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Polynomial Threshold Functions for Decision Lists

Authors Vladimir Podolskii , Nikolay V. Proskurin

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Author Details

Vladimir Podolskii
  • Courant Institute of Mathematical Sciences, New York University, NY, USA
  • Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russian Federation
Nikolay V. Proskurin
  • HSE University, Moscow, Russian Federation

Funding

  • Proskurin, Nikolay V.: This paper was prepared within the framework of the HSE University Basic Research Program.

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Vladimir Podolskii and Nikolay V. Proskurin. Polynomial Threshold Functions for Decision Lists. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 52:1-52:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.52

Abstract

For S ⊆ {0,1}ⁿ a Boolean function f : S → {-1,1} is a polynomial threshold function (PTF) of degree d and weight W if there is a polynomial p with integer coefficients of degree d and with sum of absolute coefficients W such that f(x) = sign p(x) for all x ∈ S. We study a representation of decision lists as PTFs over Boolean cubes {0,1}ⁿ and over Hamming balls {0,1}ⁿ_{≤ k}. As our first result, we show that for all d = O((n/(log n))^{1/3}) any decision list over {0,1}ⁿ can be represented by a PTF of degree d and weight 2^O(n/d²). This improves the result by Klivans and Servedio [Adam R. Klivans and Rocco A. Servedio, 2006] by a log² d factor in the exponent of the weight. Our bound is tight for all d = O((n/(log n))^{1/3}) due to the matching lower bound by Beigel [Richard Beigel, 1994]. For decision lists over a Hamming ball {0,1}ⁿ_{≤ k} we show that the upper bound on weight above can be drastically improved to n^O(√k) for d = Θ(√k). We also show that similar improvement is not possible for smaller degrees by proving the lower bound W = 2^Ω(n/d²) for all d = O(√k).

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • Threshold function
  • decision list
  • Hamming ball

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