A Two-Sided Estimate for the Gaussian Noise Stability Deficit

Speaker  Ronen Eldan

Affiliation  Microsoft

Host  David Wilson

Duration  00:43:21

Date recorded  20 October 2013

The Gaussian Noise Stability of a set A in Euclidean space is the probability that for a Gaussian vector X conditioned to be in A, a small Gaussian perturbation of X will also be in A. Borel's celebrated Isoperimetric inequality states that a half-space maximizes noise stability among sets with the same Gaussian measure. 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: We show that the deficit between the noise stability of a set A and an equally probable half-space H can be controlled by a function of the distance between the corresponding centroids. As a consequence, we prove a conjecture by Mossel and Neeman, who used the total-variation distance.

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