This paper considers the problem of measure estimation under the barycentric coding model (BCM), in which an unknown measure is assumed to belong to the set of Wasserstein-2 barycenters of a finite set of known measures. Estimating a measure under this model is equivalent to estimating the unknown barycentric coordinates. We provide novel geometrical, statistical, and computational insights for measure estimation under the BCM, consisting of three main results. Our first main result leverages the Riemannian geometry of Wasserstein-2 space to provide a procedure for recovering the barycentric coordinates as the solution to a quadratic optimization problem assuming access to the true reference measures. The essential geometric insight is that the parameters of this quadratic problem are determined by inner products between the optimal displacement maps from the given measure to the reference measures defining the BCM. Our second main result then establishes an algorithm for solving for the coordinates in the BCM when all the measures are observed empirically via i.i.d. samples. We prove precise rates of convergence for this algorithm -- determined by the smoothness of the underlying measures and their dimensionality -- thereby guaranteeing its statistical consistency. Finally, we demonstrate the utility of the BCM and associated estimation procedures in three application areas: (i) covariance estimation for Gaussian measures; (ii) image processing; and (iii) natural language processing.
翻译:本文件审议了在巴氏中心编码模型(BCM)下进行测量估计的问题,在这一模型中,假定一项未知的措施属于一组瓦塞斯坦-2号已知的有限措施的瓦塞斯坦-2中继器,根据这一模型估计一项措施相当于估计未知的巴以中心坐标。我们为在BCM下进行测量估计提供了新的几何、统计和计算见解,其中包括三个主要结果。我们的第一个主要结果利用瓦塞斯坦-2号空间的里曼尼亚几何测量法,提供了一种程序,用于恢复作为假设获得真正参考措施的二次优化问题的解决方案的巴以中心坐标。基本的几何测深是,这一二次测度问题的参数是由从给定的衡量标准到界定BCMM的参考措施之间的内部产品决定的。我们的第二个主要结果是,当所有措施通过i.i.d. 样本以实证的方式观察时,解决BCM的所有措施的坐标。我们证明了这一算法的精确趋同率率 -- 由基本措施的顺利性及其维度测量尺度确定,从而保证其应用程度。