Standard simplices and pluralities are not the most noise stable
Standard simplices and pluralities are not the most noise stable
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标准单纯形和复数并不是最稳定的噪声
DOI：
10.1145/2688073.2688076
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发表时间：
2014
影响因子：
1
通讯作者：
Joe Neeman
中科院分区：
文献类型：
--
作者：
Steven M. Heilman;Elchanan Mossel;Joe Neeman
标准的单纯猜想和多数是最稳定的猜想是两个猜想，表明某些分区对于高斯和离散的噪声稳定性是最佳的。最稳定定理（2004）。噪声的0和每个规定的测量都不等于1/k，我们的结果并不矛盾。但是，相等的测量集。在等等理论中，鉴于我们的结果很自然地要求（猜想）分区实现最佳的噪声稳定性。
The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k ≥ 3, for every value ρ ≠ 0 of the noise and for every prescribed measure for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell’s result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.