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Diakonikolas, Ilias; Servedio, Rocco A.; Tan, Li-Yang; Wan, Andrew

doi:10.4086/toc.2014.v010a002

Cited by 7 publications

(2 citation statements)

References 10 publications

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“…Regarding our structural result described in Section 1.3, Diakonikolas, Servedio, Tan and Wan [12] have independently obtained similar results to ours. As an application, they prove the existence of low-weight approximators for polynomial threshold functions.…”

Section: Related and Subsequent Worksupporting

confidence: 87%

“…As an application, they prove the existence of low-weight approximators for polynomial threshold functions. In the context of approximating functions with many influential variables by low-degree PTFs, Ben-Eliezer, Lovett and Yadin [28] obtained similar structural results (with weaker parameters) independent of both our work and that of [12].…”

Section: Related and Subsequent Worksupporting

confidence: 72%

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Harsha¹,

Klivans²,

Meka³

2014

Theory of Comput.

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We give the first nontrivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial of total degree d, we prove • The average sensitivity of f is at most O(n 1−1/(4d+6)). (We also give a combinatorial proof of the bound O(n 1−1/2 d).) • The noise sensitivity of f with noise rate δ is at most O(δ 1/(4d+6)). Previously, only bounds for the degree d = 1 case were known (O(√ n) and O(√ δ), for average and noise sensitivity, respectively). We highlight some applications of our results in learning theory where our bounds immediately yield new agnostic learning algorithms and resolve an open problem of Klivans, O'Donnell and Servedio (FOCS'08).

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Section: Related and Subsequent Worksupporting

confidence: 87%

Section: Related and Subsequent Worksupporting

confidence: 72%

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Harsha¹,

Klivans²,

Meka³

2014

Theory of Comput.

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Fooling Intersections of Low-Weight Halfspaces

Servedio

Tan

2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

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A weight-t halfspace is a Boolean function f (x) = sign(w 1 x 1 + · · · + w n x n − θ) where each w i is an integer in {−t, . . . , t}. We give an explicit pseudorandom generator that δ-fools any intersection of k weight-t halfspaces with seed length poly(log n, log k, t, 1/δ). In particular, our result gives an explicit PRG that fools any intersection of any quasipoly(n) number of halfspaces of any polylog(n) weight to any 1/polylog(n) accuracy using seed length polylog(n). Prior to this work no explicit PRG with non-trivial seed length was known even for fooling intersections of n weight-1 halfspaces to constant accuracy.The analysis of our PRG fuses techniques from two different lines of work on unconditional pseudorandomness for different kinds of Boolean functions. We extend the approach of Harsha, Klivans and Meka [HKM12] for fooling intersections of regular halfspaces, and combine this approach with results of Bazzi [Baz07] and Razborov [Raz09] on bounded independence fooling CNF formulas. Our analysis introduces new coupling-based ingredients into the standard Lindeberg method for establishing quantitative central limit theorems and associated pseudorandomness results.

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Polynomial Threshold Functions, Hyperplane Arrangements, and Random Tensors

Baldi¹,

Vershynin²

2019

SIAM Journal on Mathematics of Data Science

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A simple way to generate a Boolean function is to take the sign of a real polynomial in n variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The partial case of this problem for degree d = 1 was solved by Zuev in 1989, who showed that the number T (n, 1) of linear threshold functions satisfies log 2 T (n, 1) ≈ n 2 , up to smaller order terms. However the number of polynomial threshold functions for any higher degrees, including d = 2, has remained open. We settle this problem for all fixed degrees d ≥ 1, showing that log 2 T (n, d) ≈ n n ≤d . The solution relies on connections between the theory of Boolean threshold functions, hyperplane arrangements, and random tensors. Perhaps surprisingly, it uses also a recent result of E. Abbe, A. Shpilka, and A. Wigderson on Reed-Muller codes.

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Cited by 7 publications

References 10 publications

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Fooling Intersections of Low-Weight Halfspaces

Polynomial Threshold Functions, Hyperplane Arrangements, and Random Tensors

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