Weizmann Logo
ECCC
Electronic Colloquium on Computational Complexity

Under the auspices of the Computational Complexity Foundation (CCF)

Login | Register | Classic Style



REPORTS > DETAIL:

Revision(s):

Revision #1 to TR21-047 | 29th November 2021 18:15

Random restrictions and PRGs for PTFs in Gaussian Space

RSS-Feed




Revision #1
Authors: Zander Kelley, Raghu Meka
Accepted on: 29th November 2021 18:15
Downloads: 252
Keywords: 


Abstract:

A polynomial threshold function (PTF) $f:\mathbb{R}^n \rightarrow \mathbb{R}$ 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 (PRG) 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 $d^{O(1)}\log ( n / \varepsilon)\log(1/\varepsilon)/\varepsilon^2$ random bits with output that cannot be distinguished from $n$-dimensional gaussian distribution with advantage better than $\varepsilon$ by degree $d$ PTFs. The best previous generator due to O'Donnell, Servedio, and Tan (STOC'20) had a quasi-polynomial dependence (i.e., seedlength 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.


Paper:

TR21-047 | 26th March 2021 01:45

Random restrictions and PRGs for PTFs in Gaussian Space





TR21-047
Authors: Zander Kelley, Raghu Meka
Publication: 26th March 2021 13:15
Downloads: 614
Keywords: 


Abstract:

A polynomial threshold function (PTF) $f:\mathbb{R}^n \rightarrow \mathbb{R}$ 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 (PRG) 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 $d^{O(1)}\log ( n / \varepsilon)\log(1/\varepsilon)/\varepsilon^2$ random bits with output that cannot be distinguished from $n$-dimensional gaussian distribution with advantage better than $\varepsilon$ by degree $d$ PTFs. The best previous generator due to O'Donnell, Servedio, and Tan (STOC'20) had a quasi-polynomial dependence (i.e., seedlength 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.



ISSN 1433-8092 | Imprint