... or low incidence angle (*lower*than 12*degrees*) but not*for*others angles. ...##
###
50 Algebra in Computational Complexity (Dagstuhl Seminar 14391) Manindra Agrawal, Valentine Kabanets, Thomas Thierauf, and Christopher Umans 85 Privacy and Security in an Age of Surveillance (Dagstuhl Perspectives Workshop

unpublished

*Lower*

*bounds*

*for*(homogeneous)

*depth*-4

*and*(nonhomogeneous)

*depth*-

*3*arithmetic

*circuits*In the talk, we plan to discuss an exponential

*lower*

*bound*

*for*(homogeneous)

*depth*-4

*circuits*that comes close to ... Our work builds on several recent

*lower*

*bound*results

*and*the techniques also yield exponential

*lower*

*bounds*

*for*certain (nonhomogeneous)

*depth*-

*3*

*circuits*, in particular

*depth*-

*3*

*circuits*with low bottom ... Both algorithms apply the polynomial method of Razborov

*and*Smolensky, originally conceived

*for*proving low-

*depth*

*circuit*

*lower*

*bounds*. ...

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