发表机构
Beihang University（北京航空航天大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对最优传输问题，在温和密度与矩条件下，建立了可行计划与最优映射间距离的尖锐模量，并据此直接推导出Brenier映射的定量稳定性，改进现有结果。
AI 中文摘要
当获得一个可行计划，其二次传输成本已知接近最优成本时，我们尝试在温和的密度和矩条件下确定该可行计划与真实最优映射的接近程度。源和目标可以是无界的或具有非紧支撑。我们证明，对于具有 $L^p$ 密度和 $n$ 阶矩的源，令 $s=1-1/p$ 且 $\eta=sn/[n+(d-1)s]$。在目标约束 $\int |y|^m[\log(e+|y|)]^\beta\\,d\nu\leqslant1$ 下，最坏情况下的类均匀条件性具有一个尖锐的模量，当 $t>0$ 较小时，该模量可比较于 \\[ t^{\eta(m-2)/[m(1+\eta)-\eta]} [\log(e/t)]^{-\beta(2+\eta)/[m(1+\eta)-\eta]}, \\] 此结果对 $m\geqslant2$ 和 $\beta\geqslant0$ 成立。当 $m=2$ 时，每个 $\beta>0$ 都给出一个尖锐的对数模量，而 $m=2,\\,\beta=0$ 的极端情况没有消失模量。对于平方映射误差和重心投影误差，均验证了匹配的下界。尖锐的映射误差界使我们能够直接推导常见源 Brenier 映射的定量稳定性，而无需通过势估计，从而避免信息损失。这些稳定性在相同或更弱的设置下严格改进了 Delalande-Merigot \cite{delalande2023quantitative} 和 Letrouit-Mérigot \cite{letrouit2026gluing} 的相应结果，特别是没有假设源支撑的凸性或源密度的有界性。
英文摘要
When a feasible plan is obtained whose quadratic transport cost is known to be close to the optimal cost, we try to determine how close the feasible plan is to the true optimal map under mild density and moment conditions. The source and target may be unbounded or have non-compact supports. We show that for a source with density in $L^p$ and an $n$-th moment, let $s=1-1/p$ and $η=sn/[n+(d-1)s]$. Under the target constraint $\int |y|^m[\log(e+|y|)]^β\,dν\leqslant1$, the worst-case class-uniform conditioning has a sharp modulus comparable to \[ t^{η(m-2)/[m(1+η)-η]} [\log(e/t)]^{-β(2+η)/[m(1+η)-η]}, \] whenever $t>0$ is small. This holds for $m\geqslant2$ and $β\geqslant0$. When $m=2$, every $β>0$ gives a sharp logarithmic modulus, whereas the $m=2,\,β=0$ extreme has no vanishing modulus. The matching lower bound is verified for both the squared map error and barycentric projection error. The sharp map error bounds allow us to directly derive quantitative stabilities of common source Brenier maps without passing through potential estimates, thereby avoiding loss of information. These stabilities strictly improve the corresponding results of Delalande-Merigot \cite{delalande2023quantitative} and Letrouit-Mérigot \cite{letrouit2026gluing} under identical or weaker settings, particularly no convexity of the source support or bounds of the source density are assumed.
