Optimal Covariance Inflation under Gaussian Tilts

摘要

Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body $K\subseteq\mathbb{R}^n$, let $\mu_{K,t} (\mathrm{d} x) \propto e^{-t| x | ^2} \mathbb{1}K(x),\mathrm{d} x$, and let $Q_n$ be the supremum of $|\operatorname{Cov}(\mu{K,t})|_{\mathrm{op}}$ over all such $K$ and all $t>0$. We prove the sharp bound $Q_n=\Theta(n^{2/5})$, closing the gap between the known $\Omega(n^{1/3})$ lower bound and the $O(\sqrt{n\log(en)})$ upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance $\Omega(n^{2/5})$.