[Paper Review] Eldan's Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing

Yin Tat Lee, Santosh Vempala|arXiv (Cornell University)|Dec 5, 2016

Stochastic processes and statistical mechanics42 references4 citations

TL;DR

This paper establishes a tight $ O(n^{1/4}) $ bound on the Cheeger constant for isotropic logconcave measures in $ \mathbb{R}^n $, significantly improving the prior $ O(n^{1/3}\sqrt{\log n}) $ bound. The result is achieved via Eldan's stochastic localization, a martingale-based transformation that gradually introduces a Gaussian factor, and leads to improved bounds on the thin-shell estimate, Poincaré and log-Sobolev constants, and mixing times for the ball walk algorithm.

ABSTRACT

We show that the Cheeger constant for $n$-dimensional isotropic logconcave measures is $O(n^{1/4})$, improving on the previous best bound of $O(n^{1/3}\sqrt{\log n}).$ As corollaries we obtain the same improved bound on the thin-shell estimate, Poincaré constant and Lipschitz concentration constant and an alternative proof of this bound for the isotropic (slicing) constant; it also follows that the ball walk for sampling from an isotropic logconcave density in ${\bf R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start. The proof is based on gradually transforming any logconcave density to one that has a significant Gaussian factor via a Martingale process. Extending this proof technique, we prove that the log-Sobolev constant of any isotropic logconcave density in ${\bf R}^{n}$ with support of diameter $D$ is $Ω(1/D)$, resolving a question posed by Frieze and Kannan in 1997. This is asymptotically the best possible estimate and improves on the previous bound of $Ω(1/D^{2})$ by Kannan-Lovász-Montenegro. It follows that for any isotropic logconcave density, the ball walk with step size $δ=Θ(1/\sqrt{n})$ mixes in $O\left(n^{2}D ight)$ proper steps from \emph{any }starting point. This improves on the previous best bound of $O(n^{2}D^{2})$ and is also asymptotically tight. The new bound leads to the following large deviation inequality for an $L$-Lipschitz function $g$ over an isotropic logconcave density $p$: for any $t>0$, \[ Pr_{x\sim p}\left(\left|g(x)-\bar{g} ight|\geq L\cdot t ight)\leq\exp(-\frac{c\cdot t^{2}}{t+\sqrt{n}}) \] where $\bar{g}$ is the median or mean of $g$ for $x\sim p$; this generalizes and improves on previous bounds by Paouris and by Guedon-Milman. The technique also bounds the ``small ball'' probability in terms of the Cheeger constant, and recovers the current best bound.

Motivation & Objective

• To improve the best-known upper bound on the Cheeger constant for isotropic logconcave measures in $ \mathbb{R}^n $.
• To establish tighter bounds on the thin-shell estimate, Poincaré constant, and Lipschitz concentration constant.
• To resolve Frieze and Kannan's 1997 question on the log-Sobolev constant, showing it is $ \Omega(1/D) $ for measures with diameter $ D $.
• To improve mixing time bounds for the ball walk algorithm from $ O^*(n^2 D^2) $ to $ O(n^2 D) $, which is asymptotically tight.
• To derive a new large deviation inequality for Lipschitz functions with improved tail decay.

Proposed method

• Employing Eldan's stochastic localization, a continuous martingale process that gradually transforms a logconcave density into one with a significant Gaussian factor.
• Using a recursive localization framework where each step reduces the anisotropy of the measure, leveraging moment bounds on quadratic forms.
• Applying a decomposition of covariance operators into dyadic spectral projections to control higher-order moments.
• Establishing a bound on the operator norm of the localization process via a concentration argument, showing $ \mathbb{P}(\max_t \|A_t\|_{\mathrm{op}} \geq 2) \leq 2\exp(-1/(cT)) $ for $ T \leq 1/(c k (\log n)^{1-1/k} n^{1/k}) $.
• Revisiting and refining Lemma 39 from Eldan's original work to reduce the $ \sqrt{\log n} $ loss factor to $ (\log n)^{1/4} $ when $ k=2 $.
• Deriving a new large deviation inequality: $ \mathbb{P}(|g(x) - \bar{g}| \geq L t) \leq \exp(-c t^2 / (t + \sqrt{n})) $, which generalizes and improves prior results by Paouris and Guedon-Milman.

Experimental results

Research questions

• RQ1 Can the Cheeger constant of isotropic logconcave measures be bounded more tightly than $ O(n^{1/3}\sqrt{\log n}) $?
• RQ2 What is the optimal dependence of the log-Sobolev constant on the diameter $ D $ of the support of a logconcave measure?
• RQ3 Can the mixing time of the ball walk for sampling from isotropic logconcave densities be improved beyond $ O(n^2 D^2) $ steps?
• RQ4 Does Eldan's stochastic localization technique allow for a tighter analysis of the KLS conjecture by reducing the logarithmic loss factor?
• RQ5 Can a new large deviation inequality be derived that captures both sub-Gaussian and sub-Gaussian-like tails in a unified way?

Key findings

• The Cheeger constant for any isotropic logconcave measure in $ \mathbb{R}^n $ is bounded by $ O(n^{1/4}) $, improving the previous best bound of $ O(n^{1/3}\sqrt{\log n}) $.
• The thin-shell estimate, Poincaré constant, and Lipschitz concentration constant are all bounded by $ O(n^{1/4}) $.
• The log-Sobolev constant of any isotropic logconcave density with support diameter $ D $ is $ \Omega(1/D) $, resolving a 1997 question by Frieze and Kannan.
• The ball walk with step size $ \delta = \Theta(1/\sqrt{n}) $ mixes in $ O(n^2 D) $ proper steps from any starting point, improving the prior bound of $ O(n^2 D^2) $.
• The new large deviation inequality shows that for any $ L $-Lipschitz function $ g $, $ \mathbb{P}(|g(x) - \bar{g}| \geq L t) \leq \exp(-c t^2 / (t + \sqrt{n})) $, which generalizes and improves on earlier bounds.
• The method recovers the current best bound on small-ball probabilities in terms of the Cheeger constant, confirming its optimality in the asymptotic regime.