Regularization under diffusion and anti-concentration of the information content

Abstract

Under the Ornstein-Uhlenbeck semigroup $\{U_t\}$, any non-negative measurable $f:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ exhibits a uniform tail bound better than that implied by Markov's inequality and conservation of mass: For every $\alpha \ge e^3$, and $t>0$, \[ γ_n\left(\left\{x \in \mathbb R^n : U_t f(x) > α\int f\,dγ_n\right\}\right) \leq C(t) \frac{1}α \sqrt{\frac{\log \log α}{\log α}}\] where ${\gamma }_n$ is the $n$-dimensional Gaussian measure and $C(t)$ is a constant depending only on $t$. This confirms positively the Gaussian limiting case of Talagrand's convolution conjecture (1989). This is shown to follow from a more general phenomenon. Suppose that $f:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ is {\em semi-log-convex} in the sense that for some $\beta >0$, for all $x\in {\mathbb{R}}^n$, the eigenvalues of ${\nabla }^2\log f(x)$ are at least $-\beta$. Then $f$ satisfies a tail bound asymptotically better than that implied by Markov's inequality.

The bound is improved and the proof have been significantly simplified

Cited by in corpus (10)

• Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit
• Theoretical guarantees for sampling and inference in generative models with latent diffusions
• Stability of the Shannon-Stam inequality via the Föllmer process
• Deviation inequalities for convex functions motivated by the Talagrand conjecture
• Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $L^1$
• Regularization in $L_1$ for the Ornstein--Uhlenbeck semigroup
• Regularized Brascamp--Lieb inequalities
• Stability of Talagrand's Gaussian transport-entropy inequality via the Föllmer process
• $L^{\infty }$-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model
• Semi-log-convexity of $\mathrm{M}/\mathrm{M}/\infty$ queues on ${\mathbb{Z}}_{+}$