论文标题
通过Wasserstein距离保存结构
Structure preservation via the Wasserstein distance
论文作者
论文摘要
我们表明，在随机向量上的最小假设下，$ x \ in \ mathbb {r}^d $，并且具有很高的概率，并且给定$ m $独立副本的$ x $，每个向量$的坐标分布（\ langle x_i，θ\ rangle）_ {i = 1}^m $是由MARG的分布。 x，θ\ rangle $。具体而言，我们以较高的概率表明，\ [\ sup_ {θ\ in s^{d -1}}} \ left（\ frac {1} {1} {m} {m} \ sum_ {i = 1}^m \ left | \ left | \ langle | \ langle x_i，θ\ rangle^\ rangle^\ rang^\ rang^\ sapr -lang -lang -lang -lang -lang -λ^^^2^2 \ leq c \ left（\ frac {d} {m} \ right） }^{ - 1}（u）\，du $和$ a^\ sharp $表示$ a $的单调非压制重排，此估计是最佳的。 证明是从$ x $的边际距离和其经验对应物之间的最差瓦斯尔斯坦距离的尖锐估计之后进行的，$ \ frac {1} {m} {m} \ sum_ {i = 1}^mΔ_{\ langle x_i，θ\ rangle} $。
We show that under minimal assumptions on a random vector $X\in\mathbb{R}^d$ and with high probability, given $m$ independent copies of $X$, the coordinate distribution of each vector $(\langle X_i,θ\rangle)_{i=1}^m$ is dictated by the distribution of the true marginal $\langle X,θ\rangle$. Specifically, we show that with high probability, \[\sup_{θ\in S^{d-1}} \left( \frac{1}{m}\sum_{i=1}^m \left|\langle X_i,θ\rangle^\sharp - λ^θ_i \right|^2 \right)^{1/2} \leq c \left( \frac{d}{m} \right)^{1/4},\] where $λ^θ_i = m\int_{(\frac{i-1}{m}, \frac{i}{m}]} F_{ \langle X,θ\rangle }^{-1}(u)\,du$ and $a^\sharp$ denotes the monotone non-decreasing rearrangement of $a$. Moreover, this estimate is optimal. The proof follows from a sharp estimate on the worst Wasserstein distance between a marginal of $X$ and its empirical counterpart, $\frac{1}{m} \sum_{i=1}^m δ_{\langle X_i, θ\rangle}$.
