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Forbes, Michael A.; Shpilka, Amir; Volk, Ben Lee

doi:10.4086/toc.2018.v014a018

Cited by 12 publications

(10 citation statements)

References 97 publications

(191 reference statements)

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“…Extending the result in Theorem 1.2 to hardness of equations for VP, even under the assumption that Permanent is sufficiently hard, is an extremely interesting open question. Such an extension would answer the main question investigated in [FSV18,GKSS17] and show a natural-proofslike barrier for a fairly general family of lower bound proof techniques in algebraic complexity.…”

Section: Our Resultsmentioning

confidence: 95%

“…In one of the first results on this problem, Forbes, Shpilka and Volk [FSV18] and Grochow, Kumar, Saks and Saraf [GKSS17] observe that the class VP does not have efficiently constructible equations if we were to believe that there are hitting set generators for algebraic circuits with sufficiently succinct descriptions. However, unlike the results of Razborov and Rudich [RR97], the plausibility of the pseudorandomness assumption in [FSV18,GKSS17] is not very well understood. The question of understanding the complexity of equations for VP, or in general any natural class of algebraic circuits, continues to remain open.…”

Section: Complexity Of Equations For Classes Of Polynomialsmentioning

confidence: 99%

“…For the proof of Theorem 1.2, we show that the points in the image of the map Gen Perm , can be viewed as the coefficient vectors of polynomials in VNP, or, equivalently in the terminology in [FSV18,GKSS17], that the Kabanets-Impagliazzo hitting set generator is VNP-succinct. To this end, we work with a specific instantiation of the construction of the Kabanets-Impagliazzo generator where the underlying construction of combinatorial designs is based on Reed-Solomon codes.…”

Section: An Overview Of the Proofmentioning

confidence: 99%

“…In the last few years there has been some work (e.g. [Gro15], [GKSS17,FSV18]) aimed at developing an analogue of the Natural Proofs framework for algebraic circuit lower bounds. A crucial notion in this context is that of an equation for a class of polynomials which we now define.…”

Section: Introductionmentioning

confidence: 99%

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If VNP is hard, then so are equations for it

Kumar¹,

Ramya²,

Saptharishi³

et al. 2020

Preprint

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Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size.In a recent work of Chatterjee and the authors [CKR + 20], it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

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Section: Our Resultsmentioning

confidence: 95%

Section: Complexity Of Equations For Classes Of Polynomialsmentioning

confidence: 99%

Section: An Overview Of the Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

If VNP is hard, then so are equations for it

Kumar¹,

Ramya²,

Saptharishi³

et al. 2020

Preprint

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“…The quantity ∆(V ) is a very coarse complexity measure. A recent line of work regarding algebraic natural proofs [FSV18,GKSS17] suggests to study the arithmetic circuit complexity of equations for varieties V that correspond to polynomials with small circuit complexity. Having ∆(V ) growing like a polynomial in n is a necessary (but not a sufficient) condition for a variety V to have an algebraic natural proof for non-containment.…”

Section: Equations For Varities In Algebraic Complexity Theorymentioning

confidence: 99%

A Polynomial Degree Bound on Equations of Non-rigid Matrices and Small Linear Circuits

Kumar¹,

Volk²

2020

Preprint

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We show that there is a defining equation of degree at most poly(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field F, there is a non-zero n 2 -variate polynomial P ∈ F(x 1,1 , . . . , x n,n ) of degree at most poly(n) such that every matrix M which can be written as a sum of a matrix of rank at most n/100 and sparsity at most n 2 /100 satisfies P (M ) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [GHIL16] and improves the best upper bound known for this problem down from exp(n 2 ) [KLPS14, GHIL16] to poly(n).We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n 2 /200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded.Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [SV15] to construct low degree "universal" maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof.

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