From sylvester-gallai configurations to rank bounds

Saxena, Nitin; Seshadhri, C.

doi:10.1145/2528403

Cited by 18 publications

(9 citation statements)

References 42 publications

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“…The case of more than 3 multiplication gates is more complicated and satisfies a similar higher dimensional condition. This rank-bound approach for PIT of ΣΠΣ circuits was raised in [DS07] and later carried out in [KS09b,SS13]. 2 While such rank-bounds found important applications in studying PIT of depth-3 circuits, it seemed that such an approach cannot work for depth-4 ΣΠΣΠ circuits, 3 even in the simplest case where there are only 3 multiplication gates and the bottom fan-in is two, i.e., for homogeneous Σ [3] Π [d] ΣΠ [2] circuits that compute polynomials of the form…”

Section: Introductionmentioning

confidence: 99%

Sylvester-Gallai type theorems for quadratic polynomials

Shpilka

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection Q, of irreducible polynomials of degree at most 2, satisfy that for every two polynomials Q 1 , Q 2 ∈ Q there is a third polynomial Q 3 ∈ Q so that whenever Q 1 and Q 2 vanish then also Q 3 vanishes, then the linear span of the polynomials in Q has dimension O(1). We also prove a colored version of the theorem: If three finite sets of quadratic polynomials satisfy that for every two polynomials from distinct sets there is a polynomial in the third set satisfying the same vanishing condition then all polynomials are contained in an O(1)-dimensional space.This answers affirmatively two conjectures of Gupta [Gup14] that were raised in the context of solving certain depth-4 polynomial identities.To obtain our main theorems we prove a new result classifying the possible ways that a quadratic polynomial Q can vanish when two other quadratic polynomials vanish. Our proofs also require robust versions of a theorem of Edelstein and Kelly (that extends the Sylvester-Gallai theorem to colored sets).

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Section: Introductionmentioning

confidence: 99%

Sylvester-Gallai type theorems for quadratic polynomials

Shpilka

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…We prove that it suffices to replace P i withP i which is obtained from P i by retaining, in the product, only linear forms that appear in at most 5 locations (roughly). This is shown using a rank bound for commutative depth-three identities [25]. We also require algorithms [16,21,17] over words to efficiently find the linear forms appearing in those 5 locations.…”

Section: White-box Algorithm For +-Regular Circuitsmentioning

confidence: 99%

“…In our study of +-regular circuits, it is important to consider certain special +-regular circuits of +-depth 2. We term these as homogeneous ΣΠ * Σ circuits (as already mentioned in Section 2), in analogy with the usual ΣΠΣ arithmetic circuits (see, e. g., [25]).…”

Section: Finally Consider the Bottommost +-Layer L +mentioning

confidence: 99%

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Arvind¹,

Joglekar²,

Mukhopadhyay³

et al. 2019

Theory of Comput.

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In this paper we show that black-box polynomial identity testing (PIT) for nvariate noncommutative polynomials f of degree D with at most t nonzero monomials can be done in randomized poly(n, logt, log D) time, and consequently in randomized poly(n, logt, s) time if f is computable by a circuit of size s. This result makes progress on a question that has been open for over a decade. Our algorithm is based on efficiently isolating a monomial using nondeterministic automata. The above result does not yield an efficient randomized PIT for noncommutative circuits in general, as noncommutative circuits of size s can compute polynomials with a doubleexponential (in s) number of monomials. As a first step, we consider a natural class of homogeneous noncommutative circuits, that we call +-regular circuits, and give a white-box polynomial-time deterministic PIT for them. These circuits can compute noncommutative polynomials with number of monomials double-exponential in the circuit size. Our algorithm combines some new structural results for +-regular circuits with known PIT results for noncommutative algebraic branching programs, a rank bound for commutative depth-3 A preliminary version of this paper appeared in the Proceedings of the 49th ACM Symp. on Theory of Computing (STOC'17) [4]. * Part of the work was done while the author was a postdoctoral fellow at Chennai Mathematical Institute, India.

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“…The study of Σ k ΠΣ formulas was initiated by Dvir and Shpilka [29], who proved that in a simple and minimal 5 Σ k ΠΣ circuit computing the zero polynomial, the rank of the linear functions i, j is bounded by a number R(k, d) that is independent of the number of variables N. The number R(k, d) is called the rank bound for this class of circuits. Karnin and Shpilka [57] showed how to use the rank condenser construction of Gabizon and Raz in order to obtain a black-box identity testing algorithm, and improved rank bounds were later obtained ( [62,91,92,93]).…”

Section: Depth-3 Formulas With Bounded Top-fan-inmentioning

confidence: 99%

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Forbes¹,

Shpilka

Volk

2018

Theory of Comput.

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We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich (1997) for Boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike in the Boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support "cryptography" secure against algebraic circuits. Following a similar result of Williams (2016) in the Boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the 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) degree with polylog(N)-size circuits. Further, we give

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From sylvester-gallai configurations to rank bounds

Cited by 18 publications

References 42 publications

Sylvester-Gallai type theorems for quadratic polynomials

Sylvester-Gallai type theorems for quadratic polynomials

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