Fréchet regression for multi-label feature selection with implicit regularization

Fr\'echet regression extends linear regression to model complex responses in metric spaces, making it particularly relevant for multi-label regression, where each instance can have multiple associated labels. However, variable selection within this framework remains underexplored. In this paper, we pro pose a novel variable selection method that employs implicit regularization instead of traditional explicit regularization approaches, which can introduce bias. Our method effectively captures nonlinear interactions between predic tors and responses while promoting model sparsity. We provide theoretical results demonstrating selection consistency and illustrate the performance of our approach through numerical examples