Noise Stability is computable and low dimensional

TL;DR

This paper proves that near-optimal partitions for noise stability in Gaussian space can be found in low-dimensional spaces, making the problem computationally feasible and impacting related areas like non-interactive simulation.

Contribution

It establishes an explicit, computable bound on the dimension needed to find near-optimal noise stability partitions in Gaussian space.

Findings

Existence of an explicit function n(ε) for low-dimensional ε-optimal partitions

Implications for the computability of non-interactive simulation problems

Reduction of high-dimensional optimization to low-dimensional settings

Abstract

Questions of noise stability play an important role in hardness of approximation in computer science as well as in the theory of voting. In many applications, the goal is to find an optimizer of noise stability among all possible partitions of $\mathbb{R}^n$ for $n \geq 1$ to $k$ parts with given Gaussian measures $\mu_1,\ldots,\mu_k$. We call a partition $\epsilon$-optimal, if its noise stability is optimal up to an additive $\epsilon$. In this paper, we give an explicit, computable function $n(\epsilon)$ such that an $\epsilon$-optimal partition exists in $\mathbb{R}^{n(\epsilon)}$. This result has implications for the computability of certain problems in non-interactive simulation, which are addressed in a subsequent work.