[Paper Review] Spectral gap for spherically symmetric log-concave probability measures, and beyond

Michel Bonnefont, Aldéric Joulin+1 more|arXiv (Cornell University)|Jun 18, 2014

Point processes and geometric inequalities14 references4 citations

TL;DR

This paper establishes sharp upper and lower bounds for the spectral gap of spherically symmetric log-concave probability measures on $\mathbb{R}^n$, showing it lies between $\frac{n-1}{\int \|x\|^2 \, d\mu}$ and $\frac{n}{\int \|x\|^2 \, d\mu}$, improving Bobkov's earlier estimate. The approach uses a Markovian framework with radial weights and weighted Poincaré inequalities, extending beyond log-concave measures via modified dynamics.

ABSTRACT

Let $μ$ be a probability measure on $ r^n$ ($n \geq 2$) with Lebesgue density proportional to $e^{-V (\Vert x\Vert )}$, where $V : r_+ o r$ is a smooth convex potential. We show that the associated spectral gap in $L^2 (μ)$ lies between $(n-1) / \int_{ r^n} \Vert x\Vert ^2 μ(dx)$ and $n / \int_{ r^n} \Vert x\Vert ^2 μ(dx)$, improving a well-known two-sided estimate due to Bobkov. Our Markovian approach is remarkably simple and is sufficiently robust to be extended beyond the log-concave case, at the price of potentially modifying the underlying dynamics in the energy, leading to weighted Poincaré inequalities. All our results are illustrated by some classical and less classical examples.

Motivation & Objective

• To improve the two-sided spectral gap estimate for spherically symmetric log-concave probability measures on $\mathbb{R}^n$.
• To provide dimension-dependent bounds that capture the exact asymptotic behavior of the spectral gap as $n \to \infty$.
• To extend the method beyond the log-concave class by introducing weighted diffusion operators with modified dynamics.
• To derive explicit quantitative estimates for classical examples, including the Gaussian measure under different radial weights.

Proposed method

• A Markovian approach is employed using a diffusion generator $-\mathcal{L}_\mu$ with drift $-\nabla V \cdot \nabla f$, where $V$ is radial and convex.
• The spectral gap is analyzed via a radial transformation, reducing the problem to a one-dimensional Schrödinger-type operator on $[0, \infty)$.
• A weighted Poincaré inequality is introduced using a radial weight function $\sigma^2(r)$, leading to a modified generator $\mathcal{L}_\mu^\sigma$.
• The key technique involves bounding the effective potential $\mathcal{V}_\nu^\sigma(r)$ in the radial Fokker-Planck equation to derive lower bounds on the spectral gap.
• Two specific weights are analyzed: $\sigma^2(r) = 1 + r^2$ and $\sigma^2(r) = (1 + r^2)^{-1}$, yielding different asymptotic behaviors.
• Lemmas are applied to estimate the spectral gap via integrals over the radial measure, leveraging moment estimates and comparison inequalities.

Experimental results

Research questions

• RQ1 What are the optimal upper and lower bounds for the spectral gap of spherically symmetric log-concave probability measures in terms of the second moment?
• RQ2 How does the spectral gap scale with dimension $n$ for such measures, and can the asymptotic behavior be precisely captured?
• RQ3 Can the method be extended beyond the log-concave class by modifying the underlying diffusion dynamics?
• RQ4 What are the implications of these bounds for the KLS conjecture and concentration of measure?
• RQ5 How do different radial weights affect the spectral gap in weighted Poincaré inequalities?

Key findings

• The spectral gap $\lambda_1(-\mathcal{L}_\mu)$ for spherically symmetric log-concave measures satisfies $\frac{n-1}{\int \|x\|^2 \, d\mu} \leq \lambda_1(-\mathcal{L}_\mu) \leq \frac{n}{\int \|x\|^2 \, d\mu}$, improving Bobkov's bound by a constant factor.
• For the standard Gaussian measure with weight $\sigma^2(x) = 1 + \|x\|^2$, the spectral gap of the weighted operator satisfies $n-1 \leq \lambda_1(-\mathcal{L}_\mu^\sigma) \leq n+1$, with asymptotic behavior $\approx n$ as $n \to \infty$.
• For the weight $\sigma^2(x) = (1 + \|x\|^2)^{-1}$, the spectral gap satisfies $\frac{n-1}{n(n+3)} \leq \lambda_1(-\mathcal{L}_\mu^\sigma) \leq \frac{1}{n-2} \land 1$, indicating $\approx 1/n$ scaling in high dimensions.
• The method successfully extends beyond log-concave measures by introducing weighted dynamics, yielding new weighted Poincaré inequalities.
• The analysis confirms that the KLS conjecture holds for spherically symmetric log-concave measures with identity covariance, as the spectral gap is uniformly bounded below by a universal constant.