标题： 非线性全局弗雷歇回归通过弱条件期望用于随机对象 显示英文标题

标题： Nonlinear global Fréchet regression for random objects via weak conditional expectation

摘要： 随机对象是在一般度量空间中取值的复杂非欧几里得数据，可能不具有任何底层向量空间结构。 随着技术的迅速发展，此类数据正变得越来越丰富。 例子包括概率分布、半正定矩阵以及黎曼流形上的数据。 然而，在文献中，除了针对欧几里得预测变量的对象响应回归和分布到分布的回归外，对于对象响应与对象预测变量的通用框架的发展仍然有限。 为填补这一空白，我们引入了基于Carleman算子的弱条件Fréchet均值的概念，然后通过再生核希尔伯特空间（RKHS）嵌入提出了一个全局非线性Fréchet回归模型。 此外，我们建立了条件Fréchet均值与弱条件Fréchet均值之间在欧几里得和对象值数据之间的关系。 我们还表明，由Petersen和Mueller于2019年开发的最先进的全局Fréchet回归方法是我们的方法的一个特例，只需选择线性核即可。 我们要求预测变量的度量空间具有再生核，而响应变量的度量空间的内在几何结构被用来研究所提出估计量的渐近性质。 进行了数值研究，包括广泛的模拟和一个实际应用，以调查我们的估计量在有限样本中的性能。 显示英文摘要

摘要： Random objects are complex non-Euclidean data taking value in general metric space, possibly devoid of any underlying vector space structure. Such data are getting increasingly abundant with the rapid advancement in technology. Examples include probability distributions, positive semi-definite matrices, and data on Riemannian manifolds. However, except for regression for object-valued response with Euclidean predictors and distribution-on-distribution regression, there has been limited development of a general framework for object-valued response with object-valued predictors in the literature. To fill this gap, we introduce the notion of a weak conditional Fr\'echet mean based on Carleman operators and then propose a global nonlinear Fr\'echet regression model through the reproducing kernel Hilbert space (RKHS) embedding. Furthermore, we establish the relationships between the conditional Fr\'echet mean and the weak conditional Fr\'echet mean for both Euclidean and object-valued data. We also show that the state-of-the-art global Fr\'echet regression developed by Petersen and Mueller, 2019 emerges as a special case of our method by choosing a linear kernel. We require that the metric space for the predictor admits a reproducing kernel, while the intrinsic geometry of the metric space for the response is utilized to study the asymptotic properties of the proposed estimates. Numerical studies, including extensive simulations and a real application, are conducted to investigate the performance of our estimator in a finite sample.