The existing Fr\'echet regression is actually defined within a linear framework, since the weight function in the Fr\'echet objective function is linearly defined, and the resulting Fr\'echet regression function is identified to be a linear model when the random object belongs to a Hilbert space. Even for nonparametric and semiparametric Fr\'echet regressions, which are usually nonlinear, the existing methods handle them by local linear (or local polynomial) technique, and the resulting Fr\'echet regressions are (locally) linear as well. We in this paper introduce a type of nonlinear Fr\'echet regressions. Such a framework can be utilized to fit the essentially nonlinear models in a general metric space and uniquely identify the nonlinear structure in a Hilbert space. Particularly, its generalized linear form can return to the standard linear Fr\'echet regression through a special choice of the weight function. Moreover, the generalized linear form possesses methodological and computational simplicity because the Euclidean variable and the metric space element are completely separable. The favorable theoretical properties (e.g. the estimation consistency and presentation theorem) of the nonlinear Fr\'echet regressions are established systemically. The comprehensive simulation studies and a human mortality data analysis demonstrate that the new strategy is significantly better than the competitors.

翻译：现有Fréchet回归实际上是在线性框架内定义的，因为其目标函数中的权重函数呈线性定义，且当随机对象属于Hilbert空间时，所得Fréchet回归函数可被确认为线性模型。即便是通常非线性的非参数与半参数Fréchet回归，现有方法也通过局部线性（或局部多项式）技术处理，所得Fréchet回归同样（局部）是线性的。本文引入一种非线性Fréchet回归模型。该框架可用于在一般度量空间中拟合本质非线性模型，并在Hilbert空间中唯一识别非线性结构。特别地，其广义线性形式可通过权重函数的特殊选择回归至标准线性Fréchet回归。此外，由于欧几里得变量与度量空间元素完全可分离，该广义线性形式具有方法学与计算上的简洁性。本文系统建立了非线性Fréchet回归的优良理论性质（例如估计相合性与表示定理）。综合模拟研究与人类死亡率数据分析表明，新方法显著优于现有竞争方法。