到紧支撑对数凹目标的Brenier映射的无维度Lipschitz界
Dimension-Free Lipschitz Bounds for Brenier Maps to Compactly Supported Log-Concave Targets
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中文总结 AI 辅助
该研究针对到紧支撑对数凹目标的Brenier映射，得到无维度的Lipschitz界，消除了Kolesnikov估计中的√d损失，还适用于奇异或低维目标及有界负曲率半对数凹目标。
中文摘要 AI 辅助
我们固定整数$d\ge1$和对称正定矩阵$Q\in\mathbb{R}^{d\times d}$。设$V:\mathbb{R}^d\to\mathbb{R}$为有限值，定义$Z_\mu:=\int_{\mathbb{R}^d}e^{-V(x)}\\,dx\in(0,\infty)$，$d\mu(x):=Z_\mu^{-1}e^{-V(x)}\\,dx$，假设$\mu$具有有限二阶矩，且映射$x\longmapsto \frac12\langle Qx,x\rangle-V(x)$是凸的。设$\nu$为支撑集$K$的紧支撑对数凹概率测度，$\nabla\Phi$为从$\mu$到$\nu$的Brenier映射。对任意$v\in\mathbb{R}^d$，定义$w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle$。我们在分布意义下证明：$\partial_{vv}\Phi \le 0.587 \sqrt{\langle Qv,v\rangle}\\,w_K(v) \quad(v\in\mathbb{R}^d)$；还证明$\nabla\Phi$存在处处定义的全局Lipschitz代表元，满足$\operatorname{Lip}(\nabla\Phi) \le 0.587 \sqrt{\\|Q\\|_{\mathrm{op}}}\\,\operatorname{diam}(K)$。方向Hessian估计具有仿射协变性，全局Lipschitz估计是无维度的，该结果也适用于奇异或低维目标，尤其消除了Kolesnikov关于从高斯测度到凸体上归一化Lebesgue测度的Brenier映射估计中的$\sqrt{d}$损失；我们还证明了仅依赖支撑集的紧支撑半对数凹目标的新估计，包括具有有界负曲率的目标。
英文摘要
We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_μ:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad dμ(x):=Z_μ^{-1}e^{-V(x)}\,dx, \] and assume that $μ$ has finite second moment and that \[ x\longmapsto \frac12\langle Qx,x\rangle-V(x) \] is convex. Let $ν$ be a compactly supported log-concave probability measure with support $K$, and let $\nablaΦ$ be the Brenier map from $μ$ to $ν$. For $v\in\mathbb{R}^d$, define \[ w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle. \] We prove that \[ \partial_{vv}Φ\le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nablaΦ$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nablaΦ) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] The directional Hessian estimate is affinely covariant, whereas the global Lipschitz estimate is dimension-free. The result also applies to singular or lower-dimensional targets. In particular, it removes the $\sqrt d$ loss in Kolesnikov's estimate for the Brenier map from Gaussian measure to normalised Lebesgue measure on a convex body. We also prove new bounds that depend only on the support for compactly supported semi-log-concave targets, which includes targets with bounded negative curvature.
