Dimension Reduction for Polynomials over Gaussian Space and Applications

TL;DR

This paper introduces a novel dimension reduction technique for low-degree polynomials in Gaussian space, enabling improved bounds and decidability results for noise stability and non-interactive simulation problems.

Contribution

It develops a new dimension reduction method for polynomials in Gaussian space, leading to explicit bounds and improved results in noise stability and simulation problems.

Findings

Improved explicit bounds on dimension for noise stable partitions.

Enhanced bounds for non-interactive simulation of joint distributions.

A new technique analogous to Johnson-Lindenstrauss lemma for Gaussian polynomials.

Abstract

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 $n_0(\delta)$.