We introduce a new technique for reducing the dimension of the ambient space of lowdegree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilonoptimal noisestable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilonapproximate any joint distribution that can be noninteractively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermannlike to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap noninteractive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for lowdegree polynomials is simple and can be seen as a generalization of the JohnsonLindenstrauss lemma and could be of independent interest.
Dimension Reduction for Polynomials over Gaussian Space and Applications
We introduce a new technique for reducing the dimension of the ambient space of lowdegree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilonoptimal noisestable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilonapproximate any joint distribution that can be noninteractively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermannlike to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap noninteractive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for lowdegree polynomials is simple and can be seen as a generalization of the JohnsonLindenstrauss lemma and could be of independent interest.
 Award ID(s):
 1741137
 Publication Date:
 NSFPAR ID:
 10075656
 Journal Name:
 33rd Computational Complexity Conference, CCC 2018
 Page Range or eLocationID:
 281  2837
 Sponsoring Org:
 National Science Foundation
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We introduce a new technique for reducing the dimension of the ambient space of lowdegree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilonoptimal noisestable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilonapproximate any joint distribution that can be noninteractively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermannlike to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap noninteractive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for lowdegree polynomials is simple and can be seen as a generalization of the JohnsonLindenstrauss lemma and could be of independent interest.

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