Eldan的随机局部化和KLS超平面猜想:一种改进的扩展下界

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) Pub Date : 2016-12-05 DOI:10.1109/FOCS.2017.96

Y. Lee, S. Vempala

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引用次数: 90

摘要

我们证明了n维各向同性对数凹测量的KLS常数为O(n^{1/4})，改进了当前的最佳界O(n^{1/3}√{\log n})。作为推论，我们得到了薄壳估计的改进界、庞加莱常数和lipschitz浓度常数以及各向同性常数的改进界的另一种证明;也可以得出，从\R^{n}的各向同性对数凹密度中采样的球行走从温暖开始在O^{*}(n^{2.5})步内收敛。

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Eldan's Stochastic Localization and the KLS Hyperplane Conjecture: An Improved Lower Bound for Expansion

We show that the KLS constant for n-dimensional isotropic logconcavemeasures is O(n^{1/4}), improving on the current best bound ofO(n^{1/3}√{\log n}). As corollaries we obtain the same improvedbound on the thin-shell estimate, Poincar\e constant and Lipschitzconcentration constant and an alternative proof of this bound forthe isotropic constant; it also follows that the ball walk for samplingfrom an isotropic logconcave density in \R^{n} converges in O^{*}(n^{2.5})steps from a warm start.

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2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

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