Dimension Reduction for Polynomials over Gaussian Space and Applications

We introduce a new technique for reducing the dimension of the ambient space of low-degree 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 ε-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to ε-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like 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 non-interactive simulation problem posed by Ghazi, Kamath & Sudan (FOCS 2016). Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest. ∗MIT, Supported in parts by NSF CCF-1650733 and CCF-1420692. Email: [email protected] †MIT. Supported in parts by NSF CCF-1420956, CCF-1420692, CCF-1218547 and CCF-1650733. Email: [email protected] ‡UC Berkeley. Research supported by Okawa Research Grant and NSF CCF-1408643. Email: [email protected]