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COCO
2003
Springer

Extremal properties of polynomial threshold functions

10 years 6 months ago
Extremal properties of polynomial threshold functions
In this paper we give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following: • Almost every Boolean function has PTF degree at most n 2 + O( √ n log n). Together with results of Anthony and Alon, this establishes a conjecture of Wang and Williams [32] and Aspnes, Beigel, Furst, and Rudich [4] up to lower order terms. • Every Boolean function has PTF density at most (1 − 1 O(n) )2n . This improves a result of Gotsman [14]. • Every Boolean function has weak PTF density at most o(1)2n . This gives a negative answer to a question posed by Saks [28]. • PTF degree log2 m +1 is necessary and sufficient for Boolean functions with sparsity m. This answers a question of Beigel [7]. We also give new extremal bounds on polynomials which approximate Boolean functions in the ∞ norm.
Ryan O'Donnell, Rocco A. Servedio
Added 06 Jul 2010
Updated 06 Jul 2010
Type Conference
Year 2003
Where COCO
Authors Ryan O'Donnell, Rocco A. Servedio
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