On the probabilistic degree of OR over the reals

Bhandari, Siddharth; Harsha, Prahladh; Molli, Tulasimohan; Srinivasan, Srikanth

doi:10.1002/rsa.20991

Cited by 3 publications

(4 citation statements)

References 14 publications

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“…Or equivalently, that 𝑔 can be obtained from 𝑓 by setting some inputs to 0 and 1 respectively. 10 We will use the following obvious fact freely.…”

Section: E F I N I T I O N 4 4 (Restrictions)mentioning

confidence: 99%

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A Robust Version of Heged\H{u}s's Lemma, with Applications

Srinivasan¹

2023

TheoretiCS

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Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of some fixed Hamming weight $k\in [q,n-q]$ must also vanish at all points in $\{0,1\}^n$ of weight $k + q$. This lemma was used by Heged\H{u}s (2009) to give a solution to \emph{Galvin's problem}, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrube\v{s}, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-$2$ threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Heged\H{u}s's lemma. Informally, this version says that if a polynomial of degree $o(q)$ vanishes at most points of weight $k$, then it vanishes at many points of weight $k+q$. We prove this lemma and give three different applications.

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“…Or equivalently, that 𝑔 can be obtained from 𝑓 by setting some inputs to 0 and 1 respectively. 10 We will use the following obvious fact freely.…”

Section: E F I N I T I O N 4 4 (Restrictions)mentioning

confidence: 99%

“…In the case of characteristic 0 (or growing with 𝑛), such gaps look hard to close since we don't even understand completely the probabilistic degree of simple functions like the OR function [34,22,10]. However, in positive ( xed) characteristic, there are no obvious barrriers.…”

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confidence: 99%

A Robust Version of Heged\H{u}s's Lemma, with Applications

Srinivasan¹

2023

TheoretiCS

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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 [MNV16,HS16,BHMS18]. 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 in Section 2).…”

Section: Remarkmentioning

confidence: 99%

On the Probabilistic Degrees of Symmetric Boolean functions

Srinivasan¹,

Tripathi²,

Venkitesh³

2019

Preprint

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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 functionshave 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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“…In the case of characteristic 0 (or growing with n), such gaps look hard to close since we don't even understand completely the probabilistic degree of simple functions like the OR function [MNV16,HS16,BHMS18]. However, in positive (fixed) characteristic, there are no obvious barrriers.…”

Section: Introductionmentioning

confidence: 99%

A Robust Version of Hegedűs's Lemma, with Applications

Srinivasan¹

2022

Preprint

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Hegedűs's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field F of characteristic p > 0 and for q a power of p, the lemma says that any multilinear polynomial P ∈ F[x 1 , . . . , x n ] of degree less than q that vanishes at all points in {0, 1} n of Hamming weight k ∈ [q, n − q] must also vanish at all points in {0, 1} n of weight k + q. This lemma was used by Hegedűs (2009) to give a solution to Galvin's problem, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrubeš, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-2 threshold circuits for computing some symmetric functions.In this paper, we formulate a robust version of Hegedűs's lemma. Informally, this version says that if a polynomial of degree o(q) vanishes at most points of weight k, then it vanishes at many points of weight k + q. We prove this lemma and give the following three different applications.• Degree lower bounds for the coin problem: The δ-Coin Problem is the problem of distinguishing between a coin that is heads with probability ((1/2) + δ) and a coin that is heads with probability 1/2. We show that over a field of positive (fixed) characteristic, any polynomial that solves the δ-coin problem with error ε must have degree Ω( 1 δ log(1/ε)), which is tight up to constant factors. • Probabilistic degree lower bounds: The Probabilistic degree of a Boolean function is the minimum d such that there is a random polynomial of degree d that agrees with the function at each point with high probability. We give tight lower bounds on the probabilistic degree of every symmetric Boolean function over positive (fixed) characteristic. As far as we know, this was not known even for some very simple functions such as unweighted Exact Threshold functions, and constant error.• A robust version of the combinatorial result of Hegedűs (2009) mentioned above.

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On the probabilistic degree of OR over the reals

Cited by 3 publications

References 14 publications

A Robust Version of Heged\H{u}s's Lemma, with Applications

A Robust Version of Heged\H{u}s's Lemma, with Applications

On the Probabilistic Degrees of Symmetric Boolean functions

A Robust Version of Hegedűs's Lemma, with Applications

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