arXiv 2101.040392021-01math.STstat.MLstat.TH

光滑$p$ -Wasserstein距离：结构、经验近似和统计应用

Smooth $p$-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications

Sloan Nietert, Ziv Goldfeld, Kengo Kato

备注：: updated to match ICML 2021 paper

摘要：

概率分布之间的离散度量，通常称为统计距离，在概率论、统计学和机器学习中无处不在。为了在从数据估计这些距离时对抗维数灾难，最近的工作提出了通过与高斯核的卷积来平滑测量分布中的局部不规则性。出于这个框架的可扩展性的高维，我们调查的高斯平滑$p$ -Wasserstein距离$\mathsf{W}_p^{(σ)}$的结构和统计行为，为任意$p\geq 1$。在建立了$\mathsf{W}_p^{(σ)}$的基本度量和拓扑性质之后，我们研究了$\mathsf{W}_p^{(σ)}(\hatμ_n,μ)$的渐近统计行为，其中$\hatμ_n$是$μ$的$n$独立观测值的经验分布.我们证明了$\mathsf{W}_p^{(σ)}$具有$n^{-1/2}$的参数经验收敛速度，这与当$d \geq 3$时未平滑的$\mathsf{W}_p$的$n^{-1/d}$速度相反。我们的证明依赖于控制$\mathsf{W}_p^{(σ)}$的$p$第三阶光滑Sobolev距离$\mathsf{d}_p^{(σ)}$和推导的极限分布的$\sqrt{n}\,\mathsf{d}_p^{(σ)}(\hatμ_n,μ)$，所有尺寸$d$。作为应用程序，我们提供渐近保证双样本测试和最小距离估计使用$\mathsf{W}_p^{(σ)}$，与实验$p=2$使用最大的平均差异制定的$\mathsf{d}_2^{(σ)}$。

英文摘要：

Discrepancy measures between probability distributions, often termed statistical distances, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from data, recent work has proposed smoothing out local irregularities in the measured distributions via convolution with a Gaussian kernel. Motivated by the scalability of this framework to high dimensions, we investigate the structural and statistical behavior of the Gaussian-smoothed $p$-Wasserstein distance $\mathsf{W}_p^{(σ)}$, for arbitrary $p\geq 1$. After establishing basic metric and topological properties of $\mathsf{W}_p^{(σ)}$, we explore the asymptotic statistical behavior of $\mathsf{W}_p^{(σ)}(\hatμ_n,μ)$, where $\hatμ_n$ is the empirical distribution of $n$ independent observations from $μ$. We prove that $\mathsf{W}_p^{(σ)}$ enjoys a parametric empirical convergence rate of $n^{-1/2}$, which contrasts the $n^{-1/d}$ rate for unsmoothed $\mathsf{W}_p$ when $d \geq 3$. Our proof relies on controlling $\mathsf{W}_p^{(σ)}$ by a $p$th-order smooth Sobolev distance $\mathsf{d}_p^{(σ)}$ and deriving the limit distribution of $\sqrt{n}\,\mathsf{d}_p^{(σ)}(\hatμ_n,μ)$, for all dimensions $d$. As applications, we provide asymptotic guarantees for two-sample testing and minimum distance estimation using $\mathsf{W}_p^{(σ)}$, with experiments for $p=2$ using a maximum mean discrepancy formulation of $\mathsf{d}_2^{(σ)}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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