Marginal hitting sets imply super-polynomial lower bounds for permanent

Jansen, Maurice; Santhanam, Rahul

doi:10.1145/2090236.2090275

Cited by 7 publications

(3 citation statements)

References 35 publications

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“…The hypothesis of Theorem 1.7 bear some similarities with a result of Jansen and Santhanam [JS12] who showed that, for the class of univariate polynomial of degree d that are computable by small circuits, if there is a hitting set H of size d (any set of size d + 1 would have been sufficient) that can be efficiently encoded by a small TC 0 circuit, then Perm does not have polynomial sized constant-free algebraic circuits. Our hypothesis is similar in the sense that it requires a saving of one from the trivial hitting set for the appropriate class, but we only need the standard notion of explicitness for the hitting set.…”

Section: Relating Theorem 17 and Results Of Jansen And Santhanam [Jsmentioning

confidence: 64%

Derandomization from Algebraic Hardness: Treading the Borders

Guo

Kumar

Saptharishi

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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A hitting-set generator (HSG) is a polynomial map Gen : F k → F n such that for all n-variate polynomials C of small enough circuit size and degree, if C is nonzero, then C • Gen is nonzero. In this paper, we give a new construction of such an HSG assuming that we have an explicit polynomial of sufficient hardness. Formally, we prove the following result over any field F of characteristic zero:Suppose P(z 1 , . . . , z k ) is an explicit k-variate degree d polynomial that is not computable by circuits of size s. Then, there is an explicit hitting-set generator Gen P :This is the first HSG in the algebraic setting that yields a complete derandomization of polynomial identity testing (PIT) for general circuits from a suitable algebraic hardness assumption. Unlike the prior constructions of such maps [NW94, KI04, AGS19, KST19], our construction is purely algebraic and does not rely on the notion of combinatorial designs.As a direct consequence, we show that even saving a single point from the "trivial" explicit, exponential sized hitting sets for constant-variate polynomials of low individual-degree which are computable by small circuits, implies a deterministic polynomial time algorithm for PIT. More precisely, we show the following:Let k be a large enough constant. Suppose for every s large enough, there is an explicit hitting set of size at most ((s + 1) k − 1) for the class of k-variate polynomials of individual degree s that are computable by size s circuits. Then there is an explicit hitting set of size s O(k 2 ) for the class of s-variate polynomials, of degree s, that are computable by size s circuits.

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Section: Relating Theorem 17 and Results Of Jansen And Santhanam [Jsmentioning

confidence: 64%

Derandomization from Algebraic Hardness: Treading the Borders

Guo

Kumar

Saptharishi

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…One obvious task would be to improve the parameters of our result. As a matter of fact we have already obtained some preliminary results using a different framework involving high degree univariate polynomial identity testing [16]. Still, using low-degree multivariate polynomial identity testing as the target for derandomization may be easier, but there is the problem of connecting this up to lower bounds for permanent.…”

Section: Resultsmentioning

confidence: 99%

Permanent does not have succinct polynomial size arithmetic circuits of constant depth

Jansen

Santhanam

2013

Information and Computation

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We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {Φ n } using division by constants, 3 where Φ n has size at most p(n) and depth O (1), such that Φ n computes the n × n permanent. A circuit family {Φ n } is succinct if there exists a nonuniform Boolean circuit family {C n } with O (log n) many inputs and size n o(1) such that C n can correctly answer direct connection language queries about Φ n -succinctness is a relaxation of uniformity.To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits. From this, we obtain the lower bound by giving an explicit construction, computable in the polynomial hierarchy, of a hitting set for arithmetic circuits.

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“…One very interesting technique which we have not yet been able to fit into the representationtheoretic framework and which is only partially uniform comes from Jansen and Santhanam [JS12,JS13]. The key property they use is the existence of Z hitting sets whose bit descriptions can be encoded by small uniform (or at least succinct [JS13]) circuits.…”

Section: Discussion Relation To the Gct Program And Future Directionsmentioning

confidence: 99%

Unifying Known Lower Bounds via Geometric Complexity Theory

Grochow

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on AC 0 [p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or, Steele-Yao), depth 3 algebraic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (MignonRessayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (Bürgisser-Ikenmeyer, LandsbergOttaviani) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-KayalKamath-Saptharishi; Agrawal-Vinay; Koiran), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification of known results and broad generalization of known techniques. It also shows that the framework of GCT is at least as powerful as previous methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.Although many proofs are omitted in this extended abstract, all are present in the full version [1]. Furthermore, in Section II we give a motivating example in full detail; the remaining proofs essentially follow the same outline, being only a matter of formulating previous proofs in the right way. References to the full version [1] within the extended abstract are marked with " FULL ," as in "Section 3.3 FULL ."

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Marginal hitting sets imply super-polynomial lower bounds for permanent

Cited by 7 publications

References 35 publications

Derandomization from Algebraic Hardness: Treading the Borders

Derandomization from Algebraic Hardness: Treading the Borders

Permanent does not have succinct polynomial size arithmetic circuits of constant depth

Unifying Known Lower Bounds via Geometric Complexity Theory

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