Fréchet regression with implicit denoising and multicollinearity reduction

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.