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

Agrawal, Manindra; Saha, Chandan; Saptharishi, Ramprasad; Saxena, Nitin

doi:10.1137/130910725

Cited by 21 publications

(76 citation statements)

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“…Our lower bounds hold for a (potentially) richer class of circuits. In the model considered by [1], one imposes a global upper bound k on the rank of all the Q i feeding into some polynomial C. In our model, we can take exponentially many different sets of polynomials Q i , each with bounded rank, and apply some polynomial function to each of them and then take a sum.…”

Section: Some Background and Motivationmentioning

confidence: 99%

“…In sharp contrast to this state of knowledge on lower bounds, the problem of polynomial identity testing is very poorly understood even for depth three circuits. Till a few years ago, almost all the PIT algorithms known were for extremely restricted classes of circuits and were based on diverse proof techniques (for instance, [7,23,15,22,14,37,38,36,1,10,30]). The paper by Agrawal et al [1] gave a unified proof of several of them.…”

Section: Introductionmentioning

confidence: 99%

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Kumar

Saraf²

2017

Theory of Comput.

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In recent years, there has been a flurry of activity towards proving lower bounds for homogeneous depth-4 arithmetic circuits (Gupta et al., Fournier et al., Kayal et al., Kumar-Saraf), which has brought us very close to statements that are known to imply VP = VNP. It is open if these techniques can go beyond homogeneity, and in this paper we make progress in this direction by considering depth-4 circuits of low algebraic rank, which are a natural extension of homogeneous depth-4 arithmetic circuits.A depth-4 circuit is a representation of an N-variate, degree-n polynomial P aswhere the Q i j are given by their monomial expansion. Homogeneity adds the constraint that for every i ∈ [T ], ∑ j deg(Q i j ) = n. We study an extension, where, for every i ∈ [T ],

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Section: Some Background and Motivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Untitled

Kumar

Saraf²

2017

Theory of Comput.

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“…(For depth-3 formulas over fixed finite fields, an exponential lower bound was shown by [5,6].) Indeed, the super-polynomial lower bounds obtained by [20,19,21], and also some others like [1], are based upon studying partial derivatives or associated matrices involving partial derivatives like the Jacobian or the Hessian.…”

Section: Introductionmentioning

confidence: 99%

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

Kayal

Limaye

Saha

et al. 2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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“…In particular, we obtain a polynomial-time blackbox algorithm for multilinear constant-read constant-depth formulae and, for the special case of fields with infinite characteristic, Agrawal et al (2012) recently showed that the multilinearity condition can be removed. We refer to Section 6.3 for more details about those results, and to the rest of Section 6 for more general versions and finer parameterizations of our main result and its extensions.…”

Section: Resultsmentioning

confidence: 99%

“…Recently, Agrawal et al (2012) used the notion of algebraic dependence and presented a unified approach for obtaining identity tests for constant-depth constant-read formulae. In fact, they develop a polynomial-time blackbox identity test for constantdepth constant-read formulae with addition, multiplication, and powering gates, 2 without the restriction of multilinearity.…”

Section: Identity Testing Read-k Formulaementioning

confidence: 99%

Deterministic polynomial identity tests for multilinear bounded-read formulae

2015

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We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula, each variable occurs only a constant number of times, and each subformula computes a multilinear polynomial. Our algorithm runs in time s O(1) · n k O(k) , where s denotes the size of the formula, n denotes the number of variables, and k bounds the number of occurrences of each variable. Before our work, no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time n k O(k) +O(k log n) in general, and time n k O(k 2 ) +O(kD) for depth D. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae and for multilinear depth-four formulae.

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

Cited by 21 publications

References 21 publications

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

Deterministic polynomial identity tests for multilinear bounded-read formulae

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