论文标题
半分散瓦斯坦距离的中央限制定理
Central Limit Theorems for Semidiscrete Wasserstein Distances
论文作者
论文摘要
我们证明了经验最佳运输成本的中心限制定理，$ \ sqrt {\ frac {nm} {n+m}}} \ {\ Mathcal {\ Mathcal {t} _C（p_n，q_m） - \ natercal {t} _c（p，q） $ n $点，但没有$ q $的假设。我们表明，渐近分布是一个集中的高斯流程的至上，在概率$ q $和成本的一些其他条件下，高斯是高斯。这样的结果暗示了$ p $ -wassertein距离的中心限制定理，$ p \ geq 1 $。这意味着，对于固定的$ n $，避免了维数的诅咒。为了更好地理解这种$ n $的影响，我们提供了$ e | \ Mathcal {w} _1（p，q_m）的界限 - \ Mathcal {w} _1 _1（p，q）| $，取决于$ m $和$ n $。最后，半分化框架提供了对双重公式的第二个衍生物的控制，这产生了最佳运输电位的第一个中心极限定理。结果由模拟支持，有助于可视化给定的限制和边界。我们还分析了经典自举的情况。
We prove a Central Limit Theorem for the empirical optimal transport cost, $\sqrt{\frac{nm}{n+m}}\{\mathcal{T}_c(P_n,Q_m)-\mathcal{T}_c(P,Q)\}$, in the semi discrete case, i.e when the distribution $P$ is supported in $N$ points, but without assumptions on $Q$. We show that the asymptotic distribution is the supremun of a centered Gaussian process, which is Gaussian under some additional conditions on the probability $Q$ and on the cost. Such results imply the central limit theorem for the $p$-Wassertein distance, for $p\geq 1$. This means that, for fixed $N$, the curse of dimensionality is avoided. To better understand the influence of such $N$, we provide bounds of $E|\mathcal{W}_1(P,Q_m)-\mathcal{W}_1(P,Q)|$ depending on $m$ and $N$. Finally, the semidiscrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials. The results are supported by simulations that help to visualize the given limits and bounds. We analyse also the cases where classical bootstrap works.
