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Meka, Raghu; Nguyen, Oanh; Vu, Van Khu

doi:10.4086/toc.2016.v012a011

Cited by 30 publications

(10 citation statements)

References 14 publications

(26 reference statements)

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“…A natural open question following our results is to remove the polylogarithmic factors separating our upper and lower bounds. We remark that in characteristic 0, such gaps exist even for the very simple OR function despite much effort [13,9,4]. Over positive characteristic, there is no obvious barrier, but our techniques fall short of proving tight lower bounds for natural families of functions such as the Exact Threshold functions (defined below).…”

Section: Remarkmentioning

confidence: 99%

On the Probabilistic Degrees of Symmetric Boolean Functions

Srinivasan¹,

Tripathi²,

Venkitesh³

2021

SIAM J. Discrete Math.

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The probabilistic degree of a Boolean function f : {0, 1} n → {0, 1} is defined to be the smallest d such that there is a random polynomial P of degree at most d that agrees with f at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees of Boolean functions -specifically symmetric Boolean functions -have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems.In this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).

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Section: Remarkmentioning

confidence: 99%

On the Probabilistic Degrees of Symmetric Boolean Functions

Srinivasan¹,

Tripathi²,

Venkitesh³

2021

SIAM J. Discrete Math.

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“…Similar to previous works [8,11], our choice of hard distribution is motivated by the polynomial constructions in the upper bound. Our hard distribution  is defined in terms of the following two distributions.…”

Section: Lower Bound On Hyperplane Covering Degree Of Ormentioning

confidence: 99%

“…Similar to previous works [10,6], our choice of hard distribution is motivated by the polynomial constructions in the upper bound. We first need the following definitions to define the hard distribution D ε .…”

Section: Lower Bound On Hyperplane Covering Degree Of Ormentioning

confidence: 99%

“…However, until recently, it was not clear whether any dependence on n is necessary in P-eg (OR n ) over the reals. 2 In recent papers of Meka, Nguyen and Vu [11] and Harsha and Srinivasan [8], it was shown using anti-concentration of low-degree polynomials that the P-eg 1∕4 (OR n ) =Ω( √ log n). The main objective of this paper is to obtain a better understanding of the -error probabilistic degree of OR n , P-eg (OR n ).…”

Section: Introductionmentioning

confidence: 99%

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On the probabilistic degree of OR over the reals

Bhandari

Harsha

Molli

et al. 2021

Random Struct Algorithms

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We study the probabilistic degree over R of the OR function on n variables. For ∈ (0, 1∕3), the -error probabilistic degree of any Boolean function f : {0, 1} n → {0, 1} over R is the smallest nonnegative integer d such that the following holds: there exists a distribution P of polynomialsIt is known from the works of Tarui (Theoret. Comput. Sci. 1993) andBeigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the -error probabilistic degree of the OR function is at most O(log n ⋅ log(1∕ )). Our first observation is that this can be improved to O))), thus matching the above upper bound (up to poly-logarithmic factors).

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“…However, till not long ago, it was unclear if any dependence on n is required over the reals 2 . In recent papers of Meka, Nguyen and Vu [MNV16] and the second and last author [HS18], it was shown using anti-concentration of low-degree polynomials that the P-deg…”

Section: Introductionmentioning

confidence: 99%

On the Probabilistic Degree of OR over the Reals

Bhandari,

Harsha,

Molli

et al. 2018

Preprint

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We study the probabilistic degree over R of the OR function on n variables. For ε ∈ (0, 1/3), the ε-error probabilistic degree of any Boolean function f : {0, 1} n → {0, 1} over R is the smallest non-negative integer d such that the following holds: there exists a distribution of polynomials P ∈ R[x1, . . . , xn] entirely supported on polynomials of degree at most d such that for all z ∈ {0, 1} n , we have PrPIt is known from the works of Tarui (Theoret. Comput. Sci. 1993) andBeigel, Reingold, and Spielman (Proc. 6th CCC 1991), that the ε-error probabilistic degree of the OR function is at most O(log n • log( 1 /ε)). Our first observation is that this can be improvedwhich is better for small values of ε.In all known constructions of probabilistic polynomials for the OR function (including the above improvement), the polynomials P in the support of the distribution P have the following special structure:where each Li(x1, . . . , xn) is a linear form in the variables x1, . . . , xn, i.e., the polynomial 1 − P (x) is a product of affine forms. We show that the ε-error probabilistic degree of OR when restricted to polynomials of the above form is Ω log, thus matching the above upper bound (up to polylogarithmic factors).

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Cited by 30 publications

References 14 publications

On the Probabilistic Degrees of Symmetric Boolean Functions

On the Probabilistic Degrees of Symmetric Boolean Functions

On the probabilistic degree of OR over the reals

On the Probabilistic Degree of OR over the Reals

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