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Robust Global Fr'echet Regression via Weight Regularization

Hao Li, Shonosuke Sugasawa, Shota Katayama

TL;DR

This paper introduces a robust global Fréchet regression by weighting observations and applying an Elastic Net penalty to downweight outliers, yielding a weighted Fréchet loss that preserves efficiency on clean data while enhancing robustness to contamination. An iterative algorithm with proven linear convergence estimates updates regression targets and observation weights, and a data-driven BIC criterion selects tuning parameters. The method is developed for matrix/network responses with Frobenius distance and distribution responses with Wasserstein distance, with closed-form update steps for both settings. Through simulations and real-data analyses (network payloads, taxi networks, and mortality distributions), the approach demonstrates substantial robustness gains, practical parameter tuning guidance, and broad applicability to non-Euclidean outcomes.

Abstract

The Fréchet 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 Fréchet 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.

Robust Global Fr'echet Regression via Weight Regularization

TL;DR

This paper introduces a robust global Fréchet regression by weighting observations and applying an Elastic Net penalty to downweight outliers, yielding a weighted Fréchet loss that preserves efficiency on clean data while enhancing robustness to contamination. An iterative algorithm with proven linear convergence estimates updates regression targets and observation weights, and a data-driven BIC criterion selects tuning parameters. The method is developed for matrix/network responses with Frobenius distance and distribution responses with Wasserstein distance, with closed-form update steps for both settings. Through simulations and real-data analyses (network payloads, taxi networks, and mortality distributions), the approach demonstrates substantial robustness gains, practical parameter tuning guidance, and broad applicability to non-Euclidean outcomes.

Abstract

The Fréchet 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 Fréchet 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.

Paper Structure

This paper contains 21 sections, 2 theorems, 28 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Robust Global Fréchet Regression
  3. Global Fréchet regression
  4. Robust Fréchet regression with weight regularization
  5. On the penalty for weight parameters
  6. Optimization algorithm and its convergence property
  7. Selection of the tuning parameter
  8. Illustrative Models
  9. Robust regression for network and matrix response with Frobenius metric
  10. Robust regression for distribution response with Wasserstein distance
  11. Numerical Studies
  12. Simulation experiment with matrix response
  13. Demonstration using the New York Yellow Taxi network data
  14. Simulation experiment with distribution response
  15. Illustration of distribution response with mortality Data
  16. ...and 6 more sections

Key Result

Proposition 1

The optimal weight $({\widetilde{W}}_1(u),\ldots,{\widetilde{W}}_n(u))={\rm argmin}_{w\in [0,1]^n} Q(u,w)$ is obtained as

Figures (3)

  • Figure 1: The adaptive weight function ${\widetilde{W}}_i(u)$ as a function of the weighted distance $g(X_i, x) d^2(Y_i, u)$ under four choices of $\lambda, \gamma\in \{1,2\}$.
  • Figure 2: True networks(first), robust fitted networks(second), network regression method proposed by zhou2022network(third), and non-robust fitted networks (fourth) on May 16, 2020, corresponds to the day when the number of new COVID-19 cases was 134.
  • Figure 3: The absolute error heat map between the true network value, the network regression method proposed by zhou2022network, and the predicted network value on May 16, 2020, corresponds to the day when the number of new COVID-19 cases was 134.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2