Eldan’s stochastic localization and the KLS conjecture: Isoperimetry, concentration and mixing | Annals of Mathematics

IF 5.7 1区 数学 Q1 MATHEMATICS Annals of Mathematics Pub Date : 2024-05-01 DOI:10.4007/annals.2024.199.3.2
Yin Tat Lee, Santosh S. Vempala
{"title":"Eldan’s stochastic localization and the KLS conjecture: Isoperimetry, concentration and mixing | Annals of Mathematics","authors":"Yin Tat Lee, Santosh S. Vempala","doi":"10.4007/annals.2024.199.3.2","DOIUrl":null,"url":null,"abstract":"<p>We analyze the Poincaré and Log-Sobolev constants of logconcave densities in $\\mathbb{R}^{n}$. For the Poincaré constant, we give an improved estimate of $O(\\sqrt{n})$ for any isotropic logconcave density. For the Log-Sobolev constant, we prove a bound of $\\Omega (1/D)$, where $D$ is the diameter of the support of the density, and show that this is asymptotically the best possible bound, resolving a question posed by Frieze and Kannan in 1997. These bounds have several interesting consequences. Improved bounds on the thin-shell and Cheeger/KLS constants are immediate. The ball walk to sample an isotropic logconcave density in $\\mathbb{R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start, and the speedy version of the ball walk, studied by Kannan, Lov\\’aasz and Simonovits mixes in $O^{*}(n^{2}D)$ proper steps from any start, also a tight bound. As another consequence, we obtain a unified and improved large deviation inequality for the concentration of any $L$-Lipshitz function over an isotropic logconcave density (studied by many), generalizing bounds of Paouris and Guedon-E. Milman. Our proof technique is a development of stochastic localization, first introduced by Eldan.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"43 1","pages":""},"PeriodicalIF":5.7000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4007/annals.2024.199.3.2","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

We analyze the Poincaré and Log-Sobolev constants of logconcave densities in $\mathbb{R}^{n}$. For the Poincaré constant, we give an improved estimate of $O(\sqrt{n})$ for any isotropic logconcave density. For the Log-Sobolev constant, we prove a bound of $\Omega (1/D)$, where $D$ is the diameter of the support of the density, and show that this is asymptotically the best possible bound, resolving a question posed by Frieze and Kannan in 1997. These bounds have several interesting consequences. Improved bounds on the thin-shell and Cheeger/KLS constants are immediate. The ball walk to sample an isotropic logconcave density in $\mathbb{R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start, and the speedy version of the ball walk, studied by Kannan, Lov\’aasz and Simonovits mixes in $O^{*}(n^{2}D)$ proper steps from any start, also a tight bound. As another consequence, we obtain a unified and improved large deviation inequality for the concentration of any $L$-Lipshitz function over an isotropic logconcave density (studied by many), generalizing bounds of Paouris and Guedon-E. Milman. Our proof technique is a development of stochastic localization, first introduced by Eldan.

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
埃尔丹的随机局部化和KLS猜想:等时性、浓度和混合 | 数学年鉴
我们分析了$\mathbb{R}^{n}$中对数凹密度的Poincaré常数和Log-Sobolev常数。对于 Poincaré 常数,我们给出了任何各向同性对数凹密度的改进估计值 $O(\sqrt{n})$。对于对数-索波列夫常数,我们证明了一个 $\Omega (1/D)$ 的约束,其中 $D$ 是密度支持的直径,并证明这是渐近可能的最佳约束,解决了 Frieze 和 Kannan 在 1997 年提出的一个问题。这些约束有几个有趣的结果。薄壳常数和切格/KLS 常数的改进界值是立竿见影的。在$\mathbb{R}^{n}$中采样各向同性对数凹密度的球走法,从热起点开始,以$O^{*}(n^{2.5})$步收敛,而由 Kannan、Lov\'aasz 和 Simonovits 研究的球走法的快速版本,从任意起点开始,以$O^{*}(n^{2}D)$适当步混合,也是一个紧约束。作为另一个结果,我们对等向对数凹密度上任何 $L$-Lipshitz 函数的集中得到了一个统一的、改进的大偏差不等式(许多人都研究过),概括了 Paouris 和 Guedon-E.米尔曼。我们的证明技术是对埃尔丹首次提出的随机局部化的发展。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Annals of Mathematics
Annals of Mathematics 数学-数学
CiteScore
9.10
自引率
2.00%
发文量
29
审稿时长
12 months
期刊介绍: The Annals of Mathematics is published bimonthly by the Department of Mathematics at Princeton University with the cooperation of the Institute for Advanced Study. Founded in 1884 by Ormond Stone of the University of Virginia, the journal was transferred in 1899 to Harvard University, and in 1911 to Princeton University. Since 1933, the Annals has been edited jointly by Princeton University and the Institute for Advanced Study.
期刊最新文献
Nonlinear inviscid damping for a class of monotone shear flows in a finite channel | Annals of Mathematics Torsion-free abelian groups are Borel complete | Annals of Mathematics Eldan’s stochastic localization and the KLS conjecture: Isoperimetry, concentration and mixing | Annals of Mathematics Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves | Annals of Mathematics On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes | Annals of Mathematics
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1