Fr\'echet regression extends linear regression to model complex responses in metric spaces, making it particularly relevant for multi-label regression, where eachinstance can have multiple associated labels. However, addressing noise and dependencies among predictors within this framework remains un derexplored. In this paper, we present an extension of the Global Fr\'echet re gression model that enables explicit modeling of relationships between input variables and multiple responses. To address challenges arising from noise and multicollinearity, we propose a novel framework based on implicit regu larization, which preserves the intrinsic structure of the data while effectively capturing complex dependencies. Our approach ensures accurate and efficient modeling without the biases introduced by traditional explicit regularization methods. Theoretical guarantees are provided, and the performance of the proposed method is demonstrated through numerical experiments.

翻译：Fréchet回归将线性回归扩展至度量空间中的复杂响应建模，特别适用于多标签回归问题——其中每个实例可关联多个标签。然而，在该框架内处理预测变量间的噪声与依赖关系仍缺乏深入研究。本文提出全局Fréchet回归模型的扩展形式，能够显式建模输入变量与多重响应间的关系。针对噪声与多重共线性带来的挑战，我们提出基于隐式正则化的新型框架，在保持数据内在结构的同时有效捕捉复杂依赖关系。该方法确保建模的准确性与效率，避免了传统显式正则化方法引入的偏差。我们提供了理论保证，并通过数值实验验证了所提方法的性能。