发表机构
Institute of Software, Chinese Academy of Sciences; University of Chinese Academy of Sciences; Department of Computer Science and Technology, Tsinghua University（中国科学院软件研究所; 中国科学院大学; 清华大学计算机科学与技术系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明高斯倾斜下凸体采样协方差膨胀的最优上界为Θ(n^(2/5))，填补了已知下界与上界之间的差距，并构造了达到该下界的显式凸体。
AI 中文摘要
对凸体采样的高斯退火进行协方差敏感分析，需要控制径向高斯倾斜下协方差能增长多少。对于各向同性凸体 $K\subseteq\mathbb{R}^n$，设 $\mu_{K,t} (\mathrm{d} x) \propto e^{-t\\| x \\| ^2} \mathbb{1}_K(x)\\,\mathrm{d} x$，并令 $Q_n$ 为所有此类 $K$ 和所有 $t>0$ 上 $\\|\operatorname{Cov}(\mu_{K,t})\\|_{\mathrm{op}}$ 的上确界。我们证明了尖锐界 $Q_n=\Theta(n^{2/5})$，填补了已知 $\Omega(n^{1/3})$ 下界与 $O(\sqrt{n\log(en)})$ 上界之间的差距。上界不仅适用于凸体上的均匀测度，而且适用于每个紧支撑的各向同性对数凹概率测度。它结合了二次型的无维方差界、在邻近时间的 Rényi 比较、投影矩估计以及沿高斯倾斜路径的相对熵控制。对于匹配的下界，我们构造了一个显式的无条件凸体，其轴向坐标与横向二次能量耦合。中偏差估计表明，适当的倾斜会产生方向方差 $\Omega(n^{2/5})$。
英文摘要
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 $μ_{K,t} (\mathrm{d} x) \propto e^{-t\| x \| ^2} \mathbb{1}_K(x)\,\mathrm{d} x$, and let $Q_n$ be the supremum of $\|\operatorname{Cov}(μ_{K,t})\|_{\mathrm{op}}$ over all such $K$ and all $t>0$. We prove the sharp bound $Q_n=Θ(n^{2/5})$, closing the gap between the known $Ω(n^{1/3})$ 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 $Ω(n^{2/5})$.
Comments28 pages
