
Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem
Based on the recent breakthrough of Huang (2019), we show that for any t...
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Approximate degree, secret sharing, and concentration phenomena
The ϵapproximate degree deg_ϵ(f) of a Boolean function f is the least d...
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Random restrictions and PRGs for PTFs in Gaussian Space
A polynomial threshold function (PTF) f:ℝ^n →ℝ is a function of the form...
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Concentration of Multilinear Functions of the Ising Model with Applications to Network Data
We prove neartight concentration of measure for polynomial functions of...
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Learning from satisfying assignments under continuous distributions
What kinds of functions are learnable from their satisfying assignments?...
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VanishingError Approximate Degree and QMA Complexity
The ϵapproximate degree of a function f X →{0, 1} is the least degree o...
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On Boolean Functions with Low Polynomial Degree and Higher Order Sensitivity
Boolean functions are important primitives in different domains of crypt...
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On the Probabilistic Degree of an nvariate Boolean Function
Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}^n→{0,1} that depends on all its input variables, when represented as a realvalued multivariate polynomial P(x_1,…,x_n), has degree at least log n  O(loglog n). This was improved to a tight (log n  O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for Probabilistic degree. The function f has probabilistic degree at most d if there is a random realvalued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the bestknown bounds put it between (log n)^1/2o(1) and O(log n) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a nearoptimal understanding of the probabilistic degree of nvariate functions f, modulo our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log n)^c, then the minimum possible probabilistic degree of such an f is at least (log n)^c/(c+1)o(1), and we show this is tight up to (log n)^o(1) factors.
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