发表机构
The University of Tampa; Bowling Green State University（坦帕大学; 鲍灵格林州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对高斯空间中带噪配对数据，提出柱状筛估计最优传输映射，通过有限维经验风险最小化实现，并建立了非渐近oracle不等式和极小极大下界，证明了在正则性条件下达到最优收敛速率。
AI 中文摘要
我们考虑在具有高斯参考测度的无限维希尔伯特空间中，从带噪声的配对观测数据估计最优传输映射的问题。与独立非配对样本不同，我们观测到源样本及其图像的带噪声评估。估计量是限制在紧参数集上的Cameron-Martin梯度映射的柱状筛；单个筛元素不必是传输映射。尽管噪声具有无限Cameron-Martin范数，该方法仍产生有限回归对比度，并将估计问题简化为有限维经验风险最小化。我们建立了非渐近oracle不等式，将逼近误差、随机误差和参数化的局部条件分开，并在加权坐标正则性阶数s下给出了阶为$N^{-s/(2s+1)}$的极小极大下界。仅输出正则性不足以实现柱状逼近；对于一般Sobolev势，输入正则性指数通过条件高斯Poincaré论证实现，而对于有界混沌度的势，度界起该作用，正交Hermite筛以交互阶替换指数中的1达到相同速率。该速率在对角高斯类和非线性块类上是极小极大的，其中交互在每次固定正交坐标变换下仍然存在。
英文摘要
We study the estimation of infinite-dimensional optimal transport maps from noisy paired observations. The population map pushes a Gaussian reference measure forward to a target probability measure on a function space and takes the Cameron--Martin gradient form $T=I+\nabla_{\mathcal H}ϕ$. Our estimator uses cylindrical gradient sieves based on finitely many Cameron--Martin coordinates, thereby reducing the problem to finite-dimensional empirical risk minimization. A local nonasymptotic oracle inequality separates cylindrical approximation and statistical estimation errors while accounting for the conditioning of the parametrization. The approximation analysis relies on regularity conditions governing coordinate decay and dependence on omitted input coordinates. For diagonal Gaussian and nonlinear block-interaction classes, we derive matching upper and lower bounds that establish the minimax rate $N^{-s/(2s+1)}$ in expected norm, where $s$ measures weighted coordinate regularity. We further analyze a continuous two-groups model with Gaussian--Laplace mixtures and derive a prediction-risk bound for the resulting transport-map estimator.
