Analysis Seminar

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A Two-Sided Estimate for the Gaussian Noise Stability Deficit

Speaker: Ronen Eldan

Location: Warren Weaver Hall 1302

Date: Thursday, November 21, 2013, 11 a.m.

Synopsis:

The Gaussian Noise Stability of a set \(A \subset \mathbb{R}^n\) with parameter \(0 <\rho < 1\) is defined as $$ S_\rho(A) = \mathbb{P}(X,Y \in A) $$ where \(X,Y\) are jointly Gaussian random vectors such that \(X\) and \(Y\) are standard Gaussian vectors and \(\mathbb{E} [ X_i Y_j ] = \delta_{ij} \rho\). Borell's celebrated noise stability inequality states that if \(H\) is a half-space whose Gaussian measure is equal to that of \(A\), then \(S_\rho(H) \geq S_\rho (A)\) for all \(0 < \rho < 1\).

We will present a novel short proof of this inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set \(A\) and its corresponding half-space \(H\) (namely the distance between the two centroids), we show that the deficit between the noise stability of \(A\) and \(H\) can be controlled from both below and above by essentially the same function of the distance, up to logarithmic factors.

As a consequence, we also manage to get the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variationdistance as a metric. Moreover, in the limit \(\rho \to 1\), we get an improved dimension free robustness bound for the Gaussian isoperimetricinequality.