The Fréchet regression is a useful method for modeling random objects in a general metric space given Euclidean covariates. However, the conventional approach could be sensitive to outlying objects in the sense that the distance from the regression surface is large compared to the other objects. In this study, we develop a robust version of the global Fréchet regression by incorporating weight parameters into the objective function. We then introduce the Elastic net regularization, favoring a sparse vector of robust parameters to control the influence of outlying objects. We provide a computational algorithm to iteratively estimate the regression function and weight parameters, with providing a linear convergence property. We also propose the Bayesian information criterion to select the tuning parameters for regularization, which gives adaptive robustness along with observed data. The finite sample performance of the proposed method is demonstrated through numerical studies on matrix and distribution responses.

翻译：Fr'echet回归是一种在给定欧几里得协变量的情况下，对一般度量空间中随机对象进行建模的有效方法。然而，传统方法对异常对象可能较为敏感，即这些对象到回归曲面的距离相较于其他对象显著偏大。在本研究中，我们通过将权重参数引入目标函数，提出了一种鲁棒的全局Fr'echet回归方法。随后，我们引入Elastic net正则化，倾向于获得稀疏的鲁棒参数向量，以控制异常对象的影响。我们提出了一种计算算法，用于迭代估计回归函数和权重参数，并证明了其具有线性收敛性质。同时，我们提出了贝叶斯信息准则来选择正则化的调优参数，从而能够根据观测数据实现自适应的鲁棒性。通过对矩阵响应和分布响应的数值研究，验证了所提方法在有限样本下的性能。