Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications
Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications
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发表时间：
2021-01
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通讯作者：
Sloan Nietert;Ziv Goldfeld;Kengo Kato
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作者：
Sloan Nietert;Ziv Goldfeld;Kengo Kato
概率分布之间的差异度量，通常被称为统计距离，在概率论、统计学和机器学习中无处不在。为了在估计这些数据距离时对抗维数的诅咒，最近的工作提出了通过高斯核卷积来平滑测量分布中的局部不规则性。由于该框架具有高维的可扩展性，我们研究了任意p≥1时高斯平滑p- wasserstein距离W p的结构和统计行为。在建立了W (σ) p的基本度量和拓扑性质之后，我们研究了W （μ n, μ）的渐近统计行为，其中μ n是来自μ的n个独立观测值的经验分布。我们证明了Wp的参数经验收敛速率为n−1/2，而当d≥3时，非光滑Wp的参数经验收敛速率为n−1/d。我们的证明依赖于通过p阶平滑Sobolev距离dp来控制wp，并推导出√nd (σ) p （μ n, μ）的极限分布，对于所有维度d。作为应用，我们提供了使用wp的两样本检验和最小距离估计的渐近保证，并使用d2的最大平均差异公式进行了p = 2的实验。
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 W p , for arbitrary p ≥ 1. After establishing basic metric and topological properties of W (σ) p , we explore the asymptotic statistical behavior of W p (μ̂n, μ), where μ̂n is the empirical distribution of n independent observations from μ. We prove that W p enjoys a parametric empirical convergence rate of n−1/2, which contrasts the n−1/d rate for unsmoothed Wp when d ≥ 3. Our proof relies on controlling W p by a pth-order smooth Sobolev distance d p and deriving the limit distribution of √ n d (σ) p (μ̂n, μ), for all dimensions d. As applications, we provide asymptotic guarantees for two-sample testing and minimum distance estimation using W p , with experiments for p = 2 using a maximum mean discrepancy formulation of d 2 .