Fr\'echet regression with implicit denoising and multicollinearity reduction

TL;DR

This paper extends Fréchet regression to better handle multi-label responses by modeling complex relationships and addressing noise and multicollinearity through implicit regularization, with theoretical guarantees and numerical validation.

Contribution

It introduces an extended Global Fréchet regression model with implicit regularization to effectively manage noise and multicollinearity in complex multi-label regression tasks.

Findings

The proposed method accurately models complex dependencies.

It preserves data structure while reducing bias.

Numerical experiments demonstrate improved performance.

Abstract

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.