Abstract
Abstract Using the calculus of variations, we prove the following structure theorem for noise-stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximise their noise stability must be $(m-1)$ -dimensional, if $m-1\leq n$ . In particular, the maximum noise stability of a partition of m sets in $\mathbb {R}^{n}$ of fixed Gaussian volumes is constant for all n satisfying $n\geq m-1$ . From this result, we obtain: (i) A proof of the plurality is stablest conjecture for three candidate elections, for all correlation parameters $\rho $ satisfying $0<\rho <\rho _{0}$ , where $\rho _{0}>0$ is a fixed constant (that does not depend on the dimension n), when each candidate has an equal chance of winning. (ii) A variational proof of Borell’s inequality (corresponding to the case $m=2$ ). The structure theorem answers a question of De–Mossel–Neeman and of Ghazi–Kamath–Raghavendra. Item (i) is the first proof of any case of the plurality is stablest conjecture of Khot-Kindler-Mossel-O’Donnell for fixed $\rho $ , with the case $\rho \to L1^{-}$ being solved recently. Item (i) is also the first evidence for the optimality of the Frieze–Jerrum semidefinite program for solving MAX-3-CUT, assuming the unique games conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the plurality is stablest conjecture is known to be false.
Highlights
The structure theorem answers a question of De–Mossel–Neeman and of Ghazi–Kamath–Raghavendra
One could imagine that malicious third parties or miscounting of votes might cause random vote changes, so we desire a voting method f whose output is stable to such changes
In addition to voting motivations, finding a voting method that is stable to noise has applications to the unique games conjecture [KKMO07, MOO10, KM16], to semidefinite programming algorithms such as MAX-CUT [KKMO07, IM12], to learning theory [FGRW12], etc
Summary
Using a generalization of the central limit theorem known as the invariance principle [MOO10, IM12], there is an equivalence between the discrete problem of Conjecture 1.2 and a continuous problem known as the standard simplex conjecture [IM12]. Define the Ornstein–Uhlenbeck operator with correlation ρ applied to f by. We define the noise stability of the set Ω with correlation ρ to be. Maximising the noise stability of a Euclidean partition is the continuous analogue of finding a voting method that is most stable to random corruption of votes among voting methods where each voter has a small influence on the election’s outcome. In the remaining case that ai = 1/m for all 1 ≤ i ≤ m, it is assumed that w = 0 in Conjecture 1.6.
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