We propose a new statistical model, the spiked transport model, which
formalizes the assumption that two probability distributions differ only on a
low-dimensional subspace. We study the minimax rate of estimation for the
Wasserstein distance under this model and show that this low-dimensional
structure can be exploited to avoid the curse of dimensionality. As a byproduct
of our minimax analysis, we establish a lower bound showing that, in the
absence of such structure, the plug-in estimator is nearly rate-optimal for
estimating the Wasserstein distance in high dimension. We also give evidence
for a statistical-computational gap and conjecture that any computationally
efficient estimator is bound to suffer from the curse of dimensionality.

本文提出了一种新的统计模型 —— 尖峰运输模型，该模型规范化了两个概率分布仅在低维子空间上不同的假设。我们研究了在这个模型下 Wasserstein 距离的最小二乘率，并表明这种低维结构可以避免维度灾难。通过最小二乘分析，我们得出了一个下界，表明在缺少这样的结构的情况下，插值估计量在高维度中几乎是最优的。我们还提供了统计和计算难度之间的差距的证据，并猜测任何计算上有效的估计量注定受到维数灾难的影响。