Random Restrictions and PRGs for PTFs in Gaussian Space

Authors Zander Kelley, Raghu Meka



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Author Details

Zander Kelley
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Raghu Meka
  • Department of Computer Science, University of California, Los Angeles, CA, USA

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Zander Kelley and Raghu Meka. Random Restrictions and PRGs for PTFs in Gaussian Space. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.21

Abstract

A polynomial threshold function (PTF) f: ℝⁿ → ℝ is a function of the form f(x) = sign(p(x)) where p is a polynomial of degree at most d. PTFs are a classical and well-studied complexity class with applications across complexity theory, learning theory, approximation theory, quantum complexity and more. We address the question of designing pseudorandom generators (PRGs) for polynomial threshold functions (PTFs) in the gaussian space: design a PRG that takes a seed of few bits of randomness and outputs a n-dimensional vector whose distribution is indistinguishable from a standard multivariate gaussian by a degree d PTF. Our main result is a PRG that takes a seed of d^O(1) log(n/ε) log(1/ε)/ε² random bits with output that cannot be distinguished from an n-dimensional gaussian distribution with advantage better than ε by degree d PTFs. The best previous generator due to O'Donnell, Servedio, and Tan (STOC'20) had a quasi-polynomial dependence (i.e., seed length of d^O(log d)) in the degree d. Along the way we prove a few nearly-tight structural properties of restrictions of PTFs that may be of independent interest. Similar results were obtained in [Ryan O'Donnell et al., 2021] (independently and concurrently).

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • polynomial threshold function
  • pseudorandom generator
  • multivariate gaussian

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References

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