提出新方法解决复杂数据中噪声与变量共线性问题
Fréchet regression with implicit denoising and multicollinearity reduction
- 用隐式正则化保留数据结构，建模多标签响应关系
- 理论保证下在数值实验中实现更准更稳的回归效果
- 适合处理高维多标签数据且存在噪声与共线性的场景
Fréchet回归将线性回归扩展到度量空间中的复杂响应建模，特别适用于多标签回归——每个样本可对应多个标签。然而，该框架中对预测变量间噪声和依赖关系的处理仍缺乏探索。本文提出一种全局Fréchet回归模型的扩展，能够显式建模输入变量与多重响应之间的关系。为应对噪声与共线性带来的挑战，我们引入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。该方法避免了传统显式正则化带来的偏差，具有理论保证，并通过数值实验验证了其准确性和高效性。
原文摘要 · Abstract (English)
Fréchet 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 Fréchet 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.
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