Upper and lower risk bounds for estimating the Wasserstein barycenter of random measures on the real line-猫眼课题宝

估计实线上随机测量的 Wasserstein 重心的风险上限和下限

DOI：

--

发表时间：

2018

期刊：

影响因子：

--

通讯作者：

中科院分区：

--

文献类型：

--

作者：

Jérémie Bigot;R. Gouet;T. Klein;Alfredo López

本文的重点是概率测度$nu_{1},ldots，nu_{n}$对$R$的统计分析，它们可以看作是潜在随机过程的独立实现。我们考虑具有实际意义的情况，其中随机测度$nu_{i}$是绝对连续的，密度$fun_{i}$是不可直接观测的。在这种情况下，我们可以访问真实随机变量的数据集$(X_{i,j})_{1 leq i leq n；;1 leq j leq p_{i}}$以$n$实验单元的形式组织，使得$X_{i,1},ldots,X_{i}, p_{i}}$为每$1 leq i leq n$从随机测量$nu_{i}$中抽样的iid观测值。在这种情况下，我们集中在一阶统计方法估计，从这样的数据，一个有意义的结构平均措施。为了考虑观测中的相位和振幅变化，我们认为沃瑟斯坦质心的概念是一个相关的工具。本文的主要贡献是描述了一个（可能是光滑的）经验Wasserstein质心在渐近的情况下向它的种群对偶点收敛的速度，其中$n$和$min_{1 leq i leq n} p_{i}$可能趋于无穷。从极大极小的角度讨论了这个过程的最优性。我们还强调了我们的方法与统计学中的曲线配准问题之间的联系。用一些数值实验来说明本文关于经验Wasserstein质心收敛速度的结论。

This paper is focused on the statistical analysis of probability measures $nu_{1},ldots,nu_{n}$ on $R$ that can be viewed as independent realizations of an underlying stochastic process. We consider the situation of practical importance where the random measures $nu_{i}$ are absolutely continuous with densities $fun_{i}$ that are not directly observable. In this case, instead of the densities, we have access to datasets of real random variables $(X_{i,j})_{1 leq i leq n; ; 1 leq j leq p_{i} }$ organized in the form of $n$ experimental units, such that $X_{i,1},ldots,X_{i,p_{i}}$ are iid observations sampled from a random measure $nu_{i}$ for each $1 leq i leq n$. In this setting, we focus on first-order statistics methods for estimating, from such data, a meaningful structural mean measure. For the purpose of taking into account phase and amplitude variations in the observations, we argue that the notion of Wasserstein barycenter is a relevant tool. The main contribution of this paper is to characterize the rate of convergence of a (possibly smoothed) empirical Wasserstein barycenter towards its population counterpart in the asymptotic setting where both $n$ and $min_{1 leq i leq n} p_{i}$ may go to infinity. The optimality of this procedure is discussed from the minimax point of view with respect to the Wasserstein metric. We also highlight the connection between our approach and the curve registration problem in statistics. Some numerical experiments are used to illustrate the results of the paper on the convergence rate of empirical Wasserstein barycenters.