We consider the problem of estimating the optimal transport map between a (fixed) source distribution $P$ and an unknown target distribution $Q$, based on samples from $Q$. The estimation of such optimal transport maps has become increasingly relevant in modern statistical applications, such as generative modeling. At present, estimation rates are only known in a few settings (e.g. when $P$ and $Q$ have densities bounded above and below and when the transport map lies in a H\"older class), which are often not reflected in practice. We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure $P$ satisfies a Poincar\'e inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for bounded densities and H\"older transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when $P$ is the normal distribution and the transport map is given by an infinite-width shallow neural network.
翻译:我们考虑了在(固定)来源分配量(美元)和不明目标分配量(美元)之间估计最佳运输图的问题,根据来自Q$的样本,估算出最佳运输图在(固定)来源分配量(美元)和未知目标分配量(美元)之间(美元)之间的最佳运输图的问题。对此类最佳运输图的估计在现代统计应用(例如基因模型)中已变得日益重要。目前,估计率只在少数情况下才为人所知(例如,美元和美元具有高于和低于基准的密度,而运输图位于H\“older”级),而实际中往往没有反映这一点。我们提出了一个统一的方法,以获得一般功能空间最佳运输图的估计率。我们的假设比文献中出现的假设要弱得多:我们仅要求源计量量(美元)满足Pincar\'e的不平等,而最佳的地图是光滑的连接函数的梯度,它位于一个可以控制矩阵的空域。作为特殊的例子,我们回收了已知的封闭密度密度和H\“older运输图”的估算率,但在许多环境中也取得了不为先前工作覆盖的精确的结果。例如,我们给出的正常的正常的网络的统计比率。