Robust Global Fr'echet Regression via Weight Regularization

TL;DR

This paper introduces a robust global Fréchet regression method that incorporates weight regularization and elastic net penalty to mitigate outlier influence in modeling complex data in metric spaces.

Contribution

It develops a novel robust Fréchet regression framework with weight regularization and elastic net, including an efficient algorithm and data-driven tuning parameter selection.

Findings

Demonstrates improved robustness against outliers in numerical studies

Provides an algorithm with linear convergence for estimation

Shows effectiveness on matrix and distribution response data

Abstract

The Fr\'echet regression is a useful method for modeling random objects in a general metric space given Euclidean covariates. However, the conventional approach could be sensitive to outlying objects in the sense that the distance from the regression surface is large compared to the other objects. In this study, we develop a robust version of the global Fr\'echet regression by incorporating weight parameters into the objective function. We then introduce the Elastic net regularization, favoring a sparse vector of robust parameters to control the influence of outlying objects. We provide a computational algorithm to iteratively estimate the regression function and weight parameters, with providing a linear convergence property. We also propose the Bayesian information criterion to select the tuning parameters for regularization, which gives adaptive robustness along with observed data. The finite sample performance of the proposed method is demonstrated through numerical studies on matrix and distribution responses.