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.

翻译：随机对象是复杂的非欧几里得数据，取值于一般度量空间，可能缺乏任何潜在的向量空间结构。随着技术的快速发展，此类数据日益丰富。例如包括概率分布、半正定矩阵以及黎曼流形上的数据。然而，除了解释变量为欧几里得数据、响应变量为对象值的回归以及分布-分布回归，目前文献中针对对象值解释变量与对象值响应变量的通用框架研究十分有限。为填补这一空白，我们基于Carleman算子引入弱条件Fréchet均值概念，并通过再生核希尔伯特空间嵌入提出全局非线性Fréchet回归模型。进一步，我们建立了欧几里得数据和对象值数据中条件Fréchet均值与弱条件Fréchet均值之间的关系。我们证明，通过选择线性核，Petersen和Mueller（2019）提出的最新全局Fréchet回归方法可视为我们方法的特例。我们要求解释变量度量空间具有再生核性质，同时利用响应变量度量空间的本质几何性质来研究所提估计量的渐近性质。通过广泛模拟和实际应用案例的数值研究，我们验证了有限样本下估计量的性能。