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, analogous to the Johnson-Lindenstrauss lemma. As applications, we address the following problems: 1. Computability of Approximately Optimal Noise Stable function over Gaussian space: The goal is to find a partition of $\mathbb{R}^n$ into $k$ parts, that maximizes the noise stability. An $\delta$-optimal partition is one which is within additive $\delta$ of the optimal noise stability. De, Mossel & Neeman (CCC 2017) raised the question of proving a computable bound on the dimension $n_0(\delta)$ in which we can find an $\delta$-optimal partition. While De et al. provide such a bound, using our new technique, we obtain improved explicit bounds on the dimension $n_0(\delta)$. 2. Decidability of Non-Interactive Simulation of Joint Distributions: A "non-interactive simulation" problem is specified by two distributions $P(x,y)$ and $Q(u,v)$: The goal is to determine if two players that observe sequences $X^n$ and $Y^n$ respectively where $\{(X_i, Y_i)\}_{i=1}^n$ are drawn i.i.d. from $P(x,y)$ can generate pairs $U$ and $V$ respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to $Q(u,v)$. Even when $P$ and $Q$ are extremely simple, it is open in several cases if $P$ can simulate $Q$. In the special where $Q$ is a joint distribution over $\{0,1\} \times \{0,1\}$, Ghazi, Kamath and Sudan (FOCS 2016) proved a computable bound on the number of samples $n_0(\delta)$ that can be drawn from $P(x,y)$ to get $\delta$-close to $Q$ (if it is possible at all). Recently De, Mossel & Neeman obtained such bounds when $Q$ is a distribution over $[k] \times [k]$ for any $k \ge 2$. We recover this result with improved explicit bounds on $n_0(\delta)$.