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

Manindra Agrawal and Chandan Saha and Ramprasad Saptharishi and Nitin Saxena
2011 arXiv   pre-print
In particular, we show that polynomial time 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).  ... 
arXiv:1111.0582v1 fatcat:4o7q7jv6jjaghhs2ic5uic3ucu

Jacobian Hits Circuits: Hitting Sets, Lower Bounds for Depth-$D$ Occur-$k$ Formulas and Depth-3 Transcendence Degree-$k$ Circuits

Manindra Agrawal, Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
2016 SIAM journal on computing (Print)  
In particular, we show that polynomial (in the size of the 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.  ... 
doi:10.1137/130910725 fatcat:hpsqcu3x7japnnhpwplplox6xy

Jacobian hits circuits

Manindra Agrawal, Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
2012 Proceedings of the 44th symposium on Theory of Computing - STOC '12  
In particular, we show that polynomial time 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).  ... 
doi:10.1145/2213977.2214033 dblp:conf/stoc/AgrawalSSS12 fatcat:micc6dhlqrfztncc35ddkuv3qq

Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds [article]

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2018 arXiv   pre-print
That is, whether the coefficient vectors of polylog(N)-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.  ... 
arXiv:1701.05328v2 fatcat:ueg2mlf5l5febo7hojpqri67dq

Succinct hitting sets and barriers to proving algebraic circuits lower bounds

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
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)-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.  ... 
doi:10.1145/3055399.3055496 dblp:conf/stoc/ForbesSV17 fatcat:amglevgewvdlpozsxppzhdqify

Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2018 Theory of Computing  
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  ...  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.  ... 
doi:10.4086/toc.2018.v014a018 dblp:journals/toc/ForbesSV18 fatcat:xca443ndhzfxfjnclcmx6y57py

Progress on Polynomial Identity Testing - II [article]

Nitin Saxena
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).  ... 
arXiv:1401.0976v1 fatcat:wrc3gfl2ajbmbhl2reb6lm722m

Progress on Polynomial Identity Testing-II [chapter]

Nitin Saxena
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.  ... 
doi:10.1007/978-3-319-05446-9_7 fatcat:nakjcs6t6vf7pmd4qwjxmqarga

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

Joshua A. Grochow
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.  ... 
dblp:journals/eatcs/Grochow20 fatcat:souydyndrzd5df5btakxxjwhgi

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

Chandan Saha, Bhargav Thankey, Mary Wootters, Laura Sanità
2021
This implies a quasi-polynomial time 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.  ... 
doi:10.4230/lipics.approx/random.2021.50 fatcat:sphejkalhna7jmg3p5gsi6ggia

Circuit Simulation of All-Spin Logic

Meshal Alawein
2016
Spintronic devices emerge as prime candidates 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.  ... 
doi:10.25781/kaust-u0bzd fatcat:2s42z7vfurdrrdorb4bg5jiaa4

Certifying Unstability of Switched Systems Using Sum of Squares Programming

Benoît Legat, Pablo Parrilo, Raphaël Jungers
2020 SIAM Journal of Control and Optimization  
Therefore, the differential reachability and observability Gramians can be computed trajectory-wise by using impulse and initial output responses of dΣ 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.  ... 
doi:10.1137/18m1173460 fatcat:ytlzbwk7vbampbuyo6snenz33m

On the rank of the fibers of rational elliptic surfaces

Cecília Salgado
2012 Algebra & Number Theory  
If a surface X of 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.  ... 
doi:10.2140/ant.2012.6.1289 fatcat:ufqa4stz3raihf7a3mpa35ies4

Aerodynamic Race Car [article]

Timoteo Briet Blanes
2021 Zenodo  
the Dallara 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.  ... 
doi:10.5281/zenodo.5725229 fatcat:eqrjuetv7zae5kzondzuoevfdq

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

Maria-Florina Balcan, Bodo Manthey, Heiko Röglin, Tim Roughgarden, Artur D'avila Garcez, Marco Gori, Pascal Hitzler, Luís Lamb, Bart Preneel, Phillip Rogaway, Mark Ryan, Peter (+5 others)
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.  ... 
fatcat:e5wnf6cuhvc3bceegijgut34zm
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